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νWUnw:Énw@A��Tu<=D6Wu@A<wWJ7

ν0?A>V8A8iTuWH�/N5<hν0 = E0

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� �� I�O�;�"!$#�%�'�%h)º*-,��/.10{�

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1

λ=

(1

n2− 1

p2

)1

λ0,

1

λ0= 109677cm−1 xr3V4^m�~

M�N�v&>V<=`É@iZFWS?AWpX_N5Z6àWU<�Bu>5<=<=WUn=��>V8A? 7n = 2

WU:p ≥ 3

x�ZF>58AjV@AWU@A<¨?�[ >V8A?AWg?/N58in Z6W_hD6n=D6 AZFW2~CÁZFWbBuN5n

n = 1x&n=TU<=DFWJ?AWIM�f ` N58/~YBu>V<w<=Wunw�1>58A?W7Q?AWUn�>V8i?AWun��p¿�4"M�WUn�nwTu<wDFWun

n = 3, 4, 5, . . .>V8�:g:E>V@h:EWungTC:ETJ>V AnwWu<w5TuWUnu4¯ÈD68/N5Z6Wu`-Wu8�:2Á�opD�: E+N»nw@AjVjVTU<=T+�;@AW©Z�N�:E<EN�8An=D�:EDF>58 1

λ=(

1n2 − 1

p2

)1λ0

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1

λn= Tn − Tp

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Tn =1

n2

1

λ0

xr3V4e|V~

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(Ir, ωr)Á(Iθ, ωθ)

WC:(Iϕ, ωϕ)

:EWUZFZFWUn�;@AW

IrÁIθÁIϕ

n=>5DFWu8�:g?AWUnSBu>V8inw:ªN�8;:=WungWU:ωr

Áωθ

WU:ωϕ

?AWunSv&>V8iBU:ED6>V8An_Z6DF8ATuN5DF<wWunÀ?A@+:EWUà�AnU4��8�B2N�ZFBu@iZ�nw:EN58A?/N5<w?�BU>V8A?A@iD6: 7QZ [ W �h�A<wWun=nwDF>V8�?A@�8A>V@i5WuZ�R/N5`-DFZ�:E>V8iDFWu8 n=>V@AngZFN�v&>V<=`-W �

K (Ir, Iθ, Iϕ) = − mk2

2 (Ir + Iθ + Iϕ)2

xr3V4^±�~

��8iW :=<EN58inwv&>V<w` N�:=DF>V8GB2N58i>V8ADF�;@AW 7¹Z\[eN5D6?AWÅ?iW ZFN�v&>V8ABC:EDF>58F2 (ωr, ωθ, ωϕ; I1, I2, I3) =

(ωϕ − ωθ) I1+(ωθ − ωr) I2+ωrI3Bu>V8A?i@AD6: 7S?AW¨8i>V@iVWUZFZ6Wun��N5<=DFN5 AZ6Wun�N�BU:ED6>V8hq9N58AjVZ6W

(I1, ω1)Á

(I2, ω2) , (I3, ω3):EWuZ6ZFWUnS� @iW �

I1 = Iϕ

I2 = Iθ + Iϕ

I3 = Ir + Iθ + Iϕ

xr3V4e�V~

§¨8�<wWu`-�AZ�N;�2N58�:k�/N5< e2

4πε0

ÁA>V8©:E<w>V@iVWb�/8AN5ZFWUàWU8;: �

K = − me4

32π2ε20

1

I23

xr3V4^¬�~

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I3�1WU@i:Q�A<wWu8A?A<wW+ZFWUn

�N5ZFWU@A<=n

I3 = n~ , ~ =h

2π

⇒ K (I3 = n~) = − me4

32π2ε20~2

1

n2

⇒ En = − 1

n2Ry , 1Ry = 13.6 eV

xr3V4ezV~

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ν�AN5<

ν = c/λÁ�>��

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H =p2

2m+

1

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>V8�N �K = ωI

⇒ En = ~ωn

xr3V463{KV~

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En = ~ω

(n+

1

2

) xr3V4632m�~

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Bu>V`-`-WH =

p2

2m+

1

2mω2x2 x\Kh463{~

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�1>5@A<�;@AW�ZFWUn �N5Z6Wu@A<wn >V in=Wu<t�N5 AZFWUn_?AW

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B5[ Wunt:_��>Vn=nwDF AZ6WpnwDqÁpWU:H

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qWC:pnw>V8�:�?iWung>V��Tu<EN$:EWu@i<=ng�;@ADµnEN�:=DFntv&>V8�:

[q, p] = i~3 Z

= i~x Kh4eKV~

#�� C93d/:='+`&�I_/:3d(*CEC*K 1432C*(E)H3 CE3 14Kd].-4= 143 CTS >[email protected]`Z43�� >[g�>�I�= 1hS >WCEC*3d) 5.CE-@/ CE&�(EIbO�� C.+`&�I"=H(*3dI�= =?&�-4=?3G/ CE3d/ I.&�=?(E&�I@/LF>�=H7.K,LF>W=?(<;�-@3d/2I.KG+`3d/H/H>W(E)?3G/,O

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n=>V8�:p?AWJZ�NÉv&>V<wàW

~ω

(n +

1

2

), n ∈ N.

x Kh4^m�~

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qÁpWC:H

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qWC:pnw>V8�:�Z\[^WU8An=WU`É AZ6Wb?AWung<=TUWuZ6nu4

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WU:_nEN5@iv�<wWunt:E<=D6BU:=DF>V8an=@A�A�iZFTu`-Wu8�:EN5DF<wW5Á >V8+�1>5@A<=<=N5D6:À?A>V8iB�:E<w>V@iVWU<_?AWUnSTU:ªN$:En |ϕq,p〉 :EWuZ6nY�;@AW

q|ϕq,p〉 = q|ϕq,p〉p|ϕq,p〉 = p|ϕq,p〉

x\Kh4 �;~

�YWunpTU:EN�:EngnwWu<EN�DFWu8�:p?A>V8iBHTUj�N5Z6Wu`-Wu8�:_VWUBU:=Wu@A<wn��A<w>V�A<=WUng?AWHÁ/?AWb�N5ZFWU@A<=nS�A<w>V�A<wWun

H|ϕq,p〉 =

(p2

2m+

1

2mω2q2

)|ϕq,p〉.

x\Kh4e|V~

]�@h:E<=WUàWU8�:Y?AD�:2Á�nwDqWC:pBu>5à`É@h:ªN5D6Wu8�:2Á�Z6Wpnw�1WUBU:=<=W�?iW

HnwWu<EN�D6: ¥ � % .O)u% ¶·Ã�¥&%º'�%h)H� .O%;!9�

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qWC:p8AW Bu>5à`É@h:EWu8�:+�AN5nu4

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[q, p] = a3 Z x K 4e±�~

�(N5D6n

[q, p]† = (qp)† − (pq)† = p†q† − q†p† = pq − qp = − [q, p]

⇒ a∗ = −a ⇒ a = iα , α ∈ R

x\Kh4e�V~

§¨8i�/8cÁOD6Z�v�N5@i:JD68;:=<=> ?A@AD6<=W~?/N58AnJZ�Nº:=RATu>V<wDFWQnwDÀ>58y5Wu@i:I<=WC:E<=>5@iVWU<IZ\[^R�f �1>�:ERAkUn=W-?AW�Z�N

:ERiTu>V<wDFWb?AWUn��;@/N58�:EN[q, p] = i~

x K 4e¬�~P�� &�-.)DKdg"(*=?3d)�1hS �`=?)H36)H3d14&�I@1.>WI"=��"&�I &�LN3`=?=?)0>�5@>�):U�&�(E/D1.36CTS Kd+,)?(E)?3�O� $'&�(E)DCTS >[email protected]`Z43 �eO

� �F�� ¤-)2��#�¥�¥��i02%h,�! ��A!$¶Å��.O#\�c,O% ¦

� � � � � �� j� 6� � � �0�� N58An+BUWU:w:EW·n=WUBU:ED6>V8�Á_8i>V@An�N5ZFZ6>V8Ana?ATu`->V8�:E<wWu<+�;@AW Z [ R�f �1>5:=RAkunwW

[q, p] = i~Bu>V8A?i@AD6:

W��1WuBC:ED65Wu`-Wu8�:pN5@+nw�1WUBU:E<wW~ω(n+ 1

2

) �1>5@A<H4 � N58inSBuW�BU>V8�:EW � :=W5Á/>V8º@h:EDFZ6DFnwW��AZ6@i: 5�:

x7

Z�NQ�AZFN5BuWb?AWq �

H =p2

2m+

1

2mω2x2

[x, p] = i~

x\Kh4ezV~

79Z�Wunt:gBu>V`-`->h?AW�?�[ DF8�:E<w>h?A@iDF<=W�ZFWunYBu>V`É iDF8/N5D6n=>V8in_ZFD68AT2N5D6<=WUnYn=@iD6�N58�:EWUnS?AWung>V��Tu<=N�:EWU@A<=nxWU:

p �a ≡

√mω

2~x+ i

1√2mω~

p

a† ≡√mω

2~x− i

1√2mω~

p =BU>V8��w@AjV@iTHRiWu<=`-D6:=DF�;@AWb?AW

a

x K 463��V~

�YWunp<=WUZ�N�:=DF>V8AnYnu[ DF8�VWU<=n=WU8�:�Wu8 �x =

√~

2mω

(a† + a

)

p = i

√mω~

2

(a† − a

)x K 463V3{~

�_N5ZFBU@AZF>58AnSZFWbBU>V`a`�@i:ªN�:=Wu@A<S?iW [a, a†

] 4 � [eN��A<=kUn�Z6Wu@A<wnSW �h�A<=WUn=nwDF>V8AnUÁ1D6Z1 DFWU8�: �[a, a†

]= −i 1

2~[x, p] + i

1

2~[p, x]

⇒[a, a†

]= 1

x K 463}K5~

¾ÈN5<pN5D6ZFZ6Wu@A<wnuÁiZ\[^RAN5`aD6Z6:=>V8ADFWU8anU[^TUBu<wD6: �H = −mω~

2

(a† − a

)2 1

2m+

1

2mω2 ~

2mω

(a† + a

)2

=~ω

4

[−(a† − a

)2+(a† + a

)2]

=~ω

4

[−(a†

2+ a2 − a†a− aa†

)+(a†

2+ a2 + a†a+ aa†

)]

=~ω

2

(a†a + aa†

)

=~ω

2

(a†a + a†a + 1

)

⇒ H = ~ω

(a†a+

1

2

)

x K 463{mV~

� ��*-,O�A.10}#��V�i0{#��.Ë�I�/.��.�#��,�%

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a†anw>V8�:��>VnwD6:ED�VWUnp>5@�8;@AZ6ZFWUnu4

sh@i�A�1>5n=>V8AngWU8(W��1WU:p�;@AWa†a|ϕ〉 = λ|ϕ〉 Á/N2VWUB 〈ϕ|ϕ〉 = 1

4"��8�N �λ = 〈ϕ|a†a|ϕ〉 = ‖a|ϕ〉‖2 ≥ 0

x\Kh4�32�;~Kh4�shD |ϕλ〉 Wunw:J5WuBU:=Wu@A<I�A<=>5�A<=W�?AW

a†a?AW��N5ZFWU@A<b�A<=>5�A<=W

λÁ�N5ZF>V<wn

a|ϕλ〉Wunw:J5WuBU:=Wu@A<

�A<w>V�A<wWH?iWa†a

?AWb5N�ZFWu@i<S�A<=>V�i<=Wλ− 1

4� TU`a>58Anw:=<EN�:=DF>V8 �

a†a|ϕλ〉 = λ|ϕλ〉�/N�<SR;f ��>5:ERikun=W�4

(a†a)a|ϕλ〉 =(aa† − 1

)a|ϕλ〉

= aa†a|ϕλ〉 − a|ϕλ〉= aλ|ϕλ〉 − a|ϕλ〉= (λ− 1) a|ϕλ〉.

x\Kh463{|V~

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λ − 1Áλ− 2

Á. . .

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a4 � WU@��B2N5ng?AWb�/j5@A<=WJn=>58;:%7Q?AD6nw:ED68AjV@AWU<u4

λ /∈ N¾O>V@A<

νnw@ �-nEN5`-`aWU8�:�jV<=N58A?�Á

λ − ν < 04O��8yWu8Aj5Wu8A?A<wW @A8yVWUBU:EWU@A<��i<=>V�A<wW

?AW��N5Z6Wu@A<À�A<w>V�A<=W�8ATujVN�:ED�VW��/N�<_N5�A�iZFDFBuN�:ED6>V8�?AWaν4i�(N5D6nY>V8ºN�?ATu`->V8�:E<wT��;@AW�Z6WunÀ5N�ZFWu@i<=n

�A<w>V�A<=WUng?AWN

n=>V8�:��1>VnwD6:=D6VWUnu4"�YWC:=:=WH��>Vnwn=DF iDFZFD�:ET�?A>VD6: �C:E<wWHTUB2N5<t:ETUW54λ ∈ N

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n=@iBuBuWUn=nwD6VW ?AWaWu8Aj5Wu8A?A<wW ?AWUnH5WuBC:EWu@i<=nÉ?AWQ�N5Z6Wu@A<wnI�A<w>V�A<wWun

n − 1Án − 2

Á. . .

�w@An=�;@�[ 7Z [ >V��Tu<=N�:EWU@A<

an�1>5@A<gZFWu�;@AWUZ

N (an|ϕn〉) = (n− n) |ϕn〉 = 0�(N5D6n 〈ϕ|N |ϕ〉 = 0 ⇒ ‖a|ϕ〉‖2 = 0

⇒ an+1|ϕn〉 = 0 ⇒ ap|ϕn〉 = 0nwDp ≥ n + 1

x\Kh4632±�~

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Hn=>V8�:

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2

) 4;M�[ >V��Tu<=N�:EWU@A<a†a

Wunw:YN5�A��WuZ6TgZ [ >V�1T q<EN$:EWu@i< PdBO1 V]>F ÁhWU:pDFZ�WUnw:pn=>5@iVWU8;:�8A>5:=T

N4

§¨n=n=N2fV>V8AnÀ?AWSBU>V8Anw:=<=@AD6<=W_W �h�AZFD6BuD6:=Wu`-Wu8�:¨Z6WunÈ5WuBU:=Wu@A<wn��A<=>V�i<=WunUÁVWU:�TC:E@A?iDF>V8AnÈn=DABUWSn=��WuBU:=<=WWunt:p?ATujVTU8ATu<wT54¾�N5<w:EN58�:J?�[^@i8yTC:ªN�: |ϕn〉 ?AW��N5Z6Wu@A<��i<=>V�A<wW

nÁc>V8·N5<=<wD65W27º@A8»VWUBU:=Wu@A<J�A<w>V�A<=W

an|ϕn〉nEN$:EDFntv�N5DFn=N58�:a (an|ϕn〉) = 0

41siN58in�R�f �1>�:ERAkUn=WÉnw@A�A�AZ6Tu`-Wu8�:ªN5D6<=WHn=@A<xWC:pÁ"D6ZµWunw:�DF`-��>Vntq

n=D6 AZFW�?AW-?ATUBuDF?iWu<Hnu[ DFZ¨fyN�@i8Ån=WU@AZÀTU:ªN$:�n=N�:ED6nwv�N5D6nEN58�:a|ϕ〉 = 0

4 ��>V@AnJVWU<=<w>V8An��;@AWa?/N58An

� �F�� ¤-)2��#�¥�¥��i02%h,�! ��A!$¶Å��.O#\�c,O%

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a|ϕ〉 = 0ÁAWU:p8A>�:E>V8Anrq9Z6W |ϕ0〉

4²�:ªN5��WJn=@AD�5N�8;:=W � MµWUnÉN5@i:=<=WUn�TU:EN�:EnIn=>V8�:tq9DFZ6nJ?ATujVTU8ATu<wTun�� TUà>V8�:=<=>V8AnI�;@AW 8A>58 �/N5<H<=T qBu@A<w<=WU8ABuW�4dpf ��>5:ERAkUn=W � M�N��N5Z6Wu@A<-�A<w>V�A<=W

nWUnw:º8i>V8�?ATUjVTu8ATU<=TUW54gsh>5DFWu8�: |ϕn+1〉

WC: |ψn+1〉?AWu@ �

VWUBU:EWU@A<=n��A<=>5�A<=WUnp?iW�5N�ZFWu@i<=nS�A<w>V�A<=WUnn+ 1

4"�Y>V`-àWa|ϕn+1〉

WU:a|ψn+1〉

nw>V8�:gVWuBC:EWU@A<=n�A<=>5�A<=WUnS?AW

N?AW��N5Z6Wu@A<_�i<=>V�A<wW

nÁhWU:g�;@AWJ�/N5<_R�f ��>5:ERAkUn=W

nWunt:g8A>V8+?ATUjVTu8iTu<=T�Á/DFZ1W �hDFnt:EW

λ:EWUZµ� @iW �

a|ϕn+1〉 = λa|ψn+1〉 ⇔a†a|ϕn+1〉 = λa†a|ψn+1〉 ⇔

(n+ 1) |ϕn+1〉 = λ (n+ 1) |ψn+1〉 ⇔|ϕn+1〉 = λ|ψn+1〉

⇒ n+ 1WUnw:p8A>V8©?ATUjVTu8iTu<=TUW54

x K 463}�5~

� Wu<w8ADFkU<=WJTU:EN5��W � �Y>V`-àWU8;:�Bu>58Anw:=<=@AD6<=WÉZ6Wun�5WuBC:EWu@i<=nb�A<w>V�A<=WUn�?AW��N5Z6Wu@A<��A<=>5�A<=Wn�;@AWuZ q

Bu>V8i� @iW �¾¨<=>V��>Vn=D�:ED6>V8 � n=D |ϕ〉 Wunw:J5WuBU:=Wu@A<I�A<=>5�A<=W-?AW

N?iW-�N5Z6Wu@A<b�A<w>V�A<wW

nÁa†|ϕ〉 Wunw:HVWUBU:=Wu@A<

�A<=>5�A<=WJ?AWN

?AWb�N5ZFWU@A<S�A<w>V�A<=Wn+ 1

4� Tu`->V8Ant:E<=N�:ED6>V8 �

N |ϕ〉 = n|ϕ〉 , �AN5<SR�fh��>5:=RAkunwWNa†|ϕ〉 = a†aa†|ϕ〉

= a†(a†a+ 1

)|ϕ〉

= a†(a†a)|ϕ〉 + a†|ϕ〉

= a†n|ϕ〉 + a†|ϕ〉= (n+ 1) a†|ϕ〉

x K 463{¬V~

�Y>V8AnwTu�;@AWu8iBuW � (a†)n |ϕ0〉Wunw:pVWUBU:=Wu@A<p�A<=>5�A<=WJ?AW

N?AWb�N5ZFWU@A<S�A<w>V�A<=W

n4

��>V<w` N�ZFDFn=N�:ED6>V8 � s @A�A��>Vn=>V8in |ϕn〉 8i>V<=àN5ZFD6n=T�4∥∥a†ϕn

∥∥2= 〈ϕn|aa†|ϕn〉 = 〈ϕn|a†a+ 1|ϕn〉 = (n + 1) 〈ϕn|ϕn〉

⇒ a†√n+ 1

|ϕn〉Wunt:p8A>V<=àN5Z6DFn=T

⇒ |ϕn+1〉 =a†√n + 1

|ϕn〉

x K 463{zV~

�� *-,O�A.10}#��V�i0{#��.Ë�I�/.��.�#��,�%

]�D68An=D\Á

|ϕn〉 =

(a†)n

√n!

|ϕ0〉x\Kh4LK ��~

¯ÈD68/N5ZFWUàWU8�:2Ái��>V@A<gN�ZFZFWU<_?�[ @A8�TU:ªN$: 7-Z\[eN5@h:E<=W�ÁA>V8�N �a†|ϕn〉 =

√n + 1|ϕn+1〉

x\Kh4LK 3}~¾�>5@A<gBuDF<wBu@AZ6Wu<S?/N�8AngZ [^N5@i:E<wWbn=WU8AnuÁ">V8©<=WU` N�<=�;@AWJ�;@AW �

‖a|ϕn〉‖2 = 〈ϕn|a†a|ϕn〉 = n〈ϕn|ϕn〉 = n

⇒ a|ϕn〉 =√n|ϕn−1〉

x\Kh4LK5KV~

aWC:a†

n=>V8�:pn=>V@hVWu8�:�N5�A��WuZ6Tun B >RY^]<_`L FÛ]>c [F2@^]>Y\_`L QhBEP F^L [�� _`PRPRQGOQ D _`LTQhBEP 4

��|ϕ3〉

FF

a

��|ϕ2〉a†

JJ

a

��|ϕ1〉a†

JJ

a

��|ϕ0〉a†

JJ

�º� �V�AÇ�#&0{,�¥&�i0}#&��.MµWung�N5ZFWU@A<=nS�i<=>V�A<wWung?AWJZ [ >VnwBuDFZ6Z�N�:=Wu@A< R/N5<wà>V8iDF�;@AWbn=>58;:p?A>58A8ATuWUn��AN5<

En = ~ω

(n +

1

2

), n ∈ N

x\Kh4eK5m�~

§¨ZFZ6WunS8AWJn=>58;:p�/N�np?iTujVTU8ATu<wTuWunU4cM¨[ TU:EN�:Sv&>V8A?/N�àWU8;:EN5Z |ϕ0〉nEN�:=DFnwv�N�D6: �

a|ϕ0〉 = 0x\Kh4eK��;~

MµWbVWuBC:EWU@A<��i<=>V�A<wWJ?AWb5N�ZFWu@i<S�A<=>V�i<=WEn

Wunw:�?A>V8i8ATJ�/N5<

|ϕn〉 =

(a†)n

√n!

|ϕ0〉x\Kh4LK5|V~

� �F¡��À% )º¤�Ç¨�;!$�h0{% ,O!$)q%�0

p�¨¥��y�I�O�;�"!$#�%('O% )�¢¼�i0{!$#\�V% ) �"�

MµWunSVWUBU:EWU@A<=n��A<=>5�A<=WUn�nw>V8�:�<wWuZ6DFTunSWU8�:E<=WJWU@��/N5< �a†|ϕn〉 =

√n+ 1|ϕn+1〉

WC:a|ϕn〉 =

√n|ϕn−1〉

x K 4LK5±V~opWU` N�<=�;@AW � ��[^WUnw:ÉZFN+n=>VZ6@i:ED6>V8 @CBO1H> D �^L F ?A@Å�A<=>V iZFku`-W��u4OMON+�N5ZFWU@A<Jà>{f5Wu8A8AWa?AW-8�[^D6`�q�1>5<w:EWg�;@AWUZFZFWg>V in=Wu<t�N5 AZFWg?/N�8An 8c[^D6à��>V<w:=WS�;@AWuZ"TU:EN�:Ànu[ >V i:=DFWu8�: Wu8aW �h�A<=D6` N�8;:�Z\[^>5 An=WU<w�N5 AZ6W7QZ [^N5DF?AWb?iWun

aÁa†

WU:pWu8�@i:=DFZFD6nEN58�:YZ�NQ<=kUjVZFWb?AWJBU>Và`�@i:ªN�:=DF>V8 [a, a†

]= 1

4§ �iWUà�iZFW � §¨n=n=N{f5>V8AnO?AW¨B2N�ZFBu@iZFWu<cZ�N �N5Z6Wu@A<�`->{fVWU8A8AW�?AW

x2 ?/N58AnµZ6WÈv&>V8A?/N�àWU8;:EN5Z/x 〈ϕ0|x2|ϕ0〉~ �

x2 =~

2mω

(a† + a

)2

(a† + a

)2=(a†)2

+ a2 + a†a + aa† =(a†)2

+ a2 + 2a†a+ 1

〈ϕ0|(a†)2 |ϕ0〉 = 〈ϕ0|a†|ϕ1〉 =

√2〈ϕ0|ϕ2〉 = 0

〈ϕ0|a2|ϕ0〉 = 〈ϕ0|a†a|ϕ0〉 = 0

⇒ 〈ϕ0|(a† + a

)2 |ϕ0〉 = 〈ϕ0|ϕ0〉 = 1

⇒ < x2 >=~

2mω

x K 4LKV�5~

��8ÅV>VD�:É?A>58ABº�;@AW< x2 >6= 0

'O�A.�)·¥ ��0{�h0y'O% ÇÈ¥�,O)»Ã��/)u)u%�� .�% ! �#&% 4È�YWº<wTun=@iZ6:ªN$:Wunt:�WU8©Bu>V8�:E<=N5?AD6BU:ED6>V8+àN58AD6v&WUnw:=W�N2VWUBIZ6W�B2N5nSBUZ�N5nwn=D6� @iW5ÁA>��

x = 0x�WC:g?A>V8AB

x2 = 0~Y?AN58An

ZFWÉv&>V8i?/N5`-Wu8�:ªN5Z\4 � N�8AnJZ [ TU:ªN$:b?AW��AZF@inb /N5n=nwWaTU8AWu<wjVDFW�Á�>V8·?AD�:J� @c[^D6ZÈf·N+?AWun�� UW@^LTUk_`L QhBEPkcSVU�_`PRLTQTSVURFCc N��A�1WUZFTUWung?/N58AngBUWHBuN5n�� UR@^LTU�_`LTQhBEPkc [F?>RBEQ PRL��EY^]CB 4

� �� 0��q

�N�p

6� ��?�[� �6� �� 6�� M�Nº� @iWunw:=DF>V8·�;@AD�nwW��1>VnwWQ8/N�:=@A<=WUZFZFWUàWU8�: 7ºBuW�nt:ªN5?AWQWUnw:J?AWQn=N2V>VD6<bBu>V`-àWU8�:�:E<w>V@iVWU<

?AWunp>V��Tu<=N�:EWU@A<=nqWC:p� @iDOn=N�:ED6nwv&>V8�:gZFNÉ<=kuj5ZFWb?AWJBu>5à`É@h:ªN�:=DF>V8

[q, p] = i~3 Z x Kh4eK5¬�~

§¨8+�/N5<w:=DFBU@AZFD6Wu<uÁ;BU>Và`-W�Z6Wun_àN�:=<=DFBUWunÀBu>V8inw:ED�:E@AWU8�:g@A8+W �hWu`-�AZ6W�n=D6à�AZ6W�?c[^>V��Tu<=N�:EWU@A<_?/N�8An@A8-Wun=�AN5BuW�?AWpd�DFZ6 1WU<w:uÁ5DFZ/WUnw:À8/N�:=@A<=WUZ/?AWpn=Wg?AWuàN58A?iWu<Àn=D/>V8-�1WU@i:ÀZFWUnÀ<wWu�A<wTunwWu8�:EWU<Y�/N5<�?AWUn` N$:E<=D6BuWUn

(N ×N)4 MONJn=>5ZF@i:=DF>V8-?A@ �i<=>V AZ6ku`-Wg?AW�Z\[^>5n=BuD6ZFZFN�:EWU@A<�R/N5<wà>58ADF�;@AWgn=@ijVjVku<wW��;@AW

B5[ Wunt:bDF`-�1>5n=n=D6 AZFWJnwDN

Wunw:��/8AD� @»�;@�[ >V8·Na:=<=>V@hVTÉ@A8(Wu8AnwWu`É AZ6WÉDF8h�/8AD�?ATu8i>V`É A<=N5 AZFWI?AWv&>V8ABC:EDF>58An��A<w>V�A<=WUnJ>V<w:=RA>VjV>V8AN5ZFWUn {|ϕn〉} 4��(N5D6n�BuWUZ�NºTC:ªN5D�:��A<wTU DFnwDF AZ6WQNº�A<=D6>V<=D\4c§¨8·W��1WU:uÁWu8�?ADF`-Wu8in=DF>58º�/8ADFW�Á/Z�N�:=<EN5BUWH?iWH?iWu@��>V��Tu<=N�:EWU@A<=ngn=N�:ED6nwv�N5D�:

P�<(AB) =

P�<(BA)

x Kh4eK5z�~� � 5@>�):=DC<>jI.&�I� 14Kdf�K,I@K,)H3d/H+`3,I_+`3 1.3d/DK,=H>�=0/25.)H&�5.)H3d/

� � ��*-,O�A.10}#��V�i0{#��.Ë�I�/.��.�#��,�%

� @JBu>5@A��Á{��>V@A<�?AWUn�` N�:=<=D6BuWunc?AW¨?ADF`-Wu8in=DF>58��/8AD6WNÁ{PO<

([A,B]) = 0ÁuN5ZF>V<wn��;@AW�P�<

([x, p]) =PO<(i~

3 Z) = i~N

479Z�v�N5@i:�?i>V8ABÀBu>V8AnwDF?ATU<=WU<O?iWunO>V�1TU<EN�:=Wu@A<wnON5jVDFnwnEN58�:O?/N58in�@A8HWunw�/N5BUW ?AWÀ?AD6`aWU8An=D6>V8bDF8i�A8ADFW�4

MµWQBªRA>VD �(v�N5D�:��/N�<bXY>VRA<UÁ�dpWuDFnwWu8; �Wu<=j©WC:��5>V<=?/N�8 Bu>V8AnwDFnt:EW^7º<=WU�A<=TUn=WU8;:=Wu<xWU:p�AN5<�?AWUn

`aN�:E<wDFBuWUn¨?AWg?ADF`-Wu8in=DF>58-D68i�/8AD6WS?/N58An�ZFNJ AN5n=Wg?AWUn TC:ªN�:=nÀ�i<=>V�A<wWunÀ?AWa†a

4 � N58AnÀBuWU:w:EWg /N5nwW5ÁZFWUnS>V��Tu<EN$:EWu@i<=n

aWC:a†

nw>V8�:p<=Wu�i<=TunwWu8�:ETUn��/N5< �

a =

0√

1 0 0 0 · · ·0 0

√2 0 0 · · ·

0 0 0√

3 0 · · ·444 444 4440

√4

4 4 4444 444 444 444 4 4 4 4 4 4

a† =

0 0 0 0 · · ·√1 0 0 0 · · ·

0√

2 0 0 · · ·0 0

√3 0 · · ·444 444

0√

4

4 4 4444 444 444 444 4 4 4

x\Kh4em��~

N2VWUB

|ϕ0〉 =

100444

, |ϕ1〉 =

010444

, · · · x\Kh4emh3}~

� [^N5�A<wkunpZ [ W �h�A<wWun=nwDF>V8�?AWxWU:pWu8©v&>V8ABC:EDF>58�?iW

aWU:a†Á/>V8©WU8�?ATU?A@AD�: �

x =

√~

2mω

0√

1 0 0 · · ·√1 0

√2 0 · · ·

0√

2 0√

34 4 4

0 0√

3 04 4 4

444 444 4 4 4 4 4 4 4 4 4

p = i

√mω~

2

0 −√

1 0 0 · · ·√1 0 −

√2 0 · · ·

0√

2 0 −√

34 4 4

0 0√

3 04 4 4

444 444 4 4 4 4 4 4 4 4 4

x\Kh4emVKV~

� ��³ �À��*-,��/.10}#��V�h0}#��.¹��A.O��.O#\��,�% �h¡

N5DF8in=D��;@AW

xp =i~

2

1 0 −√

1 × 2 0 0 · · ·0 1 0 −

√2 × 3 0 · · ·

√1 × 2 0 1 0 −

√3 × 4

4 4 40

√2 × 3 0 1 0

4 4 4444 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

WC:

px =i~

2

−1 0 −√

1 × 2 0 0 · · ·0 −1 0 −

√2 × 3 0 · · ·

√1 × 2 0 −1 0 −

√3 × 4

4 4 40

√2 × 3 0 −1 0

4 4 4444 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

⇒ xp− px = i~

1 0 · · ·0 1

4 4 4444 4 4 4 4 4 4

x K 4emVmV~

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[q, p] = i~3 Z x Kh4^m5�;~

7(:E>V@h:EWunQ�AN5DF<wWunÉ?AWº�N5<wD�N5 iZFWunIB2N58i>V8ADF�;@AWUàWU8�:QBu>V8 �w@AjV@ATUWunu4À�YWC:=:=W+�A<=> BuTU?A@A<=W+nU[eN��A�1WUZFZ6WDGF > ]^Q Pd@^Q >RF2[F @CBE]g]>FCc >RBEP [0_`Pd@CF � >58»?ATU:=Wu<wàD68AWIZFN(x&>V@(Z6WunE~g�/N�DF<=WVx�nE~g?AWH�N5<=DFN5 AZFWUngB2N58A>$q8ADF�;@AWUàWU8�:gBu>V8��w@ijV@ATuWUn (

qi,∂L∂qi

≡ pi

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dt= {f,H} +

∂f

∂t

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Z [ Tu�;@/N$:EDF>58+BuZ�N�n=n=D6�;@AWJ�1>V@i<fÁ/n=>58�TCV>VZ6@i:ED6>V8º:EWUà��>V<=WUZFZ6W�Wunt:�<wTujVD6Wb�/N5<SZ\[^TU� @AN�:ED6>V8 �

df

dt=

1

i~

[f , H

]+∂f

∂t

x\Kh4emV�V~

>��©>V8�NQ<=Wu`-�AZFN5BuT {f,H} �/N5< 1i~

[f , H

] �/N5<pN58/N�ZF>VjVD6W�N2VWUB {q, p} = 1 ⇔ 1i~

[q, p] =3 Z�4

�� 6�c��6�� X�� F� � � ��

MµWun�Bu>V8in=Tu�;@AWU8ABuWUn��AR�f n=D6� @iWun�?AWÀBuWÀBªR/N58ijVWu`-Wu8�:�<EN5?AD6B2N5Z5?AW ��>VD68;:�?iWÀ @AWÀn=>V8�:�Bu>V8in=DF?iTCq<EN� AZFWUnu4/§¨8��/N5<w:=DFBU@AZFD6Wu<uÁ DFZ15NÉv�N5Z6ZF>VD6<��A<=TUBuDFnwWu<gBuWJ�;@�[ >V8�Wu8�:EWU8A?(�/N�< �C�N5ZFWU@A<_>V AnwWu<t5N� AZFW ��4�YW»�A<w>V AZ6ku`-W5Á N5@A�;@AWuZ�>V8�n=W <wTUv&kU<=W»WU8¹jVTu8ATU<EN5Z�Wu8¹�/N�<=Z�N�8;: ?AW D � Q PRLNF^] > ]CY^Lh_`LTQhBEP ?AW(Z�N`-TuB2N�8ADF�;@AWg�;@/N58�:ED6�;@AW5ÁhNH@i8AW�nw>VZF@h:EDF>58-BU>V`-`É@A8ATUàWU8;:�x&` N5D6n��/N5nÀ@A8AD�VWU<=n=WUZFZ6Wu`-Wu8�:��e~ÀN5B qBuWU�i:ETUW5Á/ /N�n=TuWJnw@A<S@A8©N5@i:=<=Wb��>VDF8�:Y?AW� @AW5Á/BUWuZF@iDc?AWbZFNÉàTUB2N58AD6�;@AW�>V8A?i@AZ�N�:=>VDF<wW5Á �;@AD�n=WU<EN?ATCVWuZ6>V�A��TJ?/N58AnpZFW�BªRAN5�AD6:=<=WJn=@iD6�N58�:24MµWH�1>VD68�:�?iWH @AWI?AW�dpWuD6n=Wu8; �Wu<wjºN�8AT2N58ià>VD68Ang@A8AWHBu>V8in=Tu�;@AWU8ABuWQ?iDF<=WUBU:=WH:=<=kunpn=@i<=�A<wWCq

8/N58�:=W54�§¨8�W��1WU:uÁVBu>V`-`-WxWU:p8AWSBU>V`-`É@i:EWU8�:¨�/N5nUÁVDFZhWunw:�D6à��>Vnwn=DF iZFWÀ?AWSZFWUn¨?AD�N5j5>V8/N5Z6DFn=WU<

?/N58inY@A8AW� /N5nwW�Bu>5à`É@i8AW54 +,µ0{!}%h¶�%h.�0�'O#�0��µ)2#|ϕ〉

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x?AWb�N5ZFWU@A<S�A<=>5�A<=W

x ∈ R �x|ϕ〉 = x|ϕ〉 WC: x2|ϕ〉 = x2|ϕ〉 ⇒ 〈ϕ|x2|ϕ〉 = 〈ϕ|x|ϕ〉2 x\Kh4em5¬�~

]�D68An=D\ÁA��>V@A<S@A8�TU:EN�:g�A<=>5�A<=WJ?AWxÁ/ZFNÉ� @AN58�:ED6:=T

∆x ≡√

〈ϕ|x2|ϕ〉 − 〈ϕ|x|ϕ〉2 x\Kh4em5z�~

� �F��À% ) �º% ¥��i0{#��.O)Q' � U.��V%;!{0}#�0},�'�%�'O%��% #\)u% .cÃ¨%;! �h�

Wunt:�8;@AZ6ZFW�4/¾ÈN5<SBU>V8�:E<wW5Á"WuZ6ZFW�8AWJZ\[^WUnw:g�/N5ngnwD |ϕ〉 8�[^WUnw:p�/N5ngTC:ªN�:g�A<w>V�A<wWH?iW x 4��Y>58An=D6?ATu<w>V8An�/N5<gW �hWUà�AZ6Wb@A8�TU:ªN$:

|ϕ〉 = α|ϕ1〉 + β|ϕ2〉:EWUZ��;@AW

|α|2 + |β|2 = 1x �p>V<=àN5Z6DFnEN$:EDF>58"~

x|ϕ1〉 = x1|ϕ1〉x|ϕ2〉 = x2|ϕ2〉

WU:x1 6= x2 (⇒ 〈ϕ1|ϕ2〉 = 0)

〈ϕ|x|ϕ〉 = 〈αϕ1 + βϕ2|αx1ϕ1 + βx2ϕ2〉 = |α|2 x1 + |β|2 x2

〈ϕ|x2|ϕ〉 = 〈αϕ1 + βϕ2|αx21ϕ1 + βx2

2ϕ2〉 = |α|2 x21 + |β|2 x2

2

⇒ 〈ϕ|x2|ϕ〉 − 〈ϕ|x|ϕ〉2 = |α|2 x21 + |β|2 x2

2 − |α|4 x21 − |β|4 x2

2 − 2 |α|2 |β|2 x1x2x K 4^� �V~§¨8�`É@iZ6:ED6�AZFDFN58�: |α|2 x2

1 + |β|2 x22

�/N5< |α2| + |β2| (= 1)ÁADFZ1hD6Wu8�:

∆x2 = |α|2 |β|2 x21 + |α|2 |β|2 x2

2 − 2 |α|2 |β|2 x1x2

= |α|2 |β|2 (x1 − x2)2

⇒ ∆x = |α| |β| |x1 − x2|

x K 4^�A3{~

PSRATu>5<=ku`-W � ¾�>V@i<g:=>V@i:gTUZFTu`-Wu8�:p8A>V<w` N�ZFDFnwT |ϕ〉 ?AWJZ [ Wunw�/N5BUWH?iWIdpDFZ6 1WU<w:2Áh>V8�N �

∆x · ∆p ≥ ~

2

x K 4^�;K5~

� Tu`->V8Ant:E<=N�:ED6>V8 � �Y>V8AnwDF?ATU<=>V8Anp@A8�TU:EN�: |ϕ〉 WU:p?ATU�A8ADFnwn=>V8AnpZFWUnS>V��Tu<EN$:EWu@i<=n

x0 = x− 〈ϕ|x|ϕ〉 3 Zp0 = p− 〈ϕ|p|ϕ〉 3 Z

x K 4^��mV~

��8ÉN� iDFWu8Hn �i<[x0, p0] = i~

3 Z�4 � W �iZF@AnUÁ�BU>Và`-WxWunt:�RAWU<=`-D6:=DF�;@AW5Á 〈ϕ|x|ϕ〉 ≡ 〈ϕ|xϕ〉 =

〈xϕ|ϕ〉 = 〈ϕ|xϕ〉∗ ≡ 〈x|ϕ|x〉∗ Wunw:�<wTuWUZ 4x0

Wunt:-?A>V8AB©N5@in=n=DYRAWu<wàD�:EDF�;@AW�4¨�Y>V8in=DF?iTu<=>58An?ATunw>V<=àN5D6nSZ [ TU:ªN$: �

(x0 + iλp0) |ϕ〉 λ ∈ Rx\Kh4 �V�;~

siN�8A>V<=`-WJN5@©B2N�<=<=TJ?A>5D6: �C:E<=WJ��>Vn=D�:ED�VW-x��AN5<g?ATU�/8iD6:ED6>V8+?AWJZ�NÉ8i>V<=`-W{~ �‖(x0 + iλp0) |ϕ〉‖2 ≥ 0

⇒ 〈ϕ| (x0 − iλp0) (x0 + iλp0) |ϕ〉 ≥ 0

⇒ 〈ϕ|x20|ϕ〉 + λ2〈ϕ|p2

0|ϕ〉 − iλ〈ϕ|p0x0|ϕ〉 + iλ〈ϕ|x0p0|ϕ〉 ≥ 0

⇒ 〈ϕ|x20|ϕ〉 + λ2〈ϕ|p2

0|ϕ〉 + iλ〈ϕ| [x0, p0] |ϕ〉 ≥ 0

⇒ λ2〈ϕ|p20|ϕ〉 − λ~ + 〈ϕ|x2

0|ϕ〉 ≥ 0

x K 4^�;|5~

� � ��*-,O�A.10}#��V�i0{#��.Ë�I�/.��.�#��,�%

¾O>V@A<p� @iWIBUWH��>VZ�fh8"5VàWb?iWH?iWujV<wTÉKÉn=>5D6:S:E>5@��w>V@A<wnp��>VnwD6:ED�vrÁinw>V8�?iDFn=BU<=D6àD68/N58�:S?A>VD�: �U:=<=W8ATUj�N�:ED�v

⇒ ~2 − 4〈ϕ|p20|ϕ〉〈ϕ|x2

0|ϕ〉 ≤ 0 ⇔ 〈ϕ|p20|ϕ〉〈ϕ|x2

0|ϕ〉 ≥~2

4

x\Kh4^�V±�~�(N5DFn 〈ϕ|x2

0|ϕ〉 = 〈ϕ|(x2 − 2〈ϕ|x|ϕ〉x+ 〈ϕ|x|ϕ〉2

)|ϕ〉

= 〈ϕ|x2|ϕ〉 − 2〈ϕ|x|ϕ〉2 + 〈ϕ|x|ϕ〉2= 〈ϕ|x2|ϕ〉 − 〈ϕ|x|ϕ〉2

� WJ` �UàW�Á 〈ϕ|p20|ϕ〉 = 〈ϕ|p2|ϕ〉 − 〈ϕ|p|ϕ〉2

⇒ ∆x · ∆p ≥ ~

2

x\Kh4^��V~

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WU:�;@AW

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νn=W�Bu>V`-�1>5n=W�?iW��/N�<w:ED6Bu@AZ6WunÀ?�[ Tu8AWU<=jVD6W

E = hν = ~ωWU:_?c[^D6à�A@iZFn=D6>V8

~p = ~~kÁ

N2VWuBE = c |~p| 4�M¨[ W �hDFnt:EWU8ABuWQ?�[ @A8AWQD6à�i@AZFnwDF>V8

~p = ~~kWunw:�<=TU5TuZ6TuWQ�/N5<�Z [ W��1WU:J�Y>5à�i:=>V8

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ψ (~r, t) ∝ ei(~k·~r−ωt) x�mi463{~

�Y>V`-àWU8;:©<wWuZ6DFWu< ZFWun©�/N�<EN5`-kU:=<=Wun ~k WC:ω

7 Z [ W �h�1TU<=D6Wu8ABUWyàN5BU<=>VnwBu>V�AD6�;@AW �;@AW�8i>V@AnN2V>V8An�?�[ @A8AWJ:EWuZ6ZFWb�/N�<w:ED6Bu@AZ6W5Á/nwWuZF>58�ZFN5�;@AWuZ6ZFWbZFWJ`->V@i5Wu`-Wu8�:p?AWIZFN-�AN5<w:=DFBu@iZFWbWunt:�<wTujVD��/N�<n=>V8�Tu8iWu<=j5DFW

EWU:pn=>58�D6à�A@iZFn=D6>V8

~p�

P�>5@��w>V@A<wnÉ�/N5<QN�8/N5ZF>5jVDFW�Á � W©X <=>VjVZ6DFW-n=@A�i�1>VnwW+�;@AWE = ~ω

WC:~p = ~~k

4��N5DFnÉnwD_ZFWUn�U>V8A?iWunJ?AW�` N$:EDFkU<=W ��n=>V8�:J@A8yBu>V8iBuWu�h:I��Wu<t:ED68AWu8�:H�1>5@A<b?AWunJ�/N�<w:ED6Bu@AZ6WunbBU>Và`-WÉZ6WunbTuZ6WuB q:E<w>V8AnuÁ�Z [ @A8AWI?AWun��A<=TU?ADFBC:ED6>V8AnpDF8ABU>V8�:E>V@A<w8/N5 AZ6Wun�?AW�BUWU:=:=WÉN5�i�A<=> BªRAWÉWUnw:��;@AW�ZFWunpTuZ6WuBC:E<=>58An

� � ��Å¢¼� �V�/.O#\��,�%�¤-.O'O,�¥&�i0{��#&!$%

?A>VD�VWU8;: �U:E<wWS?AD �1<EN�BU:ETUnÈ�/N�<�?AWun¨>5 �wWU:=n�?AW_��WU:=D6:EW :ªN5D6ZFZFW�Á{WU:¨�;@�[ >V8Q?A>5D6:�>5 An=WU<wVWU<¨?AWun¨D68�:EWu<rqv&Tu<wWu8ABUWunU4/�YWU:w:EWp�A<=TU?ADFBC:ED6>V8a8c[eNHTU:ETgVTU<=D��/TuW�� @c[^WU8»3{z�KV��?/N�8AnYZ\[^W �i��Tu<wDFWU8ABuWp?AWp?AD �1<EN5BC:ED6>V8�/N5<À@A8ºBU<=D6nw:ªN�Z1?AW � N2 DFn=nw>V8ºWC: �bWu<=`-Wu<UÁ N5�A<=kUnYZFNHv&>V<w` N5Z6DFn=N�:ED6>V8É?AWUn_DF?ATUWun ?AW � W�XY<=>5jVZFD6W�/N5<ps BªRA<=« ?ADF8ijVWu<U4

� �� c��6��

M¨[^N58/N5Z6>VjVDFWE = ~ω

Á~p = ~~k

WU:¨ZFN�<wWuZ�N$:EDF>58IBUZ�N5nwn=DF�;@AWY�1>V@i<È@A8AWY�/N5<w:=DFBU@AZFWÀZFD6 A<=WE = p2

2mn=@ijVjVku<wW��;@AWIZ�N�?ATu<wD6VTUW��/N�<�<=N5�A��>V<w:�N5@�:EWu`-�Anp?AW�ZFNÉv&>V8ABU:=DF>V8ψ (~r, t) ∝ ei(

~k·~r−ωt) ?A>VD�:�U:=<=W�<wWuZ6DFTuW%7�ZFNI?ATU<=D65TuW�nwWuBU>V8A?AWb?AW�BuWU:w:EW�>58A?AW��/N�<Y<EN��A�1>5<w:?7HZ�NI�N5<wD�N5 iZFWg?�[ Wun=�AN5BuW�4"�YWun<=WU` N�<=�;@AWunS>58;:SN5àWU8ATJshBªRi<=« ?ADF8Aj5Wu<?7É��>Vnw:=@AZFWU<_�;@AWbZ�NHv&>V8ABU:=DF>V8 ?�[^>58A?AWb?�[ @A8AWb�/N�<w:ED6Bu@AZ6W?/N58inpZ6W� DF?AWJ?A>VD�:SnEN$:EDFntv�N5DF<wW�Z [ Tu�;@/N$:EDF>58

i~∂ψ

∂t(~r, t) = − ~2

2m∆ψ (~r, t)

x\mi4eKV~§¨8�W �cWC:2Á

i~∂ψ

∂t= i~ (−iω)ψ = ~ωψ

WU: − ~2

2m∆ψ = − ~2

2m~∇ · i~kψ =

~2k2

2mψ

x�mi4^m�~

§¨8 <=WUà�iZ�NM�uN58�:~ω

�/N5<E

WC:~~k

�/N5<~pÁ�BuWC:=:EWITu�;@/N�:=DF>V8�Wunt:b ADFWU8(BU>Và�AN�:ED6 AZFWJN2VWUBÉZ�N

<=WUZ�N�:=DF>V8+BUZ�N5nwn=D6� @iWE = p2

2m

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V (~r)4$¾ÈN�<�?AWUnÈ<=N5DFnw>V8A8AWUàWU8;:=n��;@AW 8A>V@An

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i~∂ψ

∂t(~r, t) = − ~2

2m∆ψ (~r, t) + V (~r)ψ (~r, t)

x\mi4 �;~

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ψ (~r, t) = Ψ (~r) e−iE~t x�mi4L|5~

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− ~2

2m∆Ψ + VΨ = EΨ

x\mi4^±�~

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� �� j� 6� � � �0��

� � ��c��6��

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− ~2

2m

d2ψ

dx2+

1

2mω2x2ψ = Eψ

x�mi4e�V~

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x4

#�� IF-.=?(EC*(</?>�I"= CTS ��I_/?>W=��ψ (~r, t) = Ψ (~r) ·χ (t)

��CTS Kd;"-@>W=?(E&�I�.O�^/:3D)HK,KG+`)H( =i~ ∂χ

∂t1

χ = − ~2

2m∆Ψ

Ψ+V = E& � C<>�143d)?I@(*Md)?3aK,f�>�C*(*=?KJg9(*3dI"=N14- UV>W(*= ;�-@3aCE3d/N143d-4Z�LN3,L�]@)?3G/k143JCYS KG;�-_>�=?(E&�I 14K,5B3,I_143,I"= 143Jg[>�)?(<>W]@C*3G/14(</:=?(EI@+%=H3d/dO

� � ��Å¢¼� �V�/.O#\��,�%�¤-.O'O,�¥&�i0{��#&!$%

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q =√

mω~xÁ/WC:�>58��>VnwW

ϕ (q) = ψ (x)

dψ

dx=dϕ

dq

dq

dx=

√mω

~

dϕ

dq

d2ψ

dx2=mω

~

d2ϕ

dq2

x�mi4^¬�~

M¨[^TU�;@/N�:ED6>V8©?AD �1Tu<wWu8�:ED6WuZ6ZFW�nU[^TUBu<=D�:

− ~2

2m

mω

~

d2ϕ

dq2+

1

2mω2q2 ~

mωϕ (q) = Eϕ (q)

⇒ − ~ω

2

d2ϕ

dq2+

1

2ω~q2ϕ (q) = Eϕ (q)

⇒ − d2ϕ

dq2+ q2ϕ (q) =

2E

~ωϕ (q)

n=>VD�: d2ϕ

dq2+(λ− q2

)ϕ (q) = 0

x�mi4^z�~

>��H>V8INp��>Vn=Tλ = 2E

~ω

4$Mµ>V<wn=�;@AWq → ∞ ÁuZ [ Tu�;@/N�:=DF>V8J?AD �cTU<=Wu8�:=DFWuZ6ZFW¨WUnw:�N5�A�A<w> �iD6` N$:ED65Wu`-Wu8�:

?A>V8i8ATuWH�/N5<

d2ϕ

dq2' q2ϕ (q)

ϕ (q) ∝ e±12q2

x�mi463 ��~

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ϕ (q) = H (q) e−12q2

dϕ

dq=dH

dqe−

12q2 +H (q) (−q) e− 1

2q2

d2ϕ

dq2=d2H

dq2e−

12q2 + 2

dH

dq(−q) e− 1

2q2 −H (q) e−

12q2 + q2H (q) e−

12q2

x�mi46353}~

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nu[ TuBU<=D�:�?i>V8AB �d2H

dq2− 2q

dH

dq+ (λ− 1)H = 0

x\mi4�3}KV~

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H (q) =+∞∑

j=0

ajqj x\mh463{mV~

79Z� DFWu8�: �

dH

dq=

+∞∑

j=1

ajjqj−1 =

+∞∑

j=0

aj+1 (j + 1) qj

d2H

dq2=

+∞∑

j=1

aj+1 (j + 1) jqj−1 =

+∞∑

j=0

aj+2 (j + 2) (j + 1) qj

qdH

dq=

+∞∑

j=1

ajjqj =

+∞∑

j=0

ajjqj

⇒+∞∑

j=0

[aj+2 (j + 2) (j + 1) − 2ajj + (λ− 1) aj] qj = 0

⇒ aj+2 = aj2j + (1 − λ)

(j + 2) (j + 1)

x\mh4632��~

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:ED6>V8Anu4 �Y>5à`-WV (q) ∝ q2 Wunt:Y�/N�DF<uÁ5>V8a��Wu@h: BªRiWu<=BªRiWu<Y?iWunÀn=>VZ6@i:ED6>V8An¨�/N5D6<=WUnÀ>5@aD6à�/N�DF<=WUnu4

sh>VZ6@i:ED6>V8An_�/N�DF<=WUn �

H (q) =∑

j

a2j q2j

��N5DFna2j = a2j−2

4j − 3 − λ

2j (2j − 1)∝ a2j

j

�1>V@i< �SjV<=N58A?

⇒ a2j '1

j!

⇒∑

j

a2j q2j ' eq

2

⇒ ϕ (q) ∝ e12q2

x\mh463}|5~

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sh>VZ6@i:ED6>V8An_D6à�AN5DF<wWun �

H (q) =∑

j

a2j+1 q2j+1

a2j+1 = a2(j−1)+14j − 2 + 1 − λ

(2j + 1) (2j)∝ a2j−1

j

�1>5@A< �SD6à�AN5DF<

⇒ a2j+1 '1

j!

⇒∑

j

a2j+1 q2j+1 ' qeq

2

⇒ ϕ (q) ∝ qe12q2

x�mi4632±�~

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n:=WuZµ�;@AW

2n + 1 − λ = 0Á/n=>VD�:

λ = 2n+ 1x\mi4�3}�V~

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aj+2 = aj2 (j − n)

(j + 1) (j + 2)

x�mi4632¬�~

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nwD �SD6à�/N�DF<YWC:an+2 = an+4 = . . . = 0

sh>VZ6@i:ED6>V8An_D6à�AN5DF<wWun � 8�DF`-�/N5D6<uÁaj = 0

n=D �S�/N5D6<_WU:an+2 = an+4 = . . . = 0

MµWunpn=>5ZF@i:=DF>V8AnH (q)

n=>V8�:p?A>V8ABJ?AWUng�1>VZ�f 8�5V`-Wunu4

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λ = 2n + 1, n = 0, 1, . . .

⇒ E =

(n+

1

2

)~ω

x�mi4632z�~

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12q2 N2VWUB

Hn (q) =

{a0 + a2q

2 + · · ·+ anqn n=Dµ8©�/N5D6<

a1q + a3q3 + · · ·+ anq

n n=Dµ8©DF`-�/N5D6<N{5WuB

aj+2 = aj2 (j − n)

(j + 1) (j + 2)

x�mi4LK ��~

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1WU<=jAÁAfºBu>V`-�A<=D6n_Z�NQBU>V8Anw:EN58�:EW~ω2

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ϕ0 (x) ∝ e−mω2~x2 x\mh4LKh3{~

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� �� 6� � � � � � 6� �� c�� @� � ��

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x : ϕ (x, t) 7−→ xϕ (x, t)

p : ϕ (x, t) 7−→ −i~∂ϕ∂x

(x, t)

x\mh4LKVK5~

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)= 0

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ϕx0 (x) = lima→0 ϕ(a)x0 (x)

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ϕ(a)x0

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nwDx /∈

[x0 − a

2, x0 + a

2

]

1a

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2, x0 + a

2

]∫ +∞

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a

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x0−a2

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f(x) = f(x0) + (x− x0)f′(x0) + . . .

⇒ 1

a

∫ x0+a2

x0−a2

f (x) dx = f(x0) +O(a)

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∫ +∞

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³��F¡��I�A¥�% ,�! ¢¼��% .O.O%»%�0 �º% ¥��h0}#��. ' � U.��V%;!{0}#�0},O'O% ¡��

� �� @� �� N� �� c�� @� �� W� �� 79ZµnwWu<EN�@i:=DFZFW�?AWJZFDF<wW�?�[^N5 �>V<=?©Z�NÉ�/8©?AWJZ\[eN5�i�1WU8A?ADFBUWJ]�4^mi4Mµ>V<=nw� @c[^>V8�`-Wunw@A<=W @A8AWº>V in=Wu<t�N5 AZFW

A?/N58inÉ@A8�TU:EN�: |ψ〉 Á�ZFN(�A<=>5 /N5 AD6ZFD6:=T�?AW :E<w>V@i5Wu<

@A8AWb�N5ZFWU@A<S�A<w>V�A<=Wam

N5nwn=> BuD6TuWHN5@©VWuBC:EWU@A<��i<=>V�A<wW |ϕm〉 Wunt: pm = |〈ϕm|ψ〉|24

sh@A�A��>Vnw>V8An��;@AWIZ [ >V8�v�N�n=n=WI�AZF@in=DFWU@A<=np`-Wun=@i<=Wun�n=@A<�@A8»ntfhnt:EkU`aWI� @c[^>V8»Na�A<wTu�/N5<wT�?/N�8AnZFW©`��u`-W+TU:ªN$: |ψ〉 4�MµWun�`-Wun=@i<=Wun-n=@iBuBuWUn=nwD6VWUnº?ATC�/8ADFnwn=WU8;: @A8AW©?AD6nw:E<wDF A@h:EDF>58�?AW��N5Z6Wu@A<wnB2N5<=N5BU:=Tu<wDFn=TUWJ�/N5<gZFWUnSjV<EN58i?AWu@A<wngR/N5 AD�:E@AWUZFZFWUn �3V4S¿ÀN5Z6Wu@A<S`->{fVWu8i8AW � a =

∑i piaiKh4S¿ÀN5<wD�N58ABUW � σ2 =

∑i pi (ai − a)2

mi4p²¨B2N5<t:g:9fh��W � σ =√σ2 =

√∑i pi (ai − a)2

P�<=N5?A@AD6n=>V8in BUWun_BuN5<EN5BC:ETU<=DFnt:ED6� @iWunÀ?AW�ZFNH?AD6nw:=<=DF i@i:ED6>V8�?/N58An ZFWpZ�N58AjVN5jVWg?AWpZ�NI`-TuBuN58AD6� @iW�;@/N58�:ED6� @iW �3V4

a =∑

m

pmam =∑

m

am |〈ϕm|ψ〉|2 =∑

m

am〈ϕm|ψ〉〈ψ|ϕm〉

�(N5DFnA =

∑

m

am|ϕm〉〈ϕm|

⇒ a = 〈ψ|A|ψ〉

x&�A4L|5~

〈ψ|A|ψ〉 Wunw:À?A>58ABgZ�N eE_`DGF^U6] 1 BEJ0F^PRPdF �;@AWgZ [ >V8Q:E<w>V@iVWpZF>5<=n¨?AWgZ�N�`-Wunw@A<=W !}�;Ç¨��0{�;%?AWJZ [ >V AnwWu<t5N� AZFW

A?/N58inpZ\[^TC:ªN�: |ψ〉 4Kh4pM�NÉ5N�<=D�N�8ABuWb��Wu@i:pnwWH<wTuTUBu<=D6<=W �σ2 =

∑

i

pi(a2i − 2aia+ a2

)

=∑

i

pia2i − 2a

∑

i

piai + a2∑

i

pi

=∑

i

pia2i − 2a2 + a2

=∑

i

pia2i −

(∑

i

piai

)2

x&�A4e±V~

�b<UÁ ∑m

pma2m =

∑

m

a2m〈ϕm|ψ〉〈ψ|ϕm〉

�(N5D6nA|ϕm〉 = am|ϕm〉

⇒ A2|ϕm〉 = amA|ϕm〉 = a2m|ϕm〉

x&�A4L�5~

¡� ´µ�"!$¶y,O¥&�i0}#&�1.�¸+� .��;!$�A¥�%('O% ¥&�y¢¼�h�V�A.�#��,�%�*-,��/.10}#��,�%

MµWun+�N5Z6Wu@A<wn��i<=>V�A<wWun©?AWA2 n=>58;:�Z6Wun©B2N5<w<=TUn�?iWun©5N�ZFWu@i<=n+�A<w>V�A<=WUn�?iW

AÁSWU:©Z6Wun

5WuBU:=Wu@A<wn��A<w>V�A<wWungn=>V8�:pZFWUng` �UàWUnu4 � @�Bu>5@A��Á

A2 =∑

m

a2m|ϕm〉〈ϕm|

WC:σ2 = 〈ψ|A2|ψ〉 − 〈ψ|A|ψ〉2

x&�A4^¬�~

mi4��8©Wu8�?ATu?i@AD6:p�;@AWJZ [ TuBuN5<w:S:9f ��WIWUnw:p?A>V8A8iTH�AN5< � σ =

√〈ψ|A2|ψ〉 − 〈ψ|A|ψ〉2§¨8�àTUB2N58AD6�;@AWb�;@/N58�:ED6� @iW5Á"BuWC:�TUB2N5<t:g:9fh��WJWunt:�8A>�:ET

∆A|ψ〉 =

√〈ψ|A2|ψ〉 − 〈ψ|A|ψ〉2

x&�A4^z�~

� [^N5�A<wkun-ZFW3 �

��>Vnw:=@AZ�N$:2ÁÈnwD_Z [ >V8 N(:=<=>V@i5Tam

x�nw@A�A��>Vn=TUW�8A>58 ?iTujVTU8ATu<wTuW{~ÉZ6>V<=nÉ?iW+Z�N`-Wun=@i<=W ?AW

AÁ_ZFW»ntf nw:EkUàW·WUnw:©?/N58in�Z\[^TC:ªN�: |ϕm〉

N5�A<wkun+ZFN àWUn=@A<wW5Á_WC:©@A8AW ?AWU@��hDFku`-W`-Wun=@i<=WJ?A>V8A8iWu<EN

amN2VWUBH@i8AWI�i<=>V /N� ADFZ6D6:ETpTujVN5ZFW[7+3V4"�YWC:=:=WIBUWu<w:=D6:=@A?AWbn=Wb:=<EN5?A@iD6:g�/N5<SZ6W

v�N5D�:S�;@AW

∆A|ϕm〉 = 0x&�A463 ��~

BuWJ�;@AWHZ [ >V8+��Wu@i:�N5D6n=Tu`-Wu8�:S5Tu<=D��/Wu< �〈ϕm|A2|ϕm〉 = a2

m〈ϕm|ϕm〉 = a2m = 〈ϕm|A|ϕm〉2

x&�A46353}~

�Y>V8in=DF?iTu<=>58AnYàN5DF8�:=Wu8/N58�: ?AWu@ � >5 An=WU<w�N5 AZ6WunA

WU:B4hshD�WuZ6ZFWun @CBO121 UL F^PRL Á;>V8 ��Wu@h:YZ6Wun

?ADFN5jV>V8/N�ZFDFnwWu<J?/N58inÉ@A8AWº AN5n=W+BU>Và`�@A8AW54È��8��1WU@i:Q?A>58ABº�A<wTu�/N5<wWu<ÉZ6W ntfhnt:EkUàW+?/N�8AnÉ@A8��0{�i0aÇ�!}�"ÇO!$%·�V��¶ ¶·,O. WU:p:E<w>V@iVWU<�N25WuB�BUWu<t:ED6:=@A?AWH@A8AWH5N�ZFWu@i<p�i<=>V�A<wWI?iW

AZF>5<=ng?AWHZ�N

`-Wun=@i<=WJ?AWA

WU:p@A8AWb�N5ZFWU@A<S�A<w>V�A<=Wb?iWB

ZF>5<=nS?AWJZ�NÉ`-Wunw@A<=WJ?AWB4

�(N�DFn_nwDA

WC:BPdF @CBO121 UL F^PRL&>k_Ec Á >V8+8AW��1WU@i:S�/N5n :E<=>5@iVWU<S?AW� /N5nwWbBu>5à`É@i8AW5Á WU:SD6Z

Wunt:JWu8·jVTU8ATu<=N5Z�DF`-��>Vn=nwDF AZ6WI?AWQ�i<=Tu�AN5<=WU<bZFWÉntfhnt:EkUàW�?/N58inb?AWunJTU:EN�:EnJN25WuB�@A8AWQ�A<wTuBUDFn=D6>V8N5<w AD6:=<EN5D6<=Wu`-Wu8�:¨j5<EN58A?iWpnw@A<�Z�N��A<wTu?AD6BU:=DF>V8�?AW

AWU:B4 �YWC:=:EWpDF`-��>Vn=nwDF AD6ZFD6:=T�n=Wg:E<EN�?A@AD6:��/N5<

Z�N ]>F^D _`LTQhBEP�[�� Q Pd@CF^]gL Q LTU�[F <Y^PdY^]\_`DKQGcfYCF �

∆A|ψ〉∆B|ψ〉 ≥1

2

∣∣∣〈ψ|[A, B

]|ψ〉∣∣∣

x&�A463{KV~

� TUà>V8inw:E<=N�:ED6>V8 � ��8l<=WU` N�<=�;@AWy?�[eN� 1>V<w?G�;@AW ∀A, B RAWU<=`-D6:ED6�;@AWun 〈ψ|[A, B

]|ψ〉 WUnw:

DFàN5jVD68/N5D6<=Wp�A@A<u4"§¨8�W��1WU:uÁ[A, B

]†=(AB − BA

)†= B†A† − A†B† = BA− AB = −

[A, B

]

⇒ 〈ψ|[A, B

]†|ψ〉 = 〈ψ|

[A, B

]|ψ〉∗ = −〈ψ|

[A, B

]|ψ〉

x&�A4632m�~

³��³�� 1¥�,µ0}#��.�0{% ¶ÅÇ¨�"!}%h¥�¥&% �ÀÆµ�$�O!��µ'O#�. "%;! � ��% #�)u% .cÃ¨%;! Å%�0 �b�O!}% .�� % )U0 ¡c¦

��8�?ATC�/8AD�:gZFWung>V��Tu<=N�:EWU@A<=ngRAWU<=`-D6:=DF�;@AWun

A0 = A− 〈ψ|A|ψ〉 3 ZB0 = B − 〈ψ|B|ψ〉 3 Z

x��i4632��~

�;@ADµnEN�:=DFntv&>V8�:

〈ψ|A20|ψ〉 = 〈ψ|A2|ψ〉 − 2〈ψ|A|ψ〉〈ψ|A|ψ〉 + 〈ψ|A|ψ〉2 = ∆A2

|ψ〉[A0, B0

]=[A, B

]

A†0 = A0

x��i463}|5~

WU:p>V8©Bu>58An=D6?Aku<wW

f(λ) =∥∥∥(A0 − iλB0

)|ϕ〉∥∥∥

2

≥ 0 (∀λ)

�(N5DFnf(λ) = 〈ψ|

(A0 + iλB0

) (A0 − iλB0

)|ψ〉

nw>VD6:f(λ) = 〈ψ|A2

0|ψ〉 + λ2〈ψ|B20 |ψ〉 − iλ〈ψ|A0B0|ψ〉 + iλ〈ψ|B0A0|ψ〉

= 〈ψ|A20|ψ〉 + λ2〈ψ|B2

0 |ψ〉 + λ(−i〈ψ|

[A0, B0

]|ψ〉)

︸︷︷︸∈R

x��i463{±V~

¾�>V@i<¨� @iWSBuW_��>VZ6f 8�55àW ?A@Qn=WUBu>V8A?-?AWUjV<=T_nw>VD6:È�1>5n=D6:=D6v ∀λ Á5D6Zhv�N5@h:��;@�[ DFZi8c[eN5D�:��/N�n�?AWS<EN�BuDF8iW<=TUWuZFZ6W-x��/N�<EN5 �>VZFW�N5@hq9?iWun=nw@Anp?AWJZ [^N �hWλ~ Á/?A>V8ABJ�;@AWJn=>V8�?AD6n=Bu<wDF`-DF8/N�8;:_n=>VD�:S8ATujVN�:ED�v �

∆ =(−i〈ψ|

[A0, B0

]|ψ〉)2

− 4〈ψ|A20|ψ〉〈ψ|B2

0|ψ〉 ≤ 0

⇒(〈ψ|A2|ψ〉 − 〈ψ|A|ψ〉2

)(〈ψ|B2|ψ〉 − 〈ψ|B|ψ〉2

)≥ 1

4

∣∣∣〈ψ|[A, B

]|ψ〉∣∣∣2

⇒ ∆A|ψ〉∆B|ψ〉 ≥1

2

∣∣∣〈ψ|[A, B

]|ψ〉∣∣∣

x��i463}�5~

79Z WUnw:�?A>V8ABaDF`-�1>Vnwn=D6 AZFWQ?AW-`-Wun=@i<=Wu<In=D6`É@AZ6:EN58ATUàWU8;:HWUnw:QN25WuBº@A8iW �A<wTuBuD6n=D6>V8 N5<w AD6:=<EN5D6<=W?AWu@ ��>V An=WU<w�N5 AZ6Wung�;@ADµ8AWJBu>5à`É@h:EWu8�:p�/N5nU4

� �� @� � � � � �� 0�� 6�� F�6� � � � � �

�N� � �0� � � ;� �2�� 79ZµnwWu<EN�@i:=DFZFW�?AWJZFDF<wW�?�[^N5 �>V<=?©Z [^N5�A��Wu8i?ADFBUWb]�4 �A4��>V@in N25>V8AnÀ?AT �F7b @a�;@AWpZ [ TU5>VZF@i:=DF>V8É:=Wu`-�1>5<=WuZ6ZFW ?�[^@i8AWp�N5Z6Wu@A<¨`->{fVWU8A8AW%7J�AN5<w:=DF<¨?�[ @A8

DF8Ant:ªN58�: ?AW(<wTUv&Tu<wWu8ABUWt = 0

�1>V@h5N�D6:aDF8A?AD �cTU<=WUà`-Wu8�:��U:=<=W(?iTuBu<wD6:=W(�/N5<aZ [ Tu�;@/N�:=DF>V8 ?AWshBªRA<w«h?AD68AjVWU<uÁ/?�[^N5�A<wkungZ�N5�;@AWUZFZ6W�ZFWUnSTU:ªN$:EngTU5>VZF@AWU8�:�nwWuZF>58 �

i~d

dt|ψ(t)〉 = H(t)|ψ(t)〉

x��i463{¬V~

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>��J�/N5<µ@i8AW�Tu�;@/N�:=DF>V8b?�[ TU5>VZF@h:EDF>58H��>V@A<µZ6Wunµ>V��Tu<=N�:EWU@A<=n�N�8/N5ZF>5jV@AWZ7SZ [ TU5>VZF@h:EDF>58�:EWUà��>V<=WUZFZ6W?AWUnp�N5<wD�N5 AZ6Wun_?hfh8AN5àD6�;@AWungWu8�`-TuB2N�8ADF�;@AWbBUZ�N5nwn=DF�;@AW �

dAHdt

=1

i~

[AH , HH

]+

(∂A

∂t

)

H

x&�A4632z�~

�YWU:w:EWpTu�;@AD�5N�ZFWu8iBuWpnu[ TU:EWU8A?+N5@a�A<=> BuWUn=nw@AnY?AWp`-Wun=@i<=WpZF@AD q9`��u`-WY:EWUZ��;@�[ DFZ/WUnw: ?ATuBU<=D�:Y�/N�<ÀZ6W��>Vnw:=@AZ�N�:�mi4�¾�>5@A<ÈZ6WÀ5>VDF<UÁ$BU>V`-àWU8��u>V8in��/N�<ÈDF8�:=<=> ?A@ADF<wW�@A8ÉBu>58ABuWU�i:È:=<=kun�@i:ED6ZFW�Á D � B >RY^]<_`LNFÛ6][�� Yê`BEDKUL QhBEP 4� ��¤QÇ¨� !}�i02%h,�!+'�� 1¥�,µ0}#��.M¨[ Tu�;@/N�:=DF>V8 ?AW shBªRi<=« ?ADF8Aj5Wu<-Wunw:a@A8AW�Tu�;@/N�:=DF>V8 ?AD �1Tu<wWu8�:ED6WuZ6ZFW+?A@ �A<=WUàD6Wu<�>V<=?i<=W��/N5<

<EN��A�1>5<w:SN�@ :EWUà�AnU4 � @+BU>V@A��Áhn=D1>V8ºBu>58A8/N��6:_Z [ TU:EN�:_?i@ºnwf nw:=ku`-W[7�@A8+D68Anw:EN58�:_?A>V8i8AT5Áh�/N5<W �hWUà�AZ6W

t = t0Á/>V8©ZFWbBu>58A8/N��6: 7É:E>V@i:gD68Anw:EN58�:24"�YWUBuD��Wu<=`-WU:g?c[^D68;:=<=> ?A@AD6<=W�@A8©>V��Tu<=N�:EWU@A<

?ATUBu<=D��N58�:�Z\[^TCV>VZ6@i:ED6>V8º?A@�nwf nt:Eku`-WIN5@�BU>V@A<wnp?i@�:=Wu`-�Anu4� TC�/8AD�:EDF>58 � M¨[ >V��Tu<EN$:EWu@i<S?�[^TCV>VZ6@i:ED6>V8©Wunt:�?ATC�/8ADµ�/N�<

|ψ(t)〉 = U(t, t′)|ψ(t′)〉x&�A4LK ��~

¾¨<=>V�i<=DFTC:ETUn �3V4g79Zµn=N�:ED6nwv�N5D�:_Z [ Tu�;@/N$:EDF>58�?iD �1Tu<=WU8�:EDFWUZFZ6W

i~∂

∂tU(t, t′) = H(t)U(t, t′)

x&�A4LK 3}~� TUà>58Anw:=<EN�:=DF>V8 � � [^N5�A<wkunpZ [ Tu�;@/N�:=DF>V8+?iW�s BªRA<=« ?ADF8ijVWu<UÁ/>V8�N �

i~d

dt|ψ(t)〉 = H(t)|ψ(t)〉

⇒ i~∂

∂t

(U(t, t′)|ψ(t′)〉

)= H(t)U(t, t′)|ψ(t′)〉

⇒ i~∂

∂t

(U(t, t′)

)|ψ(t′)〉 = H(t)U(t, t′)|ψ(t′)〉

x&�A4LK5KV~

�(N�DFnbBuWC:=:=WaTU� @AN�:ED6>V8�WUnw:J <EN5D6WQ�;@AWuZ��;@AW-n=>VD�: |ψ(t′)〉 4�MµWunJ?AWU@��y>V��Tu<=N�:EWU@A<=nHn=>V8�:?A>58ABJTuj�N5@ �c4

Kh4U(t, t′)

Wunt:�D68;5Wu<wn=DF iZFW54� TUà>58Anw:=<EN�:=DF>V8 � ��8�N�WU8�W �cWC:

U(t, t′)U(t′, t)|ψ(t)〉 = U(t, t′)|ψ(t′)〉 = |ψ(t)〉 x��A4eK5m�~� WJ`��u`-W5ÁA>58(N �

U(t′, t)U(t, t′)|ψ(t′)〉 = U(t′, t)|ψ(t)〉 = |ψ(t′)〉 x&�A4eK��;~��8©Wu8�?ATu?i@AD6:p�;@AW �

U(t′, t) = U(t, t′)−1 x&�A4LK5|V~

³��³�� 1¥�,µ0}#��.�0{% ¶ÅÇ¨�"!}%h¥�¥&% �ÀÆµ�$�O!��µ'O#�. "%;! � ��% #�)u% .cÃ¨%;! Å%�0 �b�O!}% .�� % )U0 ¡

mi4U(t, t′)

WUnw:p@A8AD�:ªN5D6<=W�4� TUà>V8inw:E<=N�:ED6>V8 � �Y>V8AnwDF?ATU<=>V8in�Z6Wb�A<=> ?A@AD�: U †(t, t′)U(t, t′) �

∂

∂t

(U †(t, t′)U(t, t′)

)=∂U †(t, t′)

∂tU(t, t′) + U †(t, t′)

∂U (t, t′)

∂t

��<uÁ ∂U(t, t′)

∂t=

1

i~H(t)U(t, t′)

⇒ ∂U †(t, t′)

∂t= − 1

i~U †(t, t′)H(t)

x��i4LK5±V~

]�D68An=D\ÁA>V8�N �i~∂

∂t

(U †(t, t′)U(t, t′)

)= −U †(t, t′)H(t)U(t, t′) + U †(t, t′)H(t)U(t, t′) = 0x��i4LKV�5~

�(N5D6nU(t, t) =

3 Z= U †(t, t)

�/N5<g?ATC�/8AD6:=DF>V8c4i]�D68An=D\ÁU †(t, t)U(t, t) =

3 Z∂

∂t

(U †(t, t′)U(t, t′)

)= 0

⇒ U †(t, t′)U(t, t′) =

3 Z ∀t, t′ x��i4LK5¬V~

�Y>V`-àWHZ [ >V��Tu<EN$:EWu@i<�?�[ TU5>VZF@i:=DF>V8�Wunt:bDF8�VWU<=nwDF AZ6W5Á�>V8»Wu8(?ATu?A@iD6:��;@�[ DFZOWunt:b@A8AD�:ªN5D6<=W�4/79ZÈWunt:8AT2N�8Aà>5DF8AnSD68Anw:=<=@ABC:ED6vO?AWJ?ATUà>58;:=<=WU<p?iDF<=WUBU:=Wu`-Wu8�:p� @iWHZ\[^>58�NQTUj�N5ZFWUàWU8�:

U(t, t′)U †(t, t′) =3 Z ∀t, t′ x��i4LK5zV~

¾�>V@i<pBUWuZFNiÁA<=WCVWU8A>V8AnH7�ZFNQ?ATU�/8iD6:ED6>V8+?AWJZ [ >V��Tu<=N�:EWU@A<S?�[ TUV>5ZF@i:=DF>V8 �|ψ(t)〉 = U(t, t′)|ψ(t′)〉 x&�A4em��~

WU:p?ATU<=D65>V8AngBuWC:=:=WITU� @AN�:ED6>V8©�/N5<S<EN��A�1>5<w: 7t′4/79Zc DFWU8�: �

∂U (t, t′)

∂t′|ψ(t′)〉 + U(t, t′)

d|ψ(t′)〉dt′

= 0

⇒ i~∂U (t, t′)

∂t′|ψ(t′)〉 = −U(t, t′)H(t′)|ψ(t′)〉 x&�A4emh3}~

��8�WU8Å?ATU?A@AD6:H�;@AW-ZFWUnH?iTu<=D�VTUWun��AN5<J<EN5�A��>V<t:@7t′?AW

U(t, t′)WU:U †(t, t′)

nw>V8�:�?i>V8A8ATUWun�/N5< �

i~∂U(t, t′)

∂t′= −U (t, t′)H(t′)

i~∂U †(t, t′)

∂t′= H(t′)U †(t, t′)

x��i4em�K5~� @�Bu>V@A�cÁ/DFZc DFWU8�: �i~

∂

∂t′

(U(t, t′)U †(t, t′)

)= −U(t, t′)H(t′)U †(t, t′) + U(t, t′)H(t′)U †(t, t′) = 0

x&�A4em5m�~BuWJ�;@AD�DF`-�AZFD6�;@AW5Á �A@AD6n=�;@AW

U(t, t) =3 Z

= U †(t, t)ÁA�;@AW

U(t, t′)U †(t, t′) =3 Z ∀t, t′ 4

³ � ´µ�"!$¶y,O¥&�i0}#&�1.�¸+� .��;!$�A¥�%('O% ¥&�y¢¼�h�V�A.�#��,�%�*-,��/.10}#��,�%

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t04

§¨8º:EWU<=`-W�?�[ >V�1TU<EN�:=Wu@A<Y?�[^TCV>VZ6@i:ED6>V8�Á ZFW��1>VD68�:_?AW� @AW�?iWJd�WuD6n=WU8 �Wu<wjQWunt:g<=Tunw@A`-T�?/N58AnYZ�N�A<w>V�1>5n=D6:=DF>V8+nw@AD6�N58�:=W �¾¨<=>V��>VnwD6:ED6>V8 � M�N�n=>VZ6@i:ED6>V8+?AWJZ [ Tu�;@/N�:=DF>V8+?iD �1Tu<=WU8�:EDFWUZFZ6W

dAH(t)

dt=

1

i~

[AH(t), HH(t)

]+

(∂A

∂t

)

H

x&�A4em��;~

Wunt:

AH(t) = U †(t, t0)A(t)U(t, t0)N2VWUB

HH(t) = U †(t, t0)H(t)U(t, t0)x&�A4emV|V~

� TU`a>V8inw:E<=N�:ED6>V8 �dAHdt

=∂U †(t, t0)

∂tA(t)U(t, t0) + U †(t, t0)

∂A

∂tU(t, t0) + U †(t, t0)A(t)

∂U(t, t0)

∂t

= − 1

i~U †(t, t0)H(t)A(t)U(t, t0) + U †(t, t0)A(t)

1

i~H(t)U(t, t0)

+ U †(t, t0)∂A

∂tU(t, t0)

=1

i~

(−U †(t, t0)H(t)U(t, t0)U

†(t, t0)A(t)U(t, t0))

+1

i~

(U †(t, t0)A(t)U(t, t0)U

†(t, t0)H(t)U(t, t0))

+ U †(t, t0)∂A

∂tU(t, t0)

=1

i~

[AH(t), HH(t)

]+

(∂A

∂t

)

H x&�A4em5±�~>��©Z [ >V8�NQDF8��wWUBU:=T

U(t, t0)U†(t, t0) =

3 Z�Wu8�:=<=WHZ�N�K�WU:pZFNQm

�Tuj�N�ZFD6:=T54

¢¼% )2,O!}%»%�0-� ��1¥�,µ0}#��. 0{% ¶ÅÇ¨�"!$% ¥�¥�%�Y>V8in=DF?iTu<=>58An�@A8iW >V AnwWu<t5N� AZFW

A8iWa?ATU�1WU8A?/N58�:��/N5n�W �h�AZFD6BuD6:=Wu`-Wu8�:J?A@Å:=Wu`-�AnuÁ�WC:�WUntq

nEN2f5>V8Anp?AWJB2N5Z6Bu@AZ6Wu<SZ�NÉ�i<=>V /N� ADFZ6D6:ET�?AWb:E<w>V@iVWU<am

N�@©:=Wu`-�Ant4

� [^N5�A<wkun�shBªRA<w«h?AD68AjVWU<uÁ/BuWC:=:=WH�i<=>V /N� ADFZ6D6:ET�Wunt:�?i>V8A8ATUWI�/N�< �|〈ϕm|ψ(t)〉|2 =

∣∣∣〈ϕm|U(t, t0)|ψ(t0)〉∣∣∣2 x&�A4emV�V~

� [^N5�A<wkun¨dpWuD6n=Wu8; �Wu<wjAÁ�B�[^WUnw:ÈZ [ >V An=WU<w�N5 AZ6WA�;@AD TU5>VZF@AW�4Vs D >58Iv�N5D�:�@i8AWY`-Wunw@A<=W N5@I:=Wu`-�An

tÁ">V8�8AWI��Wu@h:p:=<=>V@hVWu<p�;@�[ @A8AWI�N5Z6Wu@A<g�A<w>V�A<wWH?iW

AH(t)4/79Zµf©Na?i>V8ABJ@A8AWH?AD �-BU@AZ6:=T � Z6Wun

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�N5ZFWU@A<=n��A<=>V�i<=Wun�?AWAH(t)

nw>V8�:wq WuZFZ6WunpZFWUn�`��u`-Wunp� @iWÉBuWUZFZFWUn�?iWA�ÉM�N <=TU�1>V8in=W�WUnw:�>V@AD Á

@A8AWJ�A<w>V�A<=D6TU:=TH�;@ADµ?iTuBu>5@AZFWJ?AWJZ [ @A8AD�:ªN5<wD6:=T�?AWU(t, t0)

4§¨8�W �cWC:2Á/n=@i�A�1>5n=>V8Anp�;@AW

A|ϕm〉 = am|ϕm〉4/]�ZF>5<=nuÁ

AH(t)(U †(t, t0)|ϕm〉

)= U †(t, t0)A U(t, t0)U

†(t, t0)︸︷︷︸� �|ϕm〉

= U †(t, t0)am|ϕm〉= am

(U †(t, t0)|ϕm〉

)

x��i4emV¬V~

]�D68An=DA:=>V@i:EWg�N5ZFWU@A<¨�A<w>V�A<=Wp?AWAWunw:YN5@Anwn=D/�N5ZFWU@A<��A<=>5�A<=Wg?AW

AH(t)Á�WU:ÀZFWSVWUBU:=Wu@A< �A<=>V�i<=W

N5n=nw>hBUDFTJWunt:U †(t, t0)|ϕm〉

4M�N(�A<=>5 /N5 AD6ZFD6:=Ta?iWºàWUn=@A<wWu<

amN5@ :EWUà�An

tWUnw:�?A>58ABº?A>V8i8ATuW�Á�?A@ �1>5DF8�:É?AWah@iW+?AW

d�WUDFn=WU8; 1WU<=j-�/N5<am(t) =

∣∣∣〈U †(t, t0)ϕm|ψ(t0)〉∣∣∣2

=∣∣∣〈ϕm|U(t, t0)|ψ(t0)〉

∣∣∣2

MµWun��1>VD68�:En+?AW· @AW�?iWÅdpWuDFnwWu8; �Wu<=j WU:�?AWÅshBªRA<w«h?AD68AjVWU<�nw>V8�:(?i>V8ABy<=DFj5>V@A<=WU@An=WUàWU8�:Tu�;@AD�5N�ZFWu8�:=ng D6ntq*7}q DFn_?iWunp�1>Vnt:E@AZFN�:EnU4

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U(t, t′)4¨§¨8

W��1WU:uÁYDFZgn=N�:ED6nwv�N5D�: Z\[^TU� @AN�:ED6>V8�?AD �cTU<=WU8;:=DFWUZFZFW©�A4eKh3V4Y�(N5D6n �;@/N58i?H

Wunt:ºDF8i?ATu��Wu8A?AN58�:+?A@:EWUà�AngZFNÉn=>VZ6@i:ED6>V8+Wunt:�?A>58A8ATuWJ�/N�< �

U(t, t′) = e−iH(t−t′)/~ x��i4emVzV~

§¨8�W �cWC:2Á

i~∂

∂tU(t, t′) = i~

(−−iH

~

)e−iH(t−t′)/~ = He−iH(t−t′)/~ = HU(t, t′)

x��i4^� �V~

� W��iZF@AnUÁHH(t) = H

4"§¨8»W��1WU:uÁH

BU>Và`�@i:EWHN2VWUB�ZF@iD�q9`��u`-W5Áh?A>V8ABIN2VWUB�n=>58(W �h��>V8AWU8hq:ED6WuZFZ6W54 � @�BU>V@A��Á

HH(t) ≡ U †(t, t′)HU(t, t′) = H U(t, t′)†U(t, t′)︸︷︷︸� �= H x��i4^�A3{~

� N58An�BuWIB2N5nUÁ1Z\[^TCV>VZ6@i:ED6>V8+:EWu`-��>V<=WUZFZFWb?c[^@A8(>V�1TU<EN�:=Wu@A<p8AWI?ATu��Wu8A?AN58�:��/N5n�W �h�AZ6DFBuD�:EWUàWU8;:

³1� ´µ�"!$¶y,O¥&�i0}#&�1.�¸+� .��;!$�A¥�%('O% ¥&�y¢¼�h�V�A.�#��,�%�*-,��/.10}#��,�%

?A@©:EWUà�AngnU[^TUBu<=D�: �

dAHdt

=1

i~

[AH(t), H

]

=1

i~

(U †(t, t0)AU(t, t0)H − HU †(t, t0)AU(t, t0)

)

=1

i~

(U †(t, t0)AHU(t, t0) − U †(t, t0)HAU(t, t0)

)

=1

i~U †(t, t0)

[A, H

]U(t, t0)

x&�A4^��KV~

¾ÈN�<gN5DFZ6ZFWu@i<=nuÁ n=DH

Wunw:pD68A?ATu��Wu8i?/N58�:g?A@+:EWUà�AnUÁAZ [ TU5>VZF@h:EDF>58ºnu[ TuBU<=D6:gnwDF`-�AZFWUàWU8�:_Wu8©v&>V8AB q:ED6>V8+?AWUn�TC:ªN�:=npnt:ªN�:=DF>V8i8/N5DF<wWunU4/sh@A�i�1>Vnw>V8AnpWu8�W��1WU:

H|ϕn〉 = En|ϕn〉x&�A4^�Vm�~

>��ZFWUn {|ϕn〉} nw>V8�: >5<w:ERi>V8A>V<wàTUn�WU:À>��Z [ >V8�Nbn=@i�A�1>5n=TgZFWSnw�1WUBU:E<wWg?ADFnwBu<=WC:À�1>V@i<¨n=DF`-�AZ6D6�/WU<ZFWUnS8A>5:EN�:ED6>V8AnuÁiWU:p�1>5n=>V8An

|ψ(t)〉 =∑

n

cn(t)|ϕn〉 x&�A4^�5�;~

M¨[^TU�;@/N�:ED6>V8Ang?AWIshBªRA<w«h?iDF8AjVWU<SBu>V8i?A@AD6: 7

i~d

dt|ψ(t)〉 = H|ψ(t)〉

⇒ i~∑

n

d

dtcn(t)|ϕn〉 =

∑

n

cn(t)En|ϕn〉

⇒ i~∑

n

d

dtcn(t)〈ϕm|ϕn〉 =

∑

n

cn(t)En〈ϕm|ϕn〉

⇒ i~dcm(t)

dt= cm(t)Em

⇒ cm(t) = cm(t0)e−iEm(t−t0)/~

⇒ |ψ(t)〉 =∑

n

e−iEn(t−t0)/~cn(t0)|ϕn〉

x&�A4^��|V~

M¨[^>5�1TU<EN�:=Wu@A<¨?�[ TU5>VZF@h:EDF>58Q�1>Vnwn=kU?AW�?iWS�AZF@An¨@i8AWg<=Wu�i<=TunwWu8�:ªN$:EDF>58anw�1WUBU:E<=N5ZFW_:=<=kUn¨n=DF`-�AZ6W54V§¨8

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W��1WU:uÁ

H|ϕn〉 = En|ϕn〉⇒ U(t, t′)|ϕn〉 = e−iH(t−t′)/~|ϕn〉 = e−iEn(t−t′)/~|ϕn〉⇒ U(t, t′)

∑

n

|ϕn〉〈ϕn|︸︷︷︸� �

=∑

n

e−iEn(t−t′)/~|ϕn〉〈ϕn|

⇒ U(t, t′) =∑

n

e−iEn(t−t′)/~|ϕn〉〈ϕn|

x��i4^��±V~

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e−iH(t−t′)/~ P � FCc^L&>k_Ec#eE_`D _� VDGF 4��>5@An_VWU<=<=>58AnS?/N58AnSZ6W�BªR/N��AD6:=<=W�nw@A<SZFWUn_�1WU<w:=@A<= /N$:EDF>58AnY?AT q�1WU8A?/N58�:=Wunp?A@�:=Wu`-�AngBu>V`-`-Wu8�:g�A<=> BuTU?AWu<g?/N�8AngBuWJB2N5nU4

��;��!}Ä ¶Å%(' � �b��!}%h.�� % )U0shD�Z\[^>V8+�i<=Wu8i?�ZFN��N5ZFWU@A<_`->}f5Wu8A8iWH?AN58Ang@A8©TU:EN�: |ψ〉 �;@AWuZ6Bu>V8i� @iWH?iWbZ [ Tu�;@/N�:=DF>V8+?�[ TU5>�q

ZF@i:=DF>V8a?AWbdpWuDFnwWu8; �Wu<=jQ?c[^@A8iWb>V An=WU<w�N5 AZ6W��;@AD�8iW�?ATu��Wu8i?+�/N5n_W �i�iZFDFBUD6:=Wu`-Wu8�:_?A@º:=Wu`-�AnuÁ DFZ DFWu8�: �

d

dt〈ψ|AH(t)|ψ〉 =

1

i~〈ψ|[AH(t), HH(t)

]|ψ〉

x��i4^�;�5~

�Y>V8AnwDF?ATU<=>V8AnpZFW�B2N5np�/N5<t:EDFBU@AZFD6Wu< ?�[^@i8AWJ�/N5<w:=DFBU@AZFWb?ATUBu<wD6:EWb�AN5<gZ [ R/N5`-DFZ6:=>V8AD6Wu8

H =p2

2m+ V (x)

[x, H

]=

1

2m

[x, p2

]

=1

2m([x, p] p+ p [x, p])

=i~

mp

[p, H

]= [p, V (x)]

�(N�DFn[p, V (x)]ϕ(x) = −i~ ∂

∂x(V (x)φ(x)) − V (x)(−i~)

∂ϕ

∂x

= −i~∂V∂x

ϕ(x)

⇒[p, H

]= −i~V ′(x)

x��i4^��¬V~

³"³ ´µ�"!$¶y,O¥&�i0}#&�1.�¸+� .��;!$�A¥�%('O% ¥&�y¢¼�h�V�A.�#��,�%�*-,��/.10}#��,�%

]�D68An=D\Ád

dt〈x(t)〉 =

1

m〈p(t)〉

d

dt〈p(t)〉 = 〈F (x)〉

x&�A4^�Vz�~

�YWunHTu�;@/N$:EDF>58AnbBu>V8inw:ED�:E@AWU8�:�Z6W L ORYCBE]�� 1 F [�� ?(O]>F^P - FCc^L � Z6Wun��N5Z6Wu@A<wnH`->{fVWU8A8AWunHnEN$:EDFntv&>V8�:?AWUnYTu�;@/N$:EDF>58AnYnwWu`É AZFN5 AZ6WunÀN5@��aTu�;@/N$:EDF>58An BUZ�N5nwn=D6� @iWunY?iW�d�N�àD6Z6:E>58�x ��W��S:E>V8/~C4hM�[ Tu�;@AD��N$qZFWU8ABuWJ8�[ Wunt:p�/N5nS:E>5:EN5ZFW�4/§¨8�W �cWC:2Á 〈F (x)〉 6= F (〈x〉) Wu8�jVTU8ATu<=N5Z 4"��[ Wunt:�8iT2N58A`->VD68AnSZFWbBuN5n��>V@A<SZ [ >VnwBuDFZ6Z�N�:=Wu@A< R/N5<wà>V8iDF�;@AW5ÁiBu>V`-àW�>V8�Z6W�VTU<=D6�AWIN�DFn=TUàWU8�:24

� �� c��6�� {|p〉}� � � 79ZµnwWu<=N-@i:=DFZ6W�?AWJZFD6<=W�?�[^N5 �>V<=?©Z\[eN5�i�1WU8A?ADFBUWb]�4e|h4M�N <=WU�A<=TUn=Wu8�:EN�:ED6>V8 ?AW�shBªRA<w«h?AD68AjVWU<QWunw:- /N�n=TuW+nw@A<�Z�N�v&>58ABU:=DF>V8�?�[ >V8A?iW

ϕ(x)�;@AD\Á�Wu8

:EWU<=`-Wung?AWJZ [ Wunw�/N5BUWH?iWIdpDFZ6 1WU<w:S?AWUngTU:ªN$:EnuÁ/WUnw:�?ATU�A8ADFWJ�/N5<

ϕ(x) = 〈x|ϕ〉 x&�A4L| ��~

��8��Wu@i:a?AW©v�NM�u>58 :E>5@i:wq 7$q v�N5D�:QTu�;@AD��N5ZFWU8;:=W© /N5n=WU< Z\[^WU8An=WU`É AZFW©?AW�Z�N ?AD6n=Bu@in=n=D6>V8�nw@A<-Z�Nv&>V8ABC:ED6>V8

ϕ(p)[Y��(PRQhF �/N5<

ϕ(p) = 〈p|ϕ〉 x&�A4L| 3}~

��[ Wunw:_BuW��;@�[ >V8+N5�i�1WUZFZFWpZ�N ]>F > ]>YCcfF^PRL _`LTQhBEP {|p〉} �ª4A�YWC:=:EW�<=Wu�i<=TunwWu8�:ªN$:EDF>58+Wunw:YDF8�:ED6àWUàWU8�:ZFD6TuW 7�Z�N�:=<EN58inwv&>V<wàTUWb?AWH¯">5@A<=D6Wu<

ϕ(p)4"§¨8�W �cWC:2Á

ϕ(p) = 〈p|ϕ〉 =

∫dx〈p|x〉〈x|ϕ〉 =

∫dxϕ(x)

e−ipx/~√2π~

=1√~ϕ(p

~

) x&�A4L|5KV~

� N58inaBUWU:=:=W(<wWu�A<wTun=WU8�:ªN�:=DF>V8�Á�Z6Wun�>V��Tu<EN$:EWu@i<=nxWU:p�A<=WU8A8AWu8�: @A8AW©v&>V<=`-W+:E<=kUnanwDF`-�AZFW�Á

nwf `-TU:=<=DF�;@AWJ?AWJZFNÉ<=Wu�i<=TunwWu8�:ªN$:EDF>58 {|x〉} 4"§¨8�W �cWC:2Á

〈p|p|ϕ〉 = (〈ϕ|p†|p〉)∗ = (〈ϕ|p|p〉)∗ = (p〈ϕ|p〉)∗ = p〈p|ϕ〉 x��A4e|5m�~

⇒ p : ϕ(p) 7−→ pϕ(p)x&�A4e|��;~

#�� 3NIbS 3G/ =6)H(*3dI�1hS >W-.=?)H3j;"-bS -.I +07@>WI.f�3,LN3,I"=6143j]@>�/?3j1.3jCYS 3d/?5@>�+,3j143��(ECE]_3d):= $h>W- CE(*3d-�143j=?)0>[g�>W(EC*CE3,)>[g�3G+ CE3d/jg�3G+%=H3,-.)0/�5@)?&�5.)?3G/{|x〉}

143aCYS &�5BK,)0>�=?3d-.)x� &�I =?)0>[g[>�(*CECE3a>[g�3G+ CE3d/jg�3G+%=H3,-.)0/�5@)?&�5.)?3G/

{|p〉}1.3CYS &�5_Kd)H>W=?3,-@)

pO

³��F��À� �º%;Ç�!}� )u% .10{�h0}#��. {|p〉} ³1�

� WH` �U`aW�Á

〈p|x|ϕ〉 =

∫dxx〈p|x〉〈x|ϕ〉

=

∫dxx

e−ipx/~√2π~

〈x|ϕ〉

=

∫dxi~

∂

∂p

(e−ipx/~√

2π~

)〈x|ϕ〉

=

∫dxi~

∂

∂p〈p|x〉〈x|ϕ〉

= i~∂

∂p〈p|ϕ〉

⇒ x : ϕ(p) 7−→ i~∂

∂pϕ(p)

x��i4L|V|5~

>��Á 7+8A>5@iVWuN5@�Áµ>V8yNº@i:=DFZFD6n=T ∫dx|x〉〈x| =

3 Z�4µ¾ÈN�<JN58/N5Z6>VjVDFW�N2VWuB�Z6WÉB2N5Z6Bu@AZ�v�N5D6:��1>V@i<bZFWBu>V`-`É@i:EN�:EWU@A<S?AW

xWU:pWu8�<=WU�A<=TUn=Wu8�:EN�:ED6>V8 {|x〉} ÁA>58(N �

[p, x] = −i~ ⇒ [x, p] = i~x��i4L|5±V~

³ � ´µ�"!$¶y,O¥&�i0}#&�1.�¸+� .��;!$�A¥�%('O% ¥&�y¢¼�h�V�A.�#��,�%�*-,��/.10}#��,�%

� � ��

�� µ� �� µ� �� +� � � �

� �� L��6��

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4MµWb�1>VD68�:g?AWJ?ATu�/N�<w:pWunt:�?i>V8ABJZFWb`->V@i5Wu`-Wu8�:p?AWJZ�NÉ�/N5<t:ED6Bu@AZ6WbZFDF i<=W54

� � � � � �D� �X�� N��

MµW��A<w>V AZFkUàWH?AWIZ�Na�AN5<w:=DFBu@iZFWHZFDF i<=WJWunt:[7aZFN-v&>VD6np:=<=D6 DFN5ZµWU:�n=@A i:=DFZ\4�79ZOWunt:�:E<wD6 D�N�Zµ�/N5<=BUW�;@AW»ZFWUn-v&>58ABU:=DF>V8Ana�A<=>5�A<=WUn ?AW�Z [ R/N5`-DFZ6:=>V8AD6Wu8�n=>58;: ?AT �F7�Bu>V8A8;@AWUn � BuW»nw>V8�:º?AWUnº>V8A?AWUn�AZ�N�8AWunU4"§�8�W��1WU:uÁ

H = p2

2m

p|p〉 = p|p〉

}H|p〉 =

p2

2m|p〉 x\|h4�3}~

³ � *-,O% ¥��,�%h)�ÂH!$�"Ã�¥�Ä ¶Å% )aÆ�#\¶�ÇÈ¥&% )É%h.¹Ê©#�¶Å% .�)u#��.Ë�

>V@©Wu8iBu>V<wW5Á"?/N58AngZ6WbZ�N58Aj5@/N5jVW�?iW�s BªRA<=« ?ADF8ijVWu<UÁ

Hϕp(x) = − ~2

2m

∂2

∂x2

(1√2π~

eipx/~)

= − ~2

2m

1√2π~

(ip

~

)2

eipx/~

=p2

2mϕp(x)

x\|h4eKV~

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WuZ6ZFWUnSBu>V8A?i@ADFnwWu8�:%7-?AWUngn=>VZ6@i:ED6>V8AnS?AWJZ [ Tu�;@/N$:EDF>58+?AWIshBªRA<=« ?AD68AjVWu<S?iTu��Wu8A?/N�8;:=W�?A@+:=Wu`-�An

i~∂

∂tϕp(x, t) = Hϕp(x, t)

x\|h4^m�~

?A>V8i8ATuWUn��/N�<

ϕp(x, t) =eipx/~√

2π~e−ip

2t/2m~x\|h4 �;~

�Y>V`-`aWQZ\[^TU�;@/N�:ED6>V8y?AWºs BªRA<=« ?ADF8ijVWu<J?ATU�1WU8A?/N58�:EWa?A@·:EWu`-�AnHWunt:�Z6DF8ATuN5DF<wW5Á1:=>V@i:=WaBU>V`É AD q8/N5D6n=>V8+Z6DF8ATuN5DF<wW�?AWUn

ϕp(x, t)Wunt:�N�@An=nwDµn=>VZ6@i:ED6>V8�4h¾�>V@A<_:=<=>V@hVWu<g@A8iWHnw>VZF@h:EDF>58+N5BuBUWu�i:EN5 AZ6W5Á

DFZcn=@ ��:p?A>V8ABJ?AWJBU>V8Anw:=<=@AD6<=WJBuWJ�;@�[ >V8»N��A�1WUZFZ6W�@A8 >k_0SVUWF^L [�� BEP [F ÁiB�[^WUnw:wq 7$q9?iDF<=Wb@i8AWJBu>V`Qq AD68/N5DFnw>V8+ZFD68AT2N�DF<=W�?�[^>58A?AWung�AZFN58AWUnp?iWHBuN5<=<wTJn=>V`-` N� AZFW �

ϕ(x, t) =

∫dp f(p)ϕp(x, t)

x |h4e|V~

� �F��À�yÂ��A!{0}#��,�¥&%�� #&ÃO!$% �ÈÂ��A��,O%�0 ' ��¤-.O'�%h) ³�

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t = 04"�(N�DFn

ϕ(x, t = 0) =

∫dp f(p)

eipx/~√2π~

⇒ 〈ϕ|ϕ〉 =

∫dx |ϕ(x)|2

=

∫dx

∫dp f(p)

eipx/~√2π~

∫dp′ f ∗(p′)

e−ip′x/~

√2π~

=

∫dp f(p)

∫dp′ f ∗(p′)δ(p− p′)

=

∫dp |f(p)|2

x\|h4e±V~

��8(N5<w<=D�VWJN5@�` �UàW�<=Tunw@AZ6:EN�:gWu8�<=WU` N�<=�;@/N58�:p�;@AW

|ϕ(t = 0)〉 =

∫dp f(p)|p〉

⇒ f(p) = 〈p|ϕ(t = 0)〉

⇒ 〈ϕ(t = 0)|ϕ(t = 0)〉 =

∫dp 〈ϕ(t = 0)|p〉〈p|ϕ(t = 0)〉 =

∫dp |f(p)|2

x\|h4L�5~

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f(p)?AWÉ8i>V<=`-W�N�@»B2N5<w<=TITuj�N5Z6W 7

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ϕ(x, t) �

ϕ(x, t) =

∫dp f(p)

eipx/~√2π~

e−ip2t/2m~

x\|h4e¬V~

��8 �1WU@i:�>V h:EWu8iDF<�@i8AWÉW �h�A<wWunwn=DF>58y�iZF@An�n=@AjVj5Wunw:=D65WQ?AWÉZ�N�v�NM�u>58»n=@AD��N58�:EW54��8»<=WU` N5<w�;@AW:E>V@h:�?c[eN5 �>V<w?��;@AW

f(p) = 〈p|ϕ(t = 0)〉 =

∫dx 〈p|x〉〈x|ϕ(t = 0)〉 =

∫dx ϕ(x, t = 0)

e−ipx/~√2π~

⇒ ϕ(x, t) =

∫dp

∫dx′ ϕ(x′, t = 0)

e−ipx′/~

√2π~

eipx/~√2π~

e−ip2t/2m~

=

∫dx′

ϕ(x′, t = 0)

2π~

∫dp e−

itp2

2m~− i(x′−x)p

~

x\|h4ezV~

M¨[^D68;:=TujV<=N5ZFW+jVN5@An=nwDFWU8A8AW©n=@A<pBu>5<=<=WUn=��>V8A? 7·ZFN ?AWu<w8ADFkU<=W+v&>V<=`�@AZFWº?iW�Z\[eN�8A8AW �hW�X N{5WuB

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A = −tm~

ÁB = x′−x

~

WC:�?i>V8ABJWuZ6ZFW��N5@i:√

2πm~

te−iπ/4e

−i(x′−x)2m~

2~2(−t)

⇒ ϕ(x, t) =

∫dx′√

m

2π~te−iπ/4e

i(x′−x)2m

2~t ϕ(x′, t = 0)

x\|h463 ��~

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t > 0n=>V@An�Z�N�v&>5<=`-Wa?A@

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ϕ(x′, t = 0)?AD6<=WUBU:EWUàWU8�:24

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U(t, 0) = e−iHt/~x\|h46353}~

� @�Bu>V@i��Á

ϕ(x, t) = 〈x|ϕ(t)〉 = 〈x|U(t, 0)|ϕ(t = 0)〉 =

∫dx′ 〈x|U(t, 0)|x′〉〈x′|ϕ(t = 0)〉︸︷︷︸

ϕ(x′,t=0)

�(N5D6n 〈x|U(t, 0)|x′〉 =

∫dp 〈x|U(t, 0)|p〉〈p|x′〉 =

∫dp e−

ip2t2m~

eipx/~√2π~

e−ipx′/~

√2π~

⇒ 〈x|U(t, 0)|x′〉 =

√m

2π~te−iπ/4e

i(x−x′)2m

2~t

x\|h463{KV~��8�<wWU:=<=>V@i5WJ ADFWU8�ZFNÉ` �UàWbW �i�i<=Wunwn=D6>V8��;@AWH�/N5<SZ6WbB2N5Z6Bu@AZµ�i<=TuBUTu?AWU8�:24ogWuàN5<=�;@AWUn �3}~SM�NQ�;@/N58�:ED�:ET 〈x|U(t, 0)|x′〉 nu[^N5�A��WuZ6ZFWbZ6W > ]CB >k_=<0_`LNFÛ] 4KV~�M�N�8A>�:EDF>58Q?AW >k_0SVUWF^L [�� BEP [F WUnw:��/N�<w:ED6Bu@AZ6DFkU<=Wu`-Wu8�:Èn=DF`-�AZ6W_Wu8�<=WU�A<=TUn=WU8;:EN�:ED6>V8 {|p〉} 4§¨8�W �cWC:2Á

f(p) = ϕ(p, t = 0)

§�:ϕ(p, t) = f(p)e−

ip2t2m~

x\|h4632m�~

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�/N5<t:EDFBU@AZFW¨Z6DF A<wW�n=W¨:=<EN5?A@iD6:µ�/N�<OZ6W�v�N5D6:�� @iWÀnwD�Z [ >V8J�A<=TU�/N5<wWÀ@i8AW��/N5<w:=DFBU@AZFW�?AN58An�@A8AW¨v&>V8ABC:ED6>V8?�[ >V8A?AWg?�[ W � :EWU8An=D6>V8 nw�/N�:=D�N5Z6WY?A>V8i8ATuW�Á Z\[^W � :EWu8in=DF>58-nw�/N�:=D�N5Z6WY�Nb�N5<wDFWU<¨N5@-Bu>V@A<wn�?A@Q:EWu`-�AnU4M¨[^W �iWUà�iZFW¨ZFWÈ�AZF@Ancn=DF`-�AZ6W��1>5@A<�DFZ6ZF@Ant:E<wWu<cBUW¨�1>5DF8�:µWunt:OBuWuZ6@ADV?A@ >k_0SVURF^L [�� BEP [F <0_Ù+cCc^QhF^P �

B2N5<gD6Zc��Wu<=`-WU:p?AWJBuN5ZFBU@AZFWU<_W �h�AZ6DFBuD�:EWUàWU8;:gZ\[^TCV>VZ6@i:ED6>V8+?AWJZ�N�v&>58ABU:=DF>V8+?�[ >V8A?iW54�Y>V8AnwDF?ATU<=>V8Ang?A>58ABJZ�N�v&>V8ABC:ED6>V8©?�[^>58A?AW

ψ(x) =1√σ

1

(2π)1/4eip0x/~e−x

2/4σ2 x | 4632��~��[^WUnw: @A8AW©v&>V8ABU:=DF>V8 ?�[ >V8A?iW��;@ADg?ATUBu<=D�: @i8AW��AN5<w:=DFBu@iZFW+?AW© D6:EWUn=nwW

v0 = p0m

Á¨WC: �;@ADgWunt:BuWU8;:=<=TUW�N5@��>VDF8�:

x = 04"§¨ZFZ6W��>Vn=nwku?AWHZFWUnS�A<=>V�i<=DFTC:ETUnpnw@AD6�N58�:=Wun �

3V4 ∫dx |ψ(x)|2 = 1

� TUà>V8inw:E<=N�:ED6>V8 �∫dx |ψ(x)|2 =

1√2πσ

∫ +∞

−∞dx e−x

2/4σ2

=1√2πσ

√2πσ2 = 1

x | 463}|5~

Kh4 〈x〉 = 0WC:

∆x|ψ〉 = σ

� TUà>V8inw:E<=N�:ED6>V8 �

〈ψ|x|ψ〉 =1√2πσ

∫ +∞

−∞dx xe−x

2/4σ2

= 0

〈ψ|x2|ψ〉 =1√2πσ

∫ +∞

−∞dx x2e−x

2/4σ2

=1√2πσ

√2π

(1

σ2

)3/2

= σ2

⇒ ∆x|ψ〉 =√

〈ψ|x2|ψ〉 − 〈ψ|x|ψ〉2 = σ x | 463{±V~>��Z [ >V8©NQ@i:ED6ZFD6n=T�Z6W�v�N5D6:S�;@AW

xe−Ax2 WUnw:p@A8AWJv&>V8ABC:EDF>58+DF`-�/N5DF<wW54

mi4ψ(p) =

√σ

~

(2

π

)1/4

e−σ2

~2 (p0−p)2 ≡ f(p)

� TUà>V8inw:E<=N�:ED6>V8 �

ψ(p) =1√σ

1

(2π)1/4

1√2π~

∫ +∞

−∞dx e−ipx/~eip0x/~e−x

2/4σ2

︸︷︷︸

=

√π

αeB

2/4α2

x | 463}�5~

# �R3d/ U�&�)HL�-@C*3G/ +,&�I@+,3,)HI@>WI"=DC<>k)HKd/?&�CE-4=H(*&�Ia1hS (*I"=HK,f�)0>WCE3d/ f">W-@/H/:(E3,[email protected]/2/?3^=?)H&�-.g�3,I"=21@>WI@/2CYS >�I.I.3,Z93��AO

�"� *-,O% ¥��,�%h)�ÂH!$�"Ã�¥�Ä ¶Å% )aÆ�#\¶�ÇÈ¥&% )É%h.¹Ê©#�¶Å% .�)u#��.Ë�

N2VWUBα2 = 1

4σ2

WU:B = i(p0−p)

~

⇒ ψ(p) =1√σ

1

(2π)1/4

1√2π~

√π2σe−

σ2

~2 (p0−p)2 x\|h4632¬�~

�A4 〈p〉 = p0

� TUà>58Anw:=<EN�:=DF>V8 � MµWJ�AZF@inSn=DF`-�AZ6W�Wunt:�?AWJnwWH�iZ�N5BUWu<SWu8�<=WU�A<=TUn=WU8;:EN�:ED6>V8 {|p〉} �

〈p〉 =

∫ +∞

−∞dp ψ(p)∗pψ(p)

= p0

∫ +∞

−∞dp∣∣ψ(p)

∣∣2

︸︷︷︸=1

+

∫ +∞

−∞dp∣∣ψ(p)

∣∣2 (p− p0)

︸︷︷︸=0

= p0

x\|h4632z�~

>�� ZFN ?AWU@��hDFku`-WÉD68;:=TujV<=N5ZFWIWunt:b8;@AZFZ6WÉB2N5<�Z [ DF8�:ETUjV<EN58�:�Wunw:�@A8AWÉv&>V8ABC:ED6>V8»DF`-�/N5D6<=WH?AWp4

�YWU:w:EW(v&>58ABU:=DF>V8 ?�[ >V8A?AW»BU>V<=<wWunw�1>V8i?�?A>V8AB 7Å@A8AW�j�N5@Anwn=D6Wu8A8AW�BuWU8;:=<=TUW Wu8p0

Wu8�<wWu�A<wTCqn=WU8�:ªN�:=DF>V8 {|p〉} Ác?�[^>B�yZ6WÉ8A>V` ?AW >k_0SVURF^L3[�� BEP [F[<0_Ù+cCc^QhF^P 4��Y>V`-àWÉZFNa:E<=N58Antv&>V<=`-TuWQ?AW¯/>V@A<wDFWu<g?�[ @A8AWIj�N5@Anwn=D6Wu8A8AWIWunt:�@A8iW�j�N5@in=n=D6Wu8A8iW5Á�WuZFZ6WHN N5@Anwn=DOZ�NÉv&>V<wàWH?�[ @A8AW�jVN5@An=nwDFWU8A8AWx�`�@AZ6:=DF�AZ6DFTuWp�/N5<g@A8iWHW �h�1>V8iWu8�:ED6WuZFZ6Wbn=D

p0 6= 0~_Wu8�<=Wu�i<=TunwWu8�:ªN$:EDF>58 {|x〉} 4

§�V>VZ6@i:ED6>V8�N5@©Bu>V@i<=ng?A@©:EWu`-�An � ��8��>V@A<w<EN5D�:J@i:ED6ZFD6n=Wu<�Z [ >V�1TU<EN�:=Wu@A<J?�[ TU5>VZF@h:EDF>58�Á�àN5DFnbD6ZWunt:pZFTuj5ku<=WUàWU8�:��iZF@AnSnwDF`-�AZFW�?c[^TC5N�ZF@AWU<S?ADF<wWuBU:=Wu`-Wu8�:

ψ(x, t) =

∫ +∞

−∞dp ψ(p)

eipx/~√2π~

e−ip2t2m~

=√σ

(2

π

)1/41

~

1√2π

∫ +∞

−∞dp e−

σ2

~2 (p0−p)2+i px~−i p2t

2m~

=√σ

(2

π

)1/41

~

1√2πeip0x/~

∫ +∞

−∞dp e−

σ2

~2 (p−p0)2+i(p−p0)x

~−i p2t

2m~

︸︷︷︸I

x\|h4LK ��~

79Zhv�N�@i:ÈBuN5ZFBU@AZFWU<ÈZ\[^D68;:=TujV<=N5ZFWI�;@ADiWUnw:¨@i8AW_j�N5@in=n=D6Wu8A8iW54��8Év�N5D�:OZFWYBªRAN58AjVWUàWU8;:�?AWY�N5<=DFN5 AZ6W

p→ p+ p0?�[ >��

I =

∫ +∞

−∞dp e−

σ2

~2 p2+i px

~−i (p+p0)2t

2m~

x\|h4LK 3}~

� �F��À�yÂ��A!{0}#��,�¥&%�� #&ÃO!$% �ÈÂ��A��,O%�0 ' ��¤-.O'�%h) ��¡

M¨[^W �i��>Vn=N58�:��N5@i: � p2(−σ2

~2 − it2m~

)+ p

(ix~− ip0t

m~

)− ip20t

2m~

Á/WC:pN5DF8AnwD

I = e−ip2

0t

2m~

√π

αe

B2

4α2

N25WuB{α2 = σ2

~2 + it2m~

B = ix~− ip0t

m~

n=>VD�:α2 =

√σ4

~4+

t2

4m2~2exp

[i arctan

(t~

2mσ2

)]

⇒ α =4

√σ4

~4+

t2

4m2~2exp

[i

2arctan

(t~

2mσ2

)]

WC: B2

4α2= −

(x

~− p0t

m~

)21

4σ2

~2 + 2itm~

= −(x− p0t

m

)2

4σ2 + 2i~tm

x | 4LKVK5~

sh>VD�:2ÁAWU8�<wWujV<w>V@A�/N58�:g:E>5@AnSZFWUngv�N�BU:EWU@A<=n �

ψ(x, t) =

(σ2

2π

)1/4e

»

− i2

arctan( t~

2mσ2 )−ip2

0t

2m~

–

4

√σ4

~4 + t2

4m2~2

× eip0x/~ × e−(x−

p0tm )

2

4σ2+2i~tm

x | 4LK5mV~

�YWU:w:EWJv&>V8ABU:=DF>V8+?c[^>V8i?AWI?iTuBu<wD6:g@A8iWH�AN5<w:=DFBu@iZFW�?AWb D6:=Wunwn=W p0m

BuWU8�:E<=TUWIWU8 p0mt4"§¨8�W��1WU:uÁ

〈ψ(x, t)|x|ψ(x, t)〉 =

(σ2

2π

)1/21√

σ4

~4 + t2

4m2~2

∫ +∞

−∞dx x

∣∣∣∣∣exp

[−(x− p0t

m

)2

4σ2 + 2i~tm

]∣∣∣∣∣

2

︸︷︷︸J

J =

∫ +∞

−∞dx

(x− p0t

m

) ∣∣∣∣∣exp

[−(x− p0t

m

)2

4σ2 + 2i~tm

]∣∣∣∣∣

2

︸︷︷︸=0

+p0t

m

∫ +∞

−∞dx

∣∣∣∣∣exp

[−(x− p0t

m

)2

4σ2 + 2i~tm

]∣∣∣∣∣

2

︸︷︷︸=1 x | 4LK��~

>�� 7»8A>V@hVW2N�@ ZFN(�A<wWu`-DFku<wW D68;:=TujV<=N5ZFWaWunt:Q8;@AZFZ6W B2N�<ÉZ [ DF8�:ETUjV<EN�8;:�Wunt:�DF`-�/N5DF<b:EN58A?AD6nÉ�;@AWZ�N�n=WUBu>V8A?iW��N5@i:I3�Wu8(n=W�<=N5�A��WuZFN58�:��;@AW

ψ(x)WUnw:�8i>V<=`aN5ZFD6n=TUW-x |h4�3}|V~_WU:��;@AW�Z\[^TCV>VZ6@i:ED6>V8

:EWU`a��>V<=WUZFZ6WYWunt:�Wu8Aj5Wu8A?A<wTuWp�/N5<�@i8->5�1TU<EN�:=Wu@A<¨@A8iD6:ªN�DF<=W �;@AD/Bu>V8in=Wu<tVWgZ�N�8i>V<=`-W54V�_N5ZFBU@AZF>V8in∆x|ψ(x,t)〉 =

√〈ψ| (x− 〈x〉)2 |ψ〉 �

〈ψ|(x− p0t

m

)2

|ψ〉 =

(σ2

2π

)1/21√

σ4

~4 + t2

4m2~2

∫ +∞

−∞dx x2

∣∣∣∣∣exp

[− x2

4σ2 + 2i~tm

]∣∣∣∣∣

2

x | 4LKV|5~

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>��©Z [ >V8©NÉv�N5D�:_ZFW�BªR/N58AjVWUàWU8�:�?iWJ�N5<wD�N5 iZFWunx− p0t

m→ x

4/MµW�:EWu<wàWb?iWHZ\[^W �h�1>58AWu8�:ED6WuZ�WUnw:?AWJZ�N�v&>5<=`-W

e−α

a+ib e−α

a−ib = e− 2aα

a2+b2x\|h4LK�±�~

?�[ >��

〈(x− p0t

m

)〉 =

(σ2

2π

)1/21√

σ4

~4 + t2

4m2~2

∫ +∞

−∞dx x2 exp

[− 8σ2x2

16σ4 + 4~2t2

m2

]

=

(σ2

2π

)1/21√

σ4

~4 + t2

4m2~2

√2πA−3/2 N{5WuB

A =σ2

σ4 + ~2t2

4m2

=

(σ2

2π

)1/2 √2πσ−3

(σ4

~4+

t2

4m2~2

)= σ2

(1 +

~2t2

4σ4m2

)

⇒ ∆x|ψ(x,t)〉 = σ

√1 +

~2t2

4σ4m2

x\|h4LK5�V~

��8·Bu>V8inw:ªN$:EWÉ?A>58ABQ�;@AWQZFWI�/N5�;@AWU:J?c[^>V8i?AW ) � � ¥��A! �#�0 N5@ Bu>5@A<=nb?A@»:=Wu`-�Anu4��YWQ�ARATU8A>�q`-ku8AWJnu[^N5�A��WuZ6ZFW D � Y^Lh_`DGF 1 F^PRL [Ù^>k_0SVURF^L [�� BEP [F 4

�� L�2� �� [�� N� � �� W�� @�� W� � �

�Y>V8in=DF?iTu<=>58An�Z\[^TU�;@/N�:ED6>V8+?AWIshBªRA<w«h?iDF8AjVWU< 7�@i8AWJ?ADF`-Wu8AnwDF>V8 �

− ~2

2mψ′′(x) + V (x)ψ(x) = Eψ(x)

x\|h4LK�¬�~

shDV

Wunt:pDF8A?ATU�1WU8A?/N58�:p?AWxÁ/>V8��1WU@i:pZ�NÉ<wTuTuBU<=D6<=W �

− ~2

2mψ′′(x) = (E − V )ψ(x)

x\|h4LK�z�~

79Z�v�N5@i:g?iDFnw:=DF8Aj5@AWu<_:=<=>VD6n_B2N5n �3V4E > V

ψ(x) = Aeipx/~ +Be−ipx/~x\|h4em��~

N2VWUB p2

2m= E − V

ÁAnw>VD6:p =

√2m(E − V )

� �F¡yÂ��/0{% .10}#&%h¥Y�I��.�)C0{�/.10aÇ��i!+¢¼��!$�V%;�/,�%�0+�I�1.O'O#&0}#&��.�)É�A, �� #�¶ #�0{% ) ��

Kh4E < V

ψ(x) = Aex/l +Be−x/lx | 4emi3{~

N2VWUB − ~2

2ml2= E − V

ÁAnw>VD6:l =

~√2m(V − E)mi4

E = V

ψ(x) = Ax+Bx | 4em�K5~

§¨8;:=<=WI?AWu@��(<=TUjVDF>58An�>��ZFWI�1>5:=Wu8�:ED6WuZ�Wunt:bBu>58Anw:EN58�:2Á�DFZ�?A>VD�:�f�N2V>VD6<�@A8iW�?AD6n=Bu>58;:=DF8;@AD�:ET54� Wu@ ��B2N5ng?AWb�/jV@i<=WJn=>V8�:p@i:=DFZFWUn �3V4

V (x) =

{V1

n=Dx < x0

V2n=D

x > x0

x | 4emVmV~

0

V (x)

x

V1

V2

x0

� �� |h4�3 � siN�@i:p?AWJ�1>5:=Wu8�:ED6WuZ�WU8x = x0

4

� N58in_BuW�BuN5nuÁhZ�NH?ATu<wD6VTUW�n=WUBu>V8i?AWb?AWψ?A>VD�:_N2V>5DF<Y@i8ºnEN5@h:2ÁiBUW��;@AD�DF`-�AZFD6� @iWp�;@AW

ψWU:ψ′ nw>V8�:�Bu>V8�:ED68;@AWunU4�§¨8QW��1WU:2Á5@A8AWS?AD6n=Bu>58;:=DF8;@AD�:ETY?iW

ψ�A<w>h?A@iDF<EN�D6:�@A8AWSn=D68AjV@AZFN5<=D�:ET

?AWψ′′ Wu8 δ′(x) Á�WC:b@A8AW�?AD6n=BU>V8�:EDF8;@AD�:ETI?AW

ψ′ �A<w>h?i@ADF<=N5D6:�@A8AWQnwDF8AjV@iZ�N5<wD6:ETHWu8δ(x)

4��<SBuWUn�nwDF8AjV@iZ�N5<wD6:ETUn_8AWJ��Wu@i5Wu8�:��AN5n �U:E<wWJBu>V`-�1WU8An=TUWunp�/N5<

V (x)4

�Y>V8ABUZF@AnwDF>V8 � shD V (x)�A<=TUn=Wu8�:=W @A8ln=N5@i:�Wu8

x0ÁSD6Z�v�N5@i:�<EN5BUBu>V<w?AWu<�Z�N v&>V8ABC:ED6>V8

?�[ >V8A?AWJWC:�n=N-?iTu<=D�VTUWHWU8x04

Kh4

V (x) = V0δ(x− x0) + V �� (x)x | 4em5��~

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� N�8AnJBuW�B2N5nd!{Á�Z�Nº?iTu<=D�VTUW-nwWuBU>V8A?AW�?A>VD�:JBu>V8�:=Wu8AD6<b@A8AWÉv&>V8ABC:ED6>V8δ(x − x0)

ÁcBuW��;@ADD6à�AZ6DF�;@AW�� @iWIZFNÉ?ATu<wD6VTUWJ�A<=WUàD6ku<=Wb?i>VD6:pN2V>5DF<S@A8�nEN5@h:24"§¨8�W �cWC:2Á

∫ x0+ε

x0−εdx ψ′′(x) =

2m

~2

∫ x0+ε

x0−εdx (V (x) − E)ψ(x)

⇒ ψ′(x0 + ε) − ψ′(x0 − ε) =2m

~2V0ψ(x0) +

2m

~2

∫ x0+ε

x0−εdx (V �� (x) − E)ψ(x)

︸︷︷︸→0 � � �� ε→0

⇒ limε→0+

[ψ′(x0 + ε) − ψ′(x0 − ε)] =2m

~2V0ψ(x0) x\|h4emV|V~

�Y>V8iBuZF@in=DF>58 � shD V (x)�A<wTun=WU8�:EWH@A8AWbn=D68AjV@AZFN5<=D�:ET�?AW�:9f �1W

V0δ(x− x0)ÁiZ�NIv&>V8iBU:ED6>V8

?�[ >V8A?iW»Wunt:+Bu>V8�:=DF8;@AW5Á ` N�DFn-Z�N�?iTu<=D�VTUW»�A<=TUn=WU8;:=W·@A8�nEN�@i: � ψ′(x+0 ) − ψ′(x−0 ) =

2m~2 V0ψ(x0)

4§¨8i�/8cÁcDFZ�WUnw:bnw>V@iVWU8�:b@i:ED6ZFWH?AW�BU>V8AnwDF?ATU<=Wu<�Z�N�ZFDF`-D6:=Wb>��

V → +∞ ?AN58An�@i8 ?A>VàN5DF8AW�4� N58in_BuW�B2N5nUÁiZ�Nbv&>V8ABC:ED6>V8 ?�[ >V8A?iW�?A>VD�: �C:E<wW�8;@AZFZ6W�?/N�8An :=>V@i: ZFW�?i>V` N�DF8AW�Á�WU:_Z\[^TC:E@A?iW�?AW�Z�NZFD6àD�:EW

V → +∞ à>V8�:=<=Wb�;@�[ DFZµnw@ ��:�?c[^D6à��>Vn=WU<ψ(x0) = 0

n=DV = +∞ ��>V@A<

x > x04

�Y>V8ABUZF@AnwDF>V8 � s D V = +∞ �1>V@i<x > x0

x&>V@x < x0

~ ÁÈZFN(v&>V8ABU:=DF>V8�?c[^>V8i?AW+?A>VD�: �U:=<=WBu>58;:=DF8;@AWJWu8

x = x0ÁAWU:pWuZ6ZFW��N5@h:

ψ(x0) = 04

�� W� �� 8�Bu>V8AnwDF?AkU<=WJ@A8��1>�:EWu8�:=DFWuZc?ATU�/8iDO�AN5<

V (x) =

{V1

n=Dx1 < x < x2

V2n=D

x < x1>V@

x > x2

x\|h4em5±�~

N2VWUBV2 > V1

ÁiWC:S>V8+n=Wb�i<=>V��>Vn=W�?AW�:E<w>V@i5Wu<SZFWUnY8AD�VWuN5@��+?c[^TU8AWu<wjVDFWE

Bu>V`-�A<=D6n=W�Wu8�:E<wWV1WU:

V24

V2 = +∞

� N�8An�BUWJB2N5nUÁ/>V8�?i>VD6:SBªRiWu<=BªRiWu<�ZFNÉn=>VZ6@i:ED6>V8+n=>5@An�ZFN�v&>V<=`-W

ψ(x) = Aeipx/~ +Be−ipx/~�1>V@i<

x1 < x < x2

x\|h4emV�V~N2VWUB

p =√

2m(E − V1)Á/WU:g>58�?A>5D6:SD6à��>Vn=WU<_ZFWUngBu>V8A?iD6:ED6>V8An

{Aeipx1/~ + Be−ipx1/~ = 0

Aeipx2/~ + Be−ipx2/~ = 0

x\|h4em5¬�~

PV ��

K`=H>�I"=�CE>j5@>�):=H(*3%)HK,f�-.CE(*Md)?3��14- 5B&W=?3dI"=?(E3,CYO

� ��³ �À%»ÂI,O#&0})Q'�%(Â��"02%h.�0}#�% ¥_�I�A!}!$� �c¦

0

V (x)

x

V1

V2

x1 x2

� �� |h4eK � ¾¨@AD6:=nS?AWJ�1>�:EWu8�:=DFWuZµBuN5<=<wT54

�YWÉnwf nt:Eku`-WQN @i8AWÉn=>5ZF@i:=DF>V8�8A>V8»8;@AZ6ZFW�nwDÈWUnw:bnwWu@AZ6Wu`-Wu8�:�nwDÈn=>58 ?ATU:=Wu<=`-DF8AN58�:�Wunt:b8;@AZ ÁB5[ Wunt:wq*7}q9?AD6<=Wbn=D

eip(x1−x2)

~ − e−ip(x1−x2)

~ = 0

⇒ sin

(p(x2 − x1)

~

)= 0

⇒√

2m(E − V1)(x2 − x1)

~= nπ

⇒ 2m(E − V1) =

(nπ~

(x2 − x1)

)2

⇒ E = V1 +~2π2

2m(x2 − x1)2n2

x | 4emVzV~

MµWungBu> W��-BuD6Wu8�:EnA

WU:B

n=>58;:p<wWuZFD6TunS�/N�<

B = −Ae2ipx1

~

x | 4^� �V~WU:pZ�N�v&>58ABU:=DF>V8©?�[ >V8A?AWJWUnw:p?A>V8A8iTuWH�/N5<

ψ(x) = A(eipx/~ − e−ipx/~e

2ipx1~

)

= Aeipx1/~(eip(x−x1)/~ − e−ip(x−x1)/~

)

= 2iAeipx1/~ sin

(p(x− x1)

~

)x | 4^�A3{~

�(N5D6np = nπ~

x2−x1

⇒ ψ(x) = 2iAeipx1/~ sin

(nπ

x− x1

x2 − x1

) x | 4^�;K5~


AWunt:p��iTJ�/N�<pZFNÉBu>V8i?AD6:=DF>V8 �

∫ x2

x1

dx |ψ(x)|2 = 1

⇒ 4A2x2 − x1

2= 1

⇒ A =1√

2(x2 − x1)

x\|h4^�Vm�~

¾ÈN�<_N5D6ZFZFWU@A<=nUÁV>V8º��Wu@h:YZFN5DFnwn=Wu<�:E>5`É 1WU< Z6WSv�N5BU:=Wu@A< ?AW��ARAN5n=Wieipx1/~

4i¯ÈD68/N5Z6Wu`-Wu8�:2Á�>V8a�1WU@i:TuBU<=D6<=WbZ�NQnw>VZF@i:=DF>V8 �

E = V1 +~2π2

2m(x2 − x1)2n2

ψn(x) =

√2

x2 − x1

sin

(nπ

x− x1

x2 − x1

)x\|h4^�5�;~

ogWuàN5<=�;@AW � n = 08�[ Wunt:p�/N5ng�1>5n=n=D6 AZFWbBuN5<

ψ(x)n=WU<EN5D�:gDF?AWU8;:=DF�;@AWUàWU8;:g8;@AZ6ZFW�4

V2 < +∞

� N�8An�BUWJB2N5nUÁ/>V8�?i>VD6:SBªRiWu<=BªRiWu<�ZFNÉn=>VZ6@i:ED6>V8+n=>5@An�ZFN�v&>V<=`-W �

ψ(x) =

Aex/lnwDx < x1

Beipx/~ + Ce−ipx/~nwDx1 < x < x2

De−x/lnwDx > x2

x\|h4^��|V~

N2VWUBl = ~√

2m(V2−E)

WU:p =

√2m(E − V1)

Á�WC:Q>�� >58�N(BªRA>VD6n=DÀZ�N�n=>VZ6@i:ED6>V8ex/l

�1>5@A<x < x1

WC:e−x/l

�1>5@A<x > x2

N��/8©?�[ >V i:=Wu8AD6<p@i8AWJn=>VZ6@i:ED6>V8+?AWJB2N5<w<=TJn=>5ààN5 AZ6W54�Y>V`-`-W DFZ�nU[eN5j5D6:©?AW·n=D6à�iZFWun ?ADFnwBu>V8�:ED68;@AD6:=TunuÁS>58Ë?i>VD6:©<=N5BuBU>V<=?AWU<©Z�N v&>V8ABC:EDF>58¹WU:�nEN

?ATU<=D65TuWHWu8x1

WC:x2ÁA?�[ >��Z6WunSTU� @AN�:ED6>V8An �

x1 :

{ψ : Aex1/l = Beipx1/~ + Ce−ipx1/~

ψ′ : Alex1/l = ip

~

(Beipx1/~ − Ce−ipx1/~

)

x2 :

{ψ : De−x2/l = Beipx2/~ + Ce−ipx2/~

ψ′ : −Dlex2/l = ip

~

(Beipx2/~ − Ce−ipx2/~

)

x\|h4^�V±�~

�YWUnaTU� @AN�:ED6>V8AnQBu>58Anw:=D6:E@iWu8�:�@A8 nwf nw:=ku`-W�Z6DF8ATuN5DF<wW RA>5à>Vj5ku8AWº?iW��;@/N$:E<=W+TU�;@/N�:ED6>V8An�7�;@/N�:=<=WQDF8iBu>V8A8;@AWUnu4µ79Z�8c[ f»N©?iWQn=>VZ6@i:ED6>V8»8A>V8·8;@AZ6ZFWÉ�;@AWQnwD¨n=>V8·?ATC:EWu<wàD68/N58�:bWUnw:J8;@AZ ÁcBuW

� ��³ �À%»ÂI,O#&0})Q'�%(Â��"02%h.�0}#�% ¥_�I�A!}!$� �

-1

-0.5

0

0.5

1

2 2.5 3 3.5 4

ψ (

x)

x

n = 1n = 2n = 3

�� |h4^m � ψ1(x)Áψ2(x)

WC:ψ3(x)

��>V@A<x1 = 2

WU:x2 = 4

4

�;@ADµBu>V8i?A@AD6:%7Q@A8AWJTU� @AN�:ED6>V8©�1>V@i<pZ\[^TU8AWu<wjVDFW �∣∣∣∣∣∣∣∣

ex1/l −eipx1/~ −e−ipx1/~ 01lex1/l − ip

~eipx1/~ ip

~e−ipx1/~ 0

0 −eipx2/~ −e−ipx2/~ e−x2/l

0 −eipx2/~ e−ipx2/~ −1lex2/l

∣∣∣∣∣∣∣∣= 0

x\|h4^��V~

¾�>Vnw>V8Ana = ip

~

WU:b = eip(x2−x1)/~

4��8(nEN�D6:p�;@�[^>58»8AWHBªR/N58AjVWI�/N5npZ�NQ�N5ZFWU@A<g?�[^@i8(?ATC:EWu<rq`aD68/N58�:JWu8�`É@AZ�:EDF�iZFD�N�8;:�@A8AWaZFDFj58AWÉ>V@�@A8AW-Bu>VZ6>V8A8AW��/N5<H@A8AW-Bu>58Anw:EN58�:EW54È��8��Wu@h:�?A>58AB`É@AZ�:EDF�iZFDFWU<�BªRAN5�;@AWgBu>VZ6>V8A8AWS?AWIx\|h4^��V~��/N5<¨@i8AWgW �h�1>58AWu8�:ED6WuZ6ZFW_ AD6Wu8�BªRA>VD6n=D6W_WU:À>V8�>V i:=DFWU8;: �

∣∣∣∣∣∣∣∣

1 1 1 01l

a −a 00 b 1

b1

0 ab −ab

−1l

∣∣∣∣∣∣∣∣= 0

x\|h4^�V¬�~

� � *-,O% ¥��,�%h)�ÂH!$�"Ã�¥�Ä ¶Å% )aÆ�#\¶�ÇÈ¥&% )É%h.¹Ê©#�¶Å% .�)u#��.Ë�

§¨8�?iTUVWUZF>V�i�/N58�:��AN5<S<EN5�i�1>V<t: 7QZ�NÉ�A<wWu`-DFku<wWbBu>VZ6>V8A8AW�Á/DFZ�hD6Wu8�: �∣∣∣∣∣∣

a −a 0b 1

b1

ab −ab

−1l

∣∣∣∣∣∣− 1

l

∣∣∣∣∣∣

1 1 0b 1

b1

ab −ab

−1l

∣∣∣∣∣∣= 0

⇒ a

(− 1

bl+a

b

)+ a

(−bl− ab

)− 1

l

(− 1

bl+a

b

)+

1

l

(−bl− ab

)= 0

⇒ −(a+

1

l

)(b

l+ ab

)+

(a− 1

l

)(a

b− 1

bl

)= 0

⇒ b2 =

(a− 1

l

a+ 1l

)2

n=>5D6:(

ip~− 1

lip~

+ 1l

)2

= ei2p(x2−x1)/~

⇒ip~− 1

lip~

+ 1l

= ±eip(x2−x1)/~

x\|h4^�Vz�~

��8�?i>VD6:S?i>V8ABJ?ADFnt:ED68AjV@AWU<S?AWu@��+B2N5n �3V4 ip

~− 1

lip~

+ 1l

= +eip(x2−x1)/~

ip

~

(1 − eip(x2−x1)/~

)=

1

l

(1 + eip(x2−x1)/~

)

⇒ ~

ipl=

1 − eip(x2−x1)/~

1 + eip(x2−x1)/~

⇒ ~

ipl=eip(x2−x1)/2~

(e−ip(x2−x1)/2~ − eip(x2−x1)/2~

)

eip(x2−x1)/2~ (e−ip(x2−x1)/2~ + eip(x2−x1)/2~)

⇒ ~

pl= tan

(p(x2 − x1)

2~

)

⇒ 1 + tan2

(p(x2 − x1)

2~

)= 1 +

~2

p2l2=

1

p2

(p2 +

~2

l2

)

x\|h4L| ��~

�(N�DFnp2 + ~2

l2= 2m(E − V1) + 2m(V2 − E) = 2m(V2 − V1)

4_¾�>Vnw>V8Anp2

0 =2m(V2 − V1)

Á/>58�N�?A>V8iB

1 + tan2

(p(x2 − x1)

2~

)=p2

0

p2

⇒ 1

cos2(p(x2−x1)

2~

) =p2

0

p2

⇒∣∣∣∣cos

(p(x2 − x1)

2~

)∣∣∣∣ =p

p0

x\|h4L| 3}~

� ��³ �À%»ÂI,O#&0})Q'�%(Â��"02%h.�0}#�% ¥_�I�A!}!$� ��

�YWU:w:EW�Bu>V8A?iD6:ED6>V8-?A>VD6: �C:E<=W�Bu>V`-�AZ6TU:ETUW��AN5< ZFNHBU>V8A?AD�:ED6>V8a��Wu<w?A@AW�ZF>V<wn=�;@�[ >V8+NI�i<=D6:ÀZ6WB2N�<=<=T7QZ�NQ?AWU<=8AD6ku<wWbZFDFj58AW�?AWax | 4L|��V~ � �

tan

(p(x2 − x1)

2~

)> 0

x\|h4L|5KV~

Kh4 ip~− 1

lip~

+ 1l

= −eip(x2−x1)/~

MµWJ<EN5D6n=>V8A8iWu`-Wu8�:gWunt:�Z6Wb` �UàWb�;@�[^N2�N58�:�WC:g>V8�:=<=>V@hVW �~

ipl=

i

tan(p(x2−x1)

2~

)

⇒ ~

pl= − cot

(p(x2 − x1)

2~

)

⇒ 1 + cot2

(p(x2 − x1)

2~

)=

1

sin2(p(x2−x1)

2~

) =p2

0

p2

⇒

∣∣∣∣sin(p(x2 − x1)

2~

)∣∣∣∣ =p

p0

cot

(p(x2 − x1)

2~

)< 0

x | 4L|5mV~

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V2 → ∞ � � N58AnQBuWC:=:=W�Z6DF`-D6:=W5Áp0 → ∞ ÁÈZFN»�1WU8�:EWº:EWU8A? VWu<wn �iÁÈWC:�ZFWunÉ�N5Z6Wu@A<wn

�1>5n=n=D6 AZFWUnS?AWpn=>V8�: � � nπ~

x2−x1

4/MµWunS�N5Z6Wu@A<wnp��>Vnwn=DF iZFWunS?iWHZ\[^TU8AWu<wjVDFWbnw>V8�:p?A>V8AB

E = V1 +p2

2m= V1 +

π2~2

2m (x2 − x1)2n

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Kh4pM�NQ8hq9D6ku`-W�n=>VZ6@i:ED6>V8©Nn− 1

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� � I 3 �B3,= �.&�IJ>N5@>�)D14K @I@( =H(*&�I $p, l, ~ > 0

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0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

p

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

p

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

p

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

p

0

0.2

0.4

0.6

0.8

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0 2 4 6 8 10 12 14

p

0

0.2

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0.8

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0 2 4 6 8 10 12 14

p

0

0.2

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0 2 4 6 8 10 12 14

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0

0.2

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p

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p� �� |h4 � � opTUn=>VZ6@i:ED6>V8ºjV<=N5�ARAD6� @iW�?AWung8AD�VWuN5@��+?c[^TU8AWu<wjVDFWb?A@©�A@iD6:EnYB2N5<w<=Tb��>V@A<

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f(p) = pp0

WU:�ZFWunÈn=DF8;@Anpx&Bu>V@i<= �Wun�BU>V8�:EDF8;@AWUnª~WU:pZ6WunSBu>5n=DF8;@AnIx&Bu>V@A<w 1WUnH7É:=DF<=WC:EnE~Y?A>V8A8AWU8�:�Z6WunSTu8iWu<=j5DFWung��>Vn=nwDF AZ6WunU4

�� [�� 6�

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V (x) =

{V1

nwDx < 0

V2 > V1nwD

x > 0

x\|h4L|5|V~

WU:pBªRAWU<=BªRA>58An�Z6WunSTU:EN�:Engnt:ªN�:=DF>V8A8AN5DF<wWunu4

� �F��À��¢¼�A!$��% '�%»Â��/0{% .10}#&% ¥ ��¡

0

V (x)

x

V1

V2

� �� |h4e| � �(N�<=BªRAWH?AWJ�1>�:EWu8�:=DFWuZ 7�Z\[^>5<=DFj5DF8AW�4

V1 < E < V2

� N58AngBUWHBuN5nuÁ/ZFNQn=>VZ6@i:ED6>V8+Wunt:�?iWHZFN�v&>V<=`-W

ψ(x) =

{Aeipx/~ +Be−ipx/~

n=Dx < 0

Ce−x/ln=Dx > 0

x | 4L|5±V~

N2VWuBl = ~√

2m(V2−E)

WU:p =

√2m(E − V1)

4�YWU:w:EW�v&>5DFnuÁ¨D6ZS8�[ f�Ny�;@AW K·Bu>V8A?iD6:ED6>V8An-N5@�� Z6DF`-D6:EWUn � ψ WU:ºn=N�?ATU<=D�VTuW�?A>VD�VWu8�: �U:E<wW

Bu>V8�:=DF8;@AWunpWu8 �i4"�YWUZ�NQBu>58A?A@AD�:�N�@��+<wWuZ�N$:EDF>58An{ψ : A+B = C

ψ′ : ip~(A− B) = −C

l

x | 4L|V�5~

�Y>V`-`aW_D6Zhf�NHK�TU� @AN�:ED6>V8An�WU:ÀmbDF8iBu>V8A8;@AWUnuÁ�D6ZhfQN�:=>V@��w>V@A<wn¨@A8AWgn=>5ZF@i:=DF>V8�4�M¨[^TU8AWu<wjVDFWg8�[ Wunt:>k_Ec SVU�_`PRLTQ �ZYCF 4§¨8�TUZFDF`-DF8AN58�:

CÁ/>V8+:E<w>V@iVWH@A8AWJ<wWuZ�N$:EDF>58+Wu8�:E<wW

AWC:B �

A +B =ipl

~(A−B)

⇒ A

(1 +

ipl

~

)= B

(−1 +

ipl

~

)

⇒ B = A1 + ipl

~

−1 + ipl~

= −A1 + ipl~

1 − ipl~

x | 4L|5¬V~

�(N5D6n1 + ipl

~=√

1 + p2l2

~2 eiϕ N2VWuB

tanϕ = pl~

ÁAWC:1 − ipl

~=√

1 + p2l2

~2 e−iϕ ÁA?�[ >��

B = −Ae2iϕ x | 4L|5zV~

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m

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m

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?iD �1ku<=WU8�:��/N�<�@A8iWI�ARAN5n=WH:ªN58A?iDFn�;@AWQZ�Nº?iDFn=�AN5<=D�:EDF>58(Bu>5à�AZ6kU:=W�?AWQZFN �/N5<t:EDFW�?AWQZ [ >V8A?AW��/N5n=n=N58�:Jn=>V@inHZFN àN5<=BªRAWIhD6Wu8�:J?A@v�N5D�:��;@AWÉZFNav&>V8ABC:ED6>V8(?�[ >V8A?iWÉWunt:J@A8AWÉW �i��>V8AWU8�:EDFWUZFZ6W�<=TUWuZ6ZFW5Á"?A>58ABÉTU�N58iWun=BUWu8�:EW�4�§¨8i�/8�Á�Z6Wv�N5D�:��;@�[ @A8AWg>V8A?AWS?iWS` �UàW_D68�:EWu8in=D6:=TS<=WU�/N5<w:�Wu8�n=WU8An�DF8�VWU<=n=WgWUnw: N5n=nw>hBUDFTUW 7�Z [ Tu�;@AD6�N5Z6Wu8ABUW?AWJZ [^N5`-�AZFD�:E@A?iW�?AWUnpBU>hW �-BuDFWU8�:En

AWU:B4

�YWUnb<=WuàN5<w� @iWun�nw>V8�:[7 Z�N- /N5nwWÉ?AWÉZFN-:ERiTu>V<wDFW��;@/N�8;:=DF�;@AWÉ?iWÉBu>VZ6ZFD6n=DF>58AngTuZ�N�nw:ED6�;@AWunÉx�>V@[`Q �AU0c^QhBEP TuZFN5nw:=DF�;@AW_�AN5<¨@A8��1>�:EWu8�:=DFWuZiDF8A?ATU�1WU8A?/N58�:�?A@Q:=Wu`-�Anª~ � >V8�BªRAWu<wBªRAW�Z6Wun�nw>VZF@h:EDF>58Annw:EN�:ED6>V8A8/N�DF<=WUnSN2VWUBH?iWungBu>V8i?AD6:=DF>V8An_N5@��+Z6DF`-D6:=WunÀ?A>V8A8ATUWunUÁ/WU:g>V8©DF8�:=Wu<=�i<=kU:=WbBuWUnpnw>VZF@h:EDF>58Annw:EN�:ED6>V8A8/N�DF<=WUn_Wu8©:E<=N5?A@AD6nEN58�:pZFWU@A<=nS�i<=>V�A<wDFTC:ETungWU8�:=Wu<=`-Wb?AWJ�/N5�;@AWC:Enp?�[ >V8A?AWUnu4¾O>V@A<À�A<=TUBuDFnwWu<ÀBuWC:=:EW�DF?ATUW5Á�BU>V8An=D6?ATu<w>V8AnÀ` N�DF8�:EWU8/N58�:ÀZFWSB2N�n >��aZ [ >V8A?AWg��Wu@i:ÀnwW��A<w>V�/N5j5Wu<

`��u`-Wb�1>V@i<x > 0 �

E > V2

� N�8An�BUWJB2N5nUÁ/Z�NÉv&>V8ABC:EDF>58+?�[^>58A?AWJWunt:�?AWJZFN�v&>V<=`-W �

ψ(x) =

{Aeip1x/~ +Be−ip1x/~

nwDx < 0

Ceip2x/~ +De−ip2x/~nwDx > 0

x\|h4e±��~

§¨ZFZ6W�?ATu��Wu8i?(?AWJ�-N5`-�AZFD�:E@A?AWUn_<=WUZFD6TuWunS�AN5<g?AWu@��BU>V8A?AD�:ED6>V8An_?AWJ<=N5BuBU>V<=?AWUàWU8�: �{ψ : A+B = C +D

ψ′ : p1(A−B) = p2(C −D)

x\|h4e±h3}~

M¨[ Wun=�AN5BuWÉ?iWun�nw>VZF@h:EDF>58AnSWunt:�?ATUn=>V<w` N5D6ng?AW�?AD6àWU8An=D6>V8�Kh4"79Zµf�Na?A>V8iBI@A8�BuWU<w:ªN�DF8»N�<= AD q:E<=N5DF<wW�?/N�8AnbZFWÉBªRi>VD �»?AWunbnw>VZF@i:=DF>V8inuÁcàN5D6n�N5@» @·?AWQZ�Nº?iDFn=BU@An=nwDF>V8 �i<=TuBUTu?AWU8�:EW5Á�Z6WÉBªRA>VD �

�,$'&�(E) �j3,I"=H)?3 >�-4=?)H3d/ ��CE3 � �� 143 14K,LN&�I_/ =H)H>W=?(E&�I 143 �2>�/?143dg�>WI"= �� >�C*(E]@>�)�&�- /?-.) C*3�� 3d]�� ! #"$ ��%�&('*)�&�� +-,�%-/.0)0��'*�1��.2� 304/� O

� �F��À��¢¼�A!$��% '�%»Â��/0{% .10}#&% ¥ ��

�;@AD¨nu[ DF`-�1>5n=WÉ8/N$:E@A<wWuZFZ6Wu`-Wu8�:bWUnw:J?AWQBªRA>5DFn=D6<�?AWUnHnw>VZF@h:EDF>58An��;@AD �U?iTuBu<wD65Wu8�: �QZ�Nº?AD �c@in=DF>58?�[ @A8��AN5�;@AWU:�?�[ >V8A?AWUng5Wu8/N�8;:�?AWJZ�NQj�N�@ABªRAWJ>V@©VWu8AN58�:�?AWJZFNÉ?A<=>VD�:EW�4¾�>V@i<p@i8��/N�� @iWU:�?�[^>58A?AWungVWU8/N58�:�?iWHZFNQj�N5@ABªRAW�Á">V8�BªRi>VDFnwDF<EN

D = 0 � DFZ1f�N�@i8AWI>58A?AWDF8ABUDF?AWU8�:EWax

eip1x/~~CÁ/@i8AWJ>V8A?AWJ<=T �/TuBªRiDFW x

e−ip1x/~~_WU:p@A8AWJ>V8A?iWJ:=<EN58in=`-DFn=W-x

eip2x/~~��{4

¾�>V@i<Y@A8+�/N�� @iWU:S?�[ >V8A?iWunY5Wu8/N58�:g?AW�Z�NI?i<=>VD�:EW5Á;>V8+BªRi>VDFnwDF<ENH?AW�`��u`-WA = 0

4hMµWun_�A<w>�q�A<=D6TU:=TungBuWUn�TC:ªN�:=npnt:ªN�:=DF>V8i8/N5DF<wWun_?iTuBu<wDF<w>V8�:p:=<=kUnp iDFWu8©Z6WunS�A<w>V�A<=D6TU:=Tung?AWJ�/N5�;@AWC:Enp?�[^>58A?AWunU4�Y>V`-àWSZ6WunÀ?AWu@��-BuN5nÀn=>V8�: n=DF`-DFZFN5DF<wWunUÁ�BU>V8An=D6?ATu<w>V8An BuWuZ6@AD/?�[ @A8 �AN5�;@AWU: ?�[ >V8A?AWUn VWU8/N58�:

?AW�ZFNIj�N5@iBªRAW54/¾¨@AD6n=�;@AWD = 0

Áh>58º�1WU@i:SBuN5ZFBU@AZFWU< Z6WunSN5`-�AZ6D6:E@i?AWunB

WC:C

WU8ºv&>V8ABC:EDF>58 ?AWA

7�Z\[eN�DF?AW�?iWunpTu�;@/N�:=DF>V8AnS?iWH<=N5BuBU>V<=?iWu`-Wu8�: �{ψ : A +B = C

ψ′ : p1(A− B) = p2C

⇒ p1(A−B) = p2(A+B)

⇒ A(p1 − p2) = B(p1 + p2)

⇒ B = Ap1 − p2

p1 + p2

WC:C = A+B = A

(1 +

p1 − p2

p1 + p2

)= A

2p1

p1 + p2

x | 4e±�K5~

�Y>V`-àWHZ [ >V8A?AWe−ip1x/~

Bu>V<w<=Wunw�1>58A? 7 @A8AW�>V8A?AWI<=T �ATuBªRAD6WÉWU:eip2x/~

7 @i8AW�>V8i?AW�:E<=N58AnrqàD6n=W�ÁADFZ�WUnw:pR/N5 iD6:E@iWuZ�?AWJ?ATC�/8AD6<p?iWun @CBfF��"@^QhF^PRL ca[F#]CY � FN: QhBEP�F^L [F*L ]<_`Pkc 1 QGcCc^QhBEP �AN5<

�Y>hW �-BuDFWU8�:�?iWH<wT �/W �hDF>V8 � R ≡ B

A�Y>hW �-BuDFWU8�:�?iWJ:=<EN58in=`-DFn=nwDF>V8 � T ≡ C

A� [^N5�A<=kUn�BUWJ� @iDµ�A<=TUBuku?iW5Á�>V8�N�?i>V8AB �

R =p1 − p2

p1 + p2, T =

2p1

p1 + p2

x | 4e±VmV~��8�nU[eN�:w:EWU8A? 7+BuW-�;@AW-BuWunH?AWU@��Bu> W��-BuD6Wu8�:EnHnEN�:=DFntv�N5n=nwWu8�:�@i8AW�<=WuZFN�:ED6>V8·?AW�Bu>V8in=Wu<t�N$q

:ED6>V8 ?i@�jVWu8i<=WR2 + T 2 = 1

4 �(N5DFn-BuWC:=:=W(<=WUZ�N�:=DF>V8 8�[ Wunt: ÇO�/) 5Tu<=D��/TuW�Á WC:ºZFN·<wWuZFN�:ED6>V8W��1WuBC:ED65Wu`-Wu8�:gVTU<=D6�ATuWbWunt:g�AZF@in_n=@A h:EDFZ6W54i¾�>V@A<SZFN�?ATC:EWu<wàD68AWu<YDFZ"v�N5@i:SD6?AWu8�:ED��/Wu<Y� @iWuZFZ6W�Wunt:Z�Ny� @AN58�:ED6:=T(BU>V8An=WU<w5TuW54_¾�>V@A<a@A8�TC:ªN�: nw:ªN$:EDF>58A8/N5D6<=W5Á¨B�[^WUnw:+ZFN��A<w>V /N5 iDFZFD�:ET+?AW��A<=TUn=WU8ABuWρ = |ψ(x)|2 �;@AD�Z\[^WUnw:24 � [eN��A<=kUn�Z\[^TU� @AN�:ED6>V8+?AWJBu>V8�:=DF8;@AD6:=T5Á/>V8©WU8�?ATU?A@AD�:g� @iW

?AD6 ~j = −∂ρ∂t

= 0x | 4e±5��~

� � I 3 ��3`=�� C<>�5@>W)?=?(E3 =H3,LN5_&�)?3dC*CE3a1.3aC<>cU�&�I@+%=H(*&�I 1hS &�[email protected]/dS KG+`)H( =e−iωt

>[g�3d+ω = p2

2m~

O'Q4( &�I�5B&�/?3k = p

~

�@C<>jU�&�I@+`=?(E&�Ic1hS &�I@143k143,g9(E3,I"=eikx−ωt ' cos(kx − ωt)

O � It = 0

�[email protected]�+`) �`=H3e143eCYS &�I@143e/?36=?)H&�-@g�33dIx = 0

3`= 3,It+ dt

+,3`=?=?3XL �dLj3^+`) �`=H3X/?3A/?3,)0>e14Kd5.CE>�+`Kd3X1.>WI_/ CE3 /?3,I@/ 143G/x

5_&"/:(*=?(*UV/dO�� &�I_+ �91hS >�5.)HMd/ C*3/?(*f�I.3k1.3�C<>F5@>W)?=?(E3e=H3,LN5B&�)H3,CEC*3 �_-.I@3eU�&�I_+%=?(E&�I�1hS &�I@143j>[g�3d+e-.I.3�5_>W)?=?(E3�/:5_>�=?(<>WCE3eipx/~

/?3�14Kd5.CE>�+`3j1.>WI@/CE36/:3dI@/2143d/x

5B&�/?( =H( UV/D>�C*&�)H/ ;"-bS [email protected]�&�I@+`=?(E&�I 1hS &�I@143^>[g�3G+X-.I@3A5_>W)?=?(E3A/?5@>�=H(E>�C*3e−ipx/~

/?3614K,5@CE>�+`3^1.>WI@/CE3e/:3dI@/2I.Kdf�>�=H( U O


��>V@A<�@i8AWÉn=>5ZF@i:=DF>V8�nw:ªN$:EDF>58A8/N5D6<=W541�(N�DFn87�3É?ADF`-Wu8in=DF>58�Á"?AD6 ~j = ∂j∂x

ÁcWU:�?AD6 ~j = 0 ⇒ j =BU:=WV4/��[ Wunt:p?A>V8ABbZ6W�Bu>5@A<EN58�: ~j �;@ADµWunt:gBu>V8AnwWu<tVT54��N5DFn ~j = ~

2mi

(ψ∗~∇ψ − ψ~∇ψ∗

) ÁAn=>VD�: 73J?ADF`-Wu8in=DF>58 � j = ~

2mi

(ψ∗ ∂ψ

∂x− ψ ∂ψ∗

∂x

) 4��_N5Z6Bu@AZ6>V8Anj(x)

�1>V@i<x < 0 �

ψ∗∂ψ

∂x=(A∗e−ip1x/~ +B∗eip1x/~

)×(ip1A

~eip2x/~ − ip1B

~e−ip1x/~

)

=ip1

~

(|A|2 − |B|2 − A∗Be−2ip1x/~ + AB∗e2ip1x/~

)

ψ∂ψ∗

∂x= − ip1

~

(|A|2 − |B|2 − AB∗e2ip1x/~ + A∗Be−2ip1x/~

)

⇒ j =p1

m

(|A|2 − |B|2

)=p1

m|A|2 (1 −R2)

x\|h4e±V|V~

WU:p��>V@A<x > 0

x�WU8�nwWH<=N5�A��WuZFN58�:g� @c[^>V8�N��1>VnwTD = 0

~ �

ψ∗∂ψ

∂x= |C|2 ip2

~

⇒ j =p2

m|C|2 =

p2

m|A|2 T 2

x\|h4e±5±�~

��8�WU8�?iTu?A@AD�:gZ�NÉ<wWuZ�N$:EDF>58 �p1(1 − R2) = p2T

2 x\|h4e±V�V~

�;@AWHZ [ >V8ºVTU<=D6�AWIN�DFn=TUàWU8�:H7QZ\[eN5D6?AW�?AWUngW �h�A<=WUn=nwDF>V8An�?AWR2 WC:

T 2 4

�� A� � �� 6� � �� N� � ��L��

�Y>V8in=DF?iTu<=>58An�?iTun=>5<=àN5DFnSZ6WbB2N5ng?�[ @A8AWJ /N5<w<=D6ku<=Wb?iWH��>5:=Wu8�:ED6WuZµ?ATC�/8ADFWb�AN5<

V (x) =

V1n=D

x < x1

V2 > V1n=D

x1 < x < x2

V1n=D

x > x2

x\|h4e±5¬�~

�Y>V8AnwDF?ATU<=>V8in_?/N58AnY@A8+�A<=WUàD6Wu<�:EWUà�AnYZFW�BuN5n_?�[ @A8+�/N�� @iWU:S?�[ >V8A?iWunY5Wu8/N58�:g?AW�Z�NIjVN5@ABªRAWN2VWUBI@A8iWHTU8AWu<wjVDFW

EBU>Và�i<=DFnwWbWu8�:E<wW

V1WU:V2

4

V1 < E < V2

� � � � �A!}!�#&Ä;!}%('�%»Â��/0{% .10}#&% ¥� ��%;0a�_,O.O.O% ¥ �c¦

0

V (x)

x

V1

V2

x1 x2

� �� | 4e± � X_N5<w<=DFkU<=Wb?AWJ��>5:EWU8�:EDFWUZ 4

� N58AngZ6Wung?AD �1Tu<wWu8�:EngnwWuBU:=Wu@A<wnuÁ"Z�NÉv&>V8iBU:ED6>V8+?�[ >V8A?AWJ?A>5D6: �U:=<=WJ?AWJZ�N�v&>5<=`-W

ψ(x) =

Aeipx/~ +Be−ipx/~n=D

x < x1

Ce−x/l +Dex/ln=D

x1 < x < x2

Eeipx/~n=D

x > x2

x | 4e±VzV~

>��ÅZ\[^>58Å8�[^N�j�N�<=?AT-�;@AW Z\[^>V8i?AWeipx/~

��>V@A<x > x2

�A@iDFn=�;@�[ >V8 nU[^D68�:ETu<wWunwn=W�7�@A8 �/N5�;@AWC:?�[ >V8A?AWUnp5Wu8/N58�:�?AWJZ�NÉj�N�@ABªRAW54MµWungBu>V8i?AD6:=DF>V8AnY?AWI<=N5BuBU>V<=?iWu`-Wu8�:pWu8

x1WC:x2

Bu>58A?A@AD6n=Wu8�:�N5@��©TU�;@/N�:ED6>V8AnSnw@AD6�N58�:EWUn �

x1 :

{ψ : Aeipx1/~ +Beipx1/~ = Ce−x1/l +Dex1/l

ψ′ : ip~

(Aeipx1/~ − Be−ipx1/~

)= 1

l

(−Ce−x1/l +Dex1/l

)

x2 :

{ψ : Eeipx2/~ = Ce−x2/l +Dex2/l

ψ′ : ip~Eeipx2/~ = 1

l

(−Ce−x2/l +Dex2/l

)

x | 4L��V~

��8I?AD6n=��>Vn=W�?A>58AB ?iWÀ�pTu�;@/N$:EDF>58An��1>V@i<È|SDF8iBu>V8A8;@AWUnu4��8I��Wu@h:�W �h�A<=D6àWU<O�g?AWUn�BU>hW �-BuDFWU8�:EnDF8ABU>V8A8;@AnpWu8�v&>V8ABU:=DF>V8©?A@»|}q9D6ku`-W54A79ZµWUnw:�?A>V8ABH��>Vn=nwDF AZ6Wb?AWIBuN5ZFBU@AZFWU<

EWC:B

WU8�v&>V8iBU:ED6>V8?AW

AÁ/WC:�N�DF8AnwD�?AWJ?ATU:=Wu<wàD68AWu<SZ6WunSBu> W��-BUDFWu8�:=n�?AWb:=<EN58AnwàD6n=nwDF>V8+WC:�?iWH<wT �/W �hDF>V8�4

¾�>V@i<�BuWUZ�NiÁ�>V8»<wTuTUBu<=D�:�Z6WIntfhnt:EkUàWÉWU8»v�N5DFn=N58�:��AN5n=nwWu<�Z6Wun�:=Wu<wàWUn�Wu8A

?/N58An�ZFWHn=WuBU>V8A?àWU`É A<wW54 ��8�n=W�<=WC:E<=>5@iVW(N5ZF>V<wn-N{5WuB»@i8�nwf nt:Eku`-W»Z6DF8ATuN5DF<wWº8A>V8�Ri>Và>5jVku8AW©?AW��;@/N�:E<wW


Tu�;@/N$:EDF>58Ang?A@�:9f ��WS~x = ~y �

Be−ipx1/~ − Ce−x1/l −Dex1/l = −Aeipx1/~

− ip~Be−ipx1/~ + C

le−x1/l − D

lex1/l = − ip

~Aeipx1/~

−Ce−x2/l −Dex2/l + Eeipx2/~ = 0Cle−x2/l − D

lex2/l + ip

~Eeipx2/~ = 0

x\|h4L� 3}~

79BuD\ÁA>V8©�N�@h:EDFZ6DFnwWu<_ZFN ] � <`DGF [F�� ]\_M1 F^]�� 1>5@A<S?ATU:=Wu<wàD68AWu<SZ6WunSDF8iBu>V8A8;@AWUnE

WC:B �

xi =det(Sxi

)

det(S)

x\|h4L�5KV~

>��det(S)

Wunw:�Z6Wº?ATC:EWu<wàD68/N58�:Q?AW+ZFN(àN�:E<wDFBuWa<=WU�A<=TUn=Wu8�:EN58�: Z6W nwf nt:Eku`-W©WU:det(Sxi

)Z6W

`��u`-W�?ATU:=Wu<wàD68/N58�:g?/N58AngZ6Wu�;@AWuZc>V8(NÉ<wWu`-�AZ�N�BuTbZFWUnSBu> W��-BuD6Wu8�:Eng?AWJZFNÉD�BU>VZF>58A8AWb�/N5<SZ6W

`-Wu`É A<wWb?AWJ?A<=>5D6:EW~y4��8�NQ?A>V8AB

det(Sxi) =

∣∣∣∣∣∣∣∣

e−ipx1/~ −e−x1/l −ex1/l −Aeipx1/~

− ip~e−ipx1/~ 1

le−x1/l −1

lex1/l − ip

~Aeipx1/~

0 −e−x2/l −ex2/l 00 1

le−x2/l −1

lex2/l 0

∣∣∣∣∣∣∣∣

x\|h4L��m�~

��8�?iTUVWUZF>V�i�1WJ�/N5<S<=N5�A��>V<w: 7�Z�NÉ�i<=Wu`-DFkU<=WbBU>VZF>V8i8AW5ÁA?c[^>��

det(Sxi) = e−ipx1/~

(− ip

~lAeipx1/~ − ip

~lAeipx1/~

)+ip

~e−ipx1/~

(−Aleipx1/~ − A

leipx1/~

)

= −4ip

~lA x\|h4L�$�;~

§�:pZFWbB2N�ZFBu@iZ�?AWdet(S)

?A>V8i8AW

det(S) =

∣∣∣∣∣∣∣∣

e−ipx1/~ −e−x1/l −ex1/l 0− ip

~e−ipx1/~ 1

le−x1/l −1

lex1/l 0

0 −e−x2/l −ex2/l eipx2/~

0 1le−x2/l −1

lex2/l ip

~eipx2/~

∣∣∣∣∣∣∣∣= . . .

= eip∆x/~[

1

l2(e∆x/l − e−∆x/l

)− ip

~l

(e∆x/l + e−∆x/l

)]

+ip

~eip∆x/~

[ip

~

(e∆x/l − e−∆x/l

)− 1

l

(e∆x/l + e−∆x/l

)]

= eip∆x/~[2 sinh

(∆x

l

)×(

1

l2− p2

~2

)+ 2 cosh

(∆x

l

)×(−2ip

~l

)]

x\|h4L�5|V~

� $'&�(E)�>WI.I@3`Z43 � O

� � � � �A!}!�#&Ä;!}%('�%»Â��/0{% .10}#&% ¥� ��%;0a�_,O.O.O% ¥ �

>��Z [ >V8©N-��>VnwT∆x ≡ x2 − x1

4/¯�DF8/N5Z6Wu`-Wu8�:2Áh>V8©>V i:=DFWu8�:

E =−4ip

~le−ip∆x/~A

2 sinh(

∆xl

)×(

1l2− p2

~2

)+ 2 cosh

(∆xl

)×(−2ip

~l

)

⇒ |T |2 =|E|2

|A|2

=16p2

~2l21

4(

1l2− p2

~2

)2

sinh2(

∆xl

)+ 16p2

~2l2cosh2

(∆xl

)

⇒ |T |2 =4p2

~2l2(1

l2− p2

~2

)2

sinh2 (x2 − x1)

l+

4p2

~2l2cosh2 (x2 − x1)

l

x | 4L�5±V~

� �u`-W�n=D�WuZ6ZFW�N+@A8AW-Tu8iWu<=j5DFW��AZF@inH AN5n=nwWa�;@AW-Z�N+ /N5<w<=D6ku<=W�xE < V2

~CÁµ@A8iWa�/N�<w:ED6Bu@AZ6W�N?A>V8ABp@A8AW �V%;!{0{�/#�.O%aÇ�!}�"Ã��iÃÈ#�¥\#�0{�-'�%+¥��+0{!}� ��%;!�)U% ! 4 ��[ Wunt:_BuWp�;@�[ >V8ºN5�i�1WUZFZFW D � F �IF^L LTU6P��PdF^D 4��YWC:=:=WQ�A<=>V AN5 ADFZ6D6:=TJ:=Wu8A? 8iT2N58A`->VD68Anp5Wu<wn � n=D�Z�N Z6>V8AjV@AWU@A<�?iW�Z�Na /N5<=<wDFkU<=W |x2 − x1|:EWU8A?�5Wu<wn

+∞ ÁA>5@�nwDµZ [ Tu8AWU<=jVD6W�:EWu8i?�5Wu<=nV1

Á"N5@A�;@AWuZµBuN5np→ 0

4

E > V2

� N58AnSBUWbB2N5nUÁiD6Zcn=@��-:g?iW�<=WU`a�AZFN5BuWU<_ZFWUn_W �h��>V8AWU8;:=DFWUZFZFWUn_<=TUWuZ6ZFWunY�/N5<_?AWUnSW �h��>V8AWu8�:=DFWuZ6ZFWUnBu>V`-�AZ6W �hWunÉ>5@ Wu8ABU>V<=W ?AW <wWu`-�AZ�N5BUWu< 1

l

�/N5< ip2~

N{5WuBp2 =

√2m(E − V2)

4¨s D >V8Å�1>VnwWp1 =

√2m(E − V1)

ÁA>V8©:E<=>5@iVWb�/8/N�ZFWu`-Wu8�:

|T |2 =4p2

1p22

(p21 + p2

2)2sin2

(p2(x2 − x1)

~

)+ 4p2

1p22 cos2

(p2(x2 − x1)

~

) x | 4L�V�5~

¦ � *-,O% ¥��,�%h)�ÂH!$�"Ã�¥�Ä ¶Å% )aÆ�#\¶�ÇÈ¥&% )É%h.¹Ê©#�¶Å% .�)u#��.Ë�

� � ��

� ��®��

� �� L��6��

��8 nU[^D68;:=Tu<wWun=nwW N5@ �A<=>5 AZFkUàW©?AW�Z�N �;@/N58�:ED��/B2N$:EDF>58 ?A@ `->V@iVWUàWU8�: ?�[ @A8AW��/N5<t:EDFBU@AZFW?/N58An�@A8 ��>5:EWU8�:EDFWUZ_BuWU8�:E<EN�Z Á�B�[^WUnw:wq 7$q9?iDF<=W ?/N58An�@A8 ��>5:EWU8;:=DFWUZ_�;@ADS8AW+?ATU�1WU8A? �;@AW©?AW©ZFN?ADFnt:ªN58iBuW7Q@A8��>VD68;:g?i>V8A8AT�4"M�[ R/N5`-DFZ�:E>V8iDFWu8 ?ATuBU<=D6�N58�:g@A8©:EWUZµnwf nw:=ku`-WInU[^TUBu<wD6: �

H =~p 2

2m+ V (r)

>��r =

√x2 + y2 + z2

x�±i463{~

§¨8ºàTUB2N58iDF�;@AW�BUZ�N5nwn=D6� @iW5Á Z�NJn=>VZ6@i:ED6>V8a?iW�BuW��A<=>5 AZFkUàWp<=WU�1>VnwW�nw@A<YZ6Wpv�N5D�:À� @iW�ZFWg`->VàWU8�:BuD68ATU:=DF�;@AWJWunt:�BU>V8AnwWu<w5T541§¨8�W �cWC:2Á/ZFWb`->V`-Wu8�:SBUDF8ATC:EDF�;@AW ~L WUnw:p?ATC�/8ADµ�/N5<

~L = ~r ∧ ~p

⇒ d~L

dt=d~r

dt∧ ~p+ ~r ∧ d~p

dt

=~p

m∧ ~p

︸︷︷︸=0

+~r ∧ ~F

x�±i4LK5~

�(N5D6n ~F (~r) = −~∇V Wunt:_?AD6<=DFj5TuWpn=@AD�5N�8;:~rZF>V<wn=�;@AW

V8AW�?iTu��Wu8A?+�;@AW�?iW

r ⇒ d~Ldt

= ~0 ⇒~L =

Bu>58Anw:EN58�:EW�4§¨8G`-TuBuN58ADF�;@AW·�;@/N58�:=DF�;@AW5ÁgD6Z�Wu8Ë�N ?iW�`��u`-W ` N�DFnºZ6W à>V`-Wu8�:©BUDF8ATC:EDF�;@AW·WUnw:�@A8

>V��Tu<EN$:EWu@i<u4��Y>V`-àWU8��u>58AnS?A>V8ABJ�/N�<pTC:E@A?iDFWu<SZ6Wung�A<w>V�A<=D6TU:=TunS?AWJBuWC:�>5�1TU<EN�:=Wu@A<U4

¦A� ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

� �� 0� �@� � � � ��6��N��N� � � �� W� �@� � � �

M¨[ >V�1TU<EN�:=Wu@A<g`->V`-Wu8�:gBuD68ATU:=DF�;@AW ~L =(Lx, Ly, Lz

) Wunt:�?iTU�/8AD��/N�<

~L = ~r ∧ ~p =

ypz − zpyzpx − xpzxpy − ypx

x�±i4^m�~

��8y5Tu<wD6�/WaN5DFnwTu`-Wu8�:H�;@AWaZ6WunJ>V��Tu<=N�:EWU@A<=nLx, Ly, Lz

n=>58;:IRAWU<=`-D6:ED6�;@AWunU4�§¨:EN5 AZ6DFn=nw>V8AnbZ6Wun<=kUjVZFWUnS?AWJBu>V`-`É@i:EN�:ED6>V8+Wu8�:=<=WJZ [ >V��Tu<EN$:EWu@i<Sà>V`-Wu8�:gBUDF8ATC:EDF�;@AWbWC:pZ [ >V�1TU<EN�:=Wu@A<S��>VnwD6:ED6>V8 �[Lx, x

]= 0

[Lx, y

]= [ypz − zpy, y] = −z [py, y] = i~z

⇒[Li, xj

]= i~εijkxk

x�nw>V`-` N�:=DF>V8 n=@A<gZFWUnSDF8A?iDFBuWUnS<=TU�1TC:ETunE~

N25WuBεijk :

ε123 = ε231 = ε312 = 1x��Wu<w`É@i:ªN$:EDF>58An_�/N5D6<=WUnp?iW-3{K5m�~

ε132 = ε321 = ε213 = −1x��Wu<w`É@i:EN�:ED6>V8An_D6à�/N�DF<=WUn_?AW�3}K5mV~

εijk = 0?/N58AngZ6WungN5@i:=<=WUn�BuN5n

x�±i4 �;~��8�:=<=>V@hVWJ?AWJ` N�8ADFkU<=W�nwDF`-DFZ�N�DF<=WpBuWUZFZ6WungN2VWUBIZ\[^>V��Tu<=N�:EWU@A<SDF`-�A@AZ6n=D6>V8 �

[Lx, px

]= 0

[Lx, py

]= [ypz, py] = i~pz

⇒[Li, pj

]= i~εijkpk

x�±i4e|V~

sh@i<�Z�N� /N5nwWI?iW�BuWUn��A<w>V�A<wDFTU:=TunUÁ�>58(��Wu@i:�?ATu`->V8�:E<wWu<p�;@AW ~L Bu>5à`É@h:EWHN2VWuB�Z [ R/N5`-DFZ6:=>�q8AD6Wu8�4

Lz = xpy − ypx[Lz, V (x, y, z)

]ϕ = x [py, V ]ϕ− y [px, V ]ϕ

= x~

i

∂V

∂yϕ− y

~

i

∂V

∂xϕ

=~

i

(x∂V

∂y− y

∂V

∂x

)ϕ

x�±i4^±�~

� �F�� ¤�Ç¨�;!}�i0{% ,�!+¢¼�1¶�%h.�0©��#�.O��0}#��,�%%�0 ¥ � ��/¶Å#\¥�0{��.�#&% . ¦/¡

�(N5D6nuÁV (x, y, z) = V (r)

Ár =

√x2 + y2 + z2

?c[^>��

x∂V

∂y− y

∂V

∂x= x

∂V

∂r

y

r− y

∂V

∂r

x

r= 0

⇒ LzBu>V`-`É@i:=WJN2VWuB

V.

⇒ LxÁLy

BU>V`-`É@i:EWU8�:pN{5WuBV.

⇒ ~L2 = L2x + L2

y + L2z

Bu>V`-`É@i:=WJN{5WuBV.

x�±i4L�5~

� WH�AZF@AnUÁA>V8�N

[Lz, px

]= [x, px] py = i~py ⇒

[Lz, p

2x

]= i~pypx + i~pxpy = 2i~pxpy

[Lz, py

]= −i~px ⇒

[Lz, p

2y

]= −2i~pxpy

[Lz, pz

]= 0 ⇒

[Lz, p

2z

]= 0

x�±i4e¬V~

⇒[Lz, ~p

2]

= 0

⇒ LxÁLy

Á ~L2 ÁABu>V`-`É@i:=Wu8�:pN2VWUB~p 2

⇒ LxÁLy

ÁLz

WC: ~L2 Bu>5à`É@h:EWu8�:pN2VWUBH.

x�±i4ezV~

¾ÈN5<pN5D6ZFZ6Wu@A<wnuÁ Z [ R/N5`-DFZ6:=>V8AD6Wu8 nU[^W �h�A<=D6àWbnwDF`-�AZFWUàWU8;:_Wu8©v&>V8ABU:=DF>V8©?AW ~L 4"§¨8�W �cWC:2ÁL2x = (ypz − zpy)

2

= y2p2z + z2p2

y − y pzz︸︷︷︸zpz−i~

py − z pyy︸︷︷︸ypy−i~

pz

= y2p2z + z2p2

y + i~ypy + i~zpz − 2ypy zpz

⇒ ~L2 =(x2 + y2

)p2z +

(y2 + z2

)p2x +

(z2 + x2

)p2y + 2i~ (xpx + ypy + zpz)

− 2ypy zpz − 2xpxypy − 2zpzxpx

= ~r 2~p 2 − z2p2z − x2p2

x − y2p2y + 2i~~r · ~p− 2ypyzpz − 2xpxypy

− 2zpzxpx

= ~r 2~p 2 −(~r · ~p

)2

+ i~~r · ~p

⇒ ~p 2 =1

r2~L2 − ~2 ∂

2

∂r2− 2

~2

r

∂

∂r

⇒ H = − ~2

2m

∂2

∂r2− ~2

mr

∂

∂r+

1

2mr2~L2 + V (r)

x\±h463��V~

¦h³ ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

>��Z [ >V8(N-@i:ED6ZFD6n=TJZFWUn��A<w>V�A<wDFTU:=Tunpn=@AD��N58�:EWun � x2p2x = (xpx)

2 = i~xpxWU:

(~r · ~∇)f = r ∂f∂r

Á�;@AD�?ATuBU>V@AZ6Wa?AW-Z [ W �h�A<wWunwn=DF>58�?A@�jV<EN�?ADFWU8;:HWU8ÅBu> >V<w?A>V8A8ATUWunHn=�ARiTu<=D6�;@AWun � ~∇f = ∂f

∂r~er +

1r∂f∂θ~eθ + 1

r sin θ∂f∂ϕ~eφ4

�Y>V`-`-W ~L2 Bu>V`-`É@i:=WaN25WuBHÁµ>V8Å��Wu@i:IZFWUnJ?AD�N5j5>V8/N5Z6DFn=WU<�?/N58inI@A8iWa /N�n=W BU>V`-`É@A8AW�4

��8(NQ?A>58ABJn=Tu�AN5<=TJZFWb�i<=>V AZ6ku`-WbWu8(KÉ�/N5<w:=DFWUn �� P�<w>V@iVWU<�Z6Wun_�N5Z6Wu@A<wnp�i<=>V�A<wWungWU:pZ6Wun_v&>V8ABC:ED6>V8AnS�A<w>V�A<=WUng?AW ~L2 4� P�<w>V@iVWU<SZFWUn_5N�ZFWu@i<=n_�A<w>V�A<wWunSWU:_ZFWun VWuBC:EWU@A<=np�A<=>5�A<=WUnS?AW�Z [ >V��Tu<=N�:EWU@A<

HWU8�<wWu`-�AZ�N}q

�2N58�: ~L2 �/N�<pnwWung�N5ZFWU@A<=nS�A<w>V�A<wWunu4ogWuàN5<=�;@AW � �YWC:=:EWQv&>V<wàW�?AWQZ [^d�N5`-DFZ�:E>V8iDFWu8�Wunt:J:E<=kUnb�A<=> BªRAW�?AW�Z�N-v&>V<=`É@iZ�N�:=DF>V8�R/N5`-DFZ�q:E>58ADFWU8A8AWQ?A@·�A<w>V AZFkUàWIBuZFN5n=nwDF�;@AW5Ác` N5D6n�DFZ�f N+@A8iWQn=@A i:=DFZ6D6:ET�4c§¨8·`-TuB2N�8ADF�;@AWQBUZ�N5nwn=DF�;@AW�ÁZ [^d�N�àD6Z6:E>58ADFWU8 nu[ TuBU<=D�: �

H =p2r

2m+

~L2

2mr2+ V (r)

x�±i4�3V3}~M�N��;@AWUnw:=DF>V8�� @iD� D6Wu8�:�8AN�:E@A<wWuZ6ZFWu`-Wu8�:7©Z [ Wunw�A<=D�:IWUnw:I?AW-nEN2V>VD6<Jn=D�>V8Å��Wu@h:ITUBu<wDF<=W-Z [^d�N}q`-DFZ6:=>V8AD6Wu8»�;@/N58�:=DF�;@AW-n=>V@AnI@A8AW�v&>V<=`-WQn=D6àD6Z�N5D6<=W�Á1WC:I��>V@A<J�;@AWUZ >V��Tu<=N�:EWU@A<

pr4OMµW�BªRA>VD �

� 8AN�:E@A<wWuZ�� x�àN5DFng:E<w>V�»8/N��\vrÁ"WU:bWU8»v�N5D6:pv�N5@��A~pnwWu<=N5D6:�?AWÉBªRA>VD6n=D6<�Z\[^>V��Tu<=N�:EWU@A< −i~ ∂∂r

4µ�(N5D6nBuWC:p>V�1TU<EN�:=Wu@A<g8�[ Wunt:p�/N5ngRAWu<wàD�:ED6Wu8�4i§�8�W��1WU:uÁ/n=Dµ>V8��>Vn=W

A = −i~ ∂∂r

ÁAD6Zc DFWU8;: �〈ϕ|A†|ψ〉 − 〈ϕ|A|ψ〉

= 〈ψ|A|ϕ〉∗ − 〈ϕ|A|ψ〉

= i~

∫ψ∂

∂rϕ∗d~r + i~

∫ϕ∗ ∂

∂rψd~r

= i~

∫ π

0

sin θdθ

∫ 2π

0

dϕ

∫ +∞

0

r2 ∂

∂r(ψϕ∗)dr

︸︷︷︸��

��

��

x�±i463{KV~

¾ÈN�<pBU>V8�:E<wW5Á/Z [ >V��Tu<=N�:EWU@A<pr = −i~1

r∂∂rr = −i~

(∂∂r

+ 1r

) Wunw:pRAWU<=`-D6:=DFWu8c4A§¨8�W �cWC:2Á〈ϕ|p†r|ψ〉 − 〈ϕ|pr|ψ〉

= i~

∫ π

0

sin θdθ

∫ 2π

0

dϕ

∫ +∞

0

r2

[(∂

∂r+

1

r

)ψϕ∗ +

(∂

∂r+

1

r

)ϕ∗ψ

]

M�NQ�/N5<w:=DFWb<=N5?AD�N�ZFW�nU[^TUBu<=D�:pN5ZF>5<=n �∫ +∞

0

r2

[(∂

∂r+

1

r

)ψϕ∗ +

(∂

∂r+

1

r

)ϕ∗ψ

]dr

=

∫ +∞

0

r2

[∂

∂r(ψϕ∗) +

2

rψϕ∗

]dr

=

∫ +∞

0

∂

∂r(r2ψϕ∗)dr

=[r2ψϕ∗]+∞

0= 0

n=Drψ → 0

�1>5@A<r → +∞ WU:

04 x�±i4632m�~

� �F�� ¤�Ç¨�;!}�i0{% ,�!+¢¼�1¶�%h.�0©��#�.O��0}#��,�%%�0 ¥ � ��/¶Å#\¥�0{��.�#&% . ¦/�

��8��Wu@i:�N5@in=n=D�5>VDF<SZ6WungBªRA>VnwWunp?ADF<wWuBC:EWu`-Wu8�:pWu8©:EWU<=`-WunS?�[ >V��Tu<=N�:EWU@A<=nU41¾�N5<g?ATU�A8AD6:=DF>V8�Á

A = −i~ ∂

∂r=~r

r.~p =

x

rpx +

y

rpy +

z

rpz

⇒ A+ = pxx

r+ py

y

r+ pz

z

r6= A

x�±i463u�;~�A@AD6n=�;@AWHZFWunSBU>V`-`É@i:ªN$:EWu@i<=n

[x, px]Á[y, py]

WC:[z, pz]

8AWJn=>V8�:p�/N5ng8;@AZ6nu4¾ÈN5< Bu>V8�:E<wW5Á Z [ >V�1TU<EN�:=Wu@A< 1

2[A + A+]

WUnw:+TChD6?AWu`-`-Wu8�:+RAWU<=`-D6:ED6�;@AW5Á�WU:ºDFZpWunw:ºTuj�N5ZH712

[~rr.~p+ ~p.~r

r

] 4"��<uÁ/��>V@A<_:=>V@i:EWbv&>V8iBU:ED6>V8f(~r)

ÁA>V8�N �1

2

[~r

r.~p+ ~p.

~r

r

]f =

1

2

~r

r.(−i~)~∇f +

1

2(−i~)~∇.

(~r

rf

) x\±h463}|5~

�(N5D6n

~∇.(~r

rf

)=

(~∇.~rr

)f +

~r

r.~∇f

� WH�AZF@AnUÁ

~∇.(g(r)~er) =1

r2

∂

∂r(r2g(r))

?�[ >��

~∇.~rr

=1

r2

∂

∂r(r2) =

2

r]�D68An=D\Á

1

2

[~r

r.~p+ ~p.

~r

r

]f =

(~r

r.~p+

1

r

)

��8�WU8�?ATU?A@AD�:g� @iWpr

WUnw:pn=D6`a�iZFWu`-Wu8�:SZFNÉv&>V<=`-W�nwf `-TU:=<=DFnwTuWH?AWJZ [ >V��Tu<EN$:EWu@i< −i~ ∂∂r

�

⇒ pr =1

2

[~r

r.~p+ ~p.

~r

r

]

¾ÈN5<pN5D6ZFZ6Wu@A<wnuÁiBU>V`-`aW

~p2r = −~2

(∂

∂r+

1

r

)(∂

∂r+

1

r

)

= −~2

[∂2

∂r2+

1

r

∂

∂r+

1

r2− 1

r2+

1

r

∂

∂r

]

= −~2

(∂2

∂r2+

2

r

∂

∂r

)

¦ � ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

Z [ DF?iWu8�:ED�:ET�?ATUà>58;:=<=TUWH�iZF@AngR/N5@h:

~p 2 =1

r2~L2 − ~2 ∂

2

∂r2− 2

~2

r

∂

∂r

x�±i4632±�~DF`-�AZ6DF�;@AW��;@AW

~p 2 = p2r +

~L2

r2

x�±i463{�V~M¨[ed�N5`-DFZ6:=>V8AD6Wu8 �;@/N�8;:=DF�;@AWH�1WU@i:p?A>V8ABJW �cWUBU:ED�VWUàWU8;:pnwWI`-WU:w:E<=Wbnw>V@AngZ�NÉv&>V<wàW �

H =p2r

2m+

~L2

2mr2+ V (r)

x�±i4�3{¬�~

� � � � � � � � �� ~L2 � �@��D�6�� c��6�� 6� ��A� ��

� WUnS�A<=>5�A<=D6TU:ETUnY�A<wTuBUDFn=WUn_WU:YjVTu8iTu<EN�ZFWunY?A@ºn=��WuBC:E<wW�?AW ~L2 ��Wu@i5Wu8�:gn=W�?ATu?A@iDF<=W�?AW�BU>V8An=D q?ATU<EN�:=DF>V8An�N5ZFj5Tu A<wDF�;@AWun�@A8ADF�;@AWUàWU8�:b /N5nwTuWunbnw@A<�Z6Wun�<wkujVZ6Wun�?iWÉBu>V`-`É@i:EN�:ED6>V8(WU8�:E<=W�Bu>V`Qq��>VnEN58�:=Wunp?AW ~L 4��<UÁ

[Lx, Ly

]= [ypz − zpy, zpx − xpz]

= [ypz, zpx] + [zpy, xpz]

= −i~ypx + i~xpy = i~Lz

⇒[Li, Lj

]= i~εijkLk ⇔ ~L ∧ ~L = i~~L

[Lx, ~L

2]

=[Lx, L

2y

]+[Lx, L

2z

]

= Lyi~Lz + i~LzLy − Lzi~Ly − i~LyLz

= 0

⇒ LxÁLy

WU:Lz

BU>V`-`É@i:EWU8�:pN{5WuB ~L2.

x�±i4632z�~

�Y>V`-àWLx

ÁLy

WU:Lz

8iWpBU>V`-`É@i:EWU8�:��/N5n¨Wu8�:=<=WpWu@��1Á�>V8�8AWg��Wu@i:��/N�nÀZ6Wun¨?ADFN5jV>V8AN5ZFD6n=Wu<On=D q`É@AZ�:ªN58iTu`-Wu8�:24$¾ÈN�<ÈBu>58;:=<=W�Á5>V8É��Wu@h:�?ADFN5jV>V8/N�ZFDFnwWu<µnwDF`É@AZ�:ªN58iTu`-Wu8�:ÈZ\[^@i8�?�[ Wu8�:E<wW_Wu@ �ÉWC: ~L2 4²�:ªN�:=ng�A<=>V�i<=Wung?AW

LzWC: ~L2 �

sh>5D6: |ψ〉 @A8�TU:EN�:g�A<=>5�A<=WJ?AWLz

WU: ~L2 �Lz|ψ〉 = ~λ|ψ〉 Á ~L2|ψ〉 = ~2β|ψ〉

x�±i4LK ��~

� �F¡��À%�ÆcÇ¨% �50{!}%»'�% ~L2 ��I�1.O)2#�'O�;!}�i0}#&��.�) +¥ "� ÃO!$#\�c,O% ) ¦�¦

798�:E<=> ?A@AD6n=>V8inpZ6WunS>V��Tu<=N�:EWU@A<=nL+

WU:L− �

{L+ = Lx + iLy

L− = Lx − iLy⇔{Lx = 1

2(L+ + L−)

Ly = 12i

(L+ − L−)

x\±h4LKh3{~WU:pTU:EN5 AZ6DFn=nw>V8AnSZ6Wu@A<wnp<wWuZFN�:ED6>V8An_?AWJBU>Và`�@i:ªN�:=DF>V8 �[

Lz, L+

]=[Lz, Lx

]+ i[Lz, Ly

]= i~Ly − i2~Lx = ~L+

[Lz, L−

]=[Lz, Lx

]− i[Lz, Ly

]= i~Ly + i2~Lx = −~L+

[L+, L−

]= −i

[Lx, Ly

]+ i[Ly, Lx

]= 2~Lz

x\±h4LKVK5~

¾ÈN5<pN5D6ZFZ6Wu@A<wnuÁiBU>V`-àWbBUWInw>V8�:�?iWunpBu>V`É iDF8/N5D6n=>V8inSZFDF8iT2N5D6<=WunY?AWLx

WU:Ly

ÁL+

WU:L−

BU>V`�q`É@i:=Wu8�:�N{5WuB ~L2 4�Y>V8AnwDF?ATU<=>V8AngZ6W��5WU:

L+|ψ〉4/�Y>Và`-W [

Lz, L+

]= ~L+

Á/DFZ1hD6Wu8�: �LzL+|ψ〉 = L+Lz|ψ〉 + ~L+|ψ〉 = ~(λ+ 1)L+|ψ〉

x\±h4LK5mV~¾ÈN5<pN5D6ZFZ6Wu@A<wnuÁ ~L2L+|ψ〉 = L+

~L2|ψ〉 = ~2βL+|ψ〉4 � WJ` �UàW�Á

LzL−|ψ〉 = ~(λ− 1)L−|ψ〉x\±h4LK��~

¾¨<=>V�A<wDFTC:ETun �3V4 ~L2 = L2

x + L2y + L2

z ⇒ ~L2 − L2z = L2

x + L2y�(N5D6n 〈ϕ|L2

x+ L2y|ϕ〉 ≥ 0

4�§¨8�W��1WU:2Á 〈ϕ|L2x|ϕ〉

Wunw:�nw@A��Tu<=D6Wu@A<#7�ZFN��AZF@An¨��WU:=D6:=WY�N5ZFWU@A<�A<w>V�A<=WJ?AW

L2x

Á/�;@ADµWUnw:p@A8�B2N5<w<=TJWU:pWunt:p?A>V8ABJ��>Vn=D�:ED65W54� @»BU>V@A��Á 〈ψ|~L2 − L2

z|ψ〉 = ~2(β − λ2) ≥ 04�¾�>V@i<

β?A>V8i8AT5Á�ZFWUnp�N5Z6Wu@A<wn��A<w>V�A<wWun

��>Vn=nwDF AZ6WunS?AWLz

nw>V8�:� �>V<w8ATuWUn �−√β ≤ L′ ≤ λ ≤ L ≤

√β

x\±h4LKV|5~>��

L(≥ 0)WUnw:pZ�NQ�AZ6@AnSjV<=N58A?AWb�N5ZFWU@A<S�A<w>V�A<=WJ��>Vn=D�:ED�VWbWU:

L′ ZFNÉ�AZF@inp��WU:=D6:=W�5N�ZFWu@i<�A<w>V�A<=WJ8ATUj�N�:=D6VW�4

Kh4�sh>VD�: |ψL〉 :EWUZµ�;@AW {~L2|ψL〉 = ~2β|ψL〉Lz|ψL〉 = ~L|ψL〉

x\±h4LK5±V~

¾¨@ADFnw�;@AWL

Wunw:�Z�Na�iZF@An�jV<EN58i?AW��N5ZFWU@A<p�A<=>V�i<=WÉ��>Vn=nwDF AZ6WI?iWLz

WU:b�A@iDFn=�;@AWL+|ψL〉Wunt:gVWuBC:EWU@A<��A<w>V�A<wWH?iW

Lz?AWb�N5ZFWU@A<S�A<=>5�A<=W

~(L+ 1)Á/>58�?i>VD6:gN25>VDF< �

L+|ψL〉 = 0 ⇒∥∥∥L+|ψL〉

∥∥∥2

= 0 ⇒ 〈ψL|L−L+|ψL〉 = 0

�(N�DFnL−L+ = L2

x + L2y + i

[Lx, Ly

]= ~L2 − L2

z − ~Lz

⇒ L−L+|ψL〉 = ~2(β − L2 − L

)= 0

⇒ β = L (L + 1)

x\±h4LKV�5~

¦ � ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

� WJ`��u`-W5ÁABU>V8AnwDF?ATU<=>V8An

|ψL′〉 :

{~L2|ψL′〉 = ~2β|ψL′〉Lz|ψL′〉 = ~L′|ψL′〉

x�±i4LK�¬�~

¾¨@AD6n=�;@AWL′ WUnw:pZ�NÉ�AZ6@Ang�1WC:ED�:EW��N5ZFWU@A<S�A<w>V�A<=Wb8iTuj�N�:=D65W5Á/>V8�?A>VD�:gN2V>VD6<

L−|ψL′〉 = 0 ⇒∥∥∥L−|ψL′〉

∥∥∥2

= 0 ⇒ 〈ψ′L|L+L−|ψ′

L〉 = 0

�(N5D6nL+L− = ~L2 − L2

z + ~Lz

⇒ L+L−|ψL′〉 = ~2(β − L′2 + L′) = 0

⇒ β = L′ (L′ − 1)

x�±i4LK�z�~

��8©?A>VD�:g?A>V8ABHN2V>VD6<

L′ (L′ − 1) = L (L + 1) ⇒{L′ = −L >V@L′ = L + 1

x�±i4em��~

M�NQ?AWu@ �iD6ku`-WJn=>VZ6@i:ED6>V8+Wunt:H7Q<wW �wWU:=Wu<g�A@AD6n=�;@AWL′ ≤ L

4ogTuBuN5�AD6:=@AZ�N$:EDF>58º�A<=>{ DFnw>VDF<wW � MµWunJVWuBC:EWU@A<=nI�A<=>V�i<=WunHBu>5`a`É@i8An7 ~L2 WU:

LzBu>V<w<=WUntq

��>V8A?/N�8;:87a@A8iWI�N5ZFWU@A<p�A<=>V�i<=W~2L(L + 1)

?iW ~L2 >58;:�?AWUn��N5ZFWU@A<=np�A<=>5�A<=WUn�?AWLz?AWJZFN�v&>V<=`-W

~λN2VWUB −L ≤ λ ≤ L

4mi4�sh>5D6:

|ψ〉 :

{~L2|ψ〉 = ~2L(L + 1)|ψ〉Lz|ψ〉 = ~L|ψ〉

x�±i4emh3}~

WC:�N��A�AZFD6�;@A>V8Anrq9ZF@iD�Z [ >V��Tu<EN$:EWu@i<L−

?AW�v�NM�u>V8y<=TU�1TC:ETUW54��8�WU8AjVWu8i?A<=Wa?AWunJVWUBU:EWU@A<=n�A<w>V�A<wWun_?AW ~L2 WU:

Lz?AW�5N�ZFWu@i<=nY�i<=>V�A<wWun

~2L(L+ 1)�1>5@A< ~L2 x

L−BU>V`a`�@i:EW�N{5WuB

~L2 ~YWU: ~(L− 1), . . . , ~(L− n)��>V@A<

Lz4

�(N�DFn −L Wunt:HZ�N �AZ6@AnJ�1WC:ED�:EW��N5Z6Wu@A<b�A<w>V�A<wWQ�1>Vnwn=D6 AZFW�4µ¾�>V@i<b8AW��/N5nbWu8ijVWu8A?i<=Wu<H?AW5WuBU:=Wu@A<wna�i<=>V�A<wWun�?AWº�N5Z6Wu@A<wnQ�A<=>V�i<=Wun�D68iv&Tu<wDFWu@i<=Wun 7 −L Á¨DFZÀv�N5@i:Q�;@AW©BUWU:=:=W©n=Tu<wDFWBU>V8�:EDFWU8A8AWH@A8�5WuBC:EWu@i<��A<w>V�A<=WH?AWJ�N5ZFWU@A<S�A<=>5�A<=W −L Á�N5@i� @iWuZOB2N5npZ [^N5�A�AZ6DFBuN�:ED6>V8º?AWhL−

?i>V8A8AWU<EN �i4

⇒ ∃n :=WuZµ�;@AWL− n = −L

⇒ L =n

2

⇒ LWunt:�WU8�:EDFWU<g>V@©?AWu`-D�q Wu8�:ED6Wu<u4 (

L = 0,1

2, 1,

3

2, 2, . . .

)x�±i4emVKV~

�Y>V8ABUZF@AnwDF>V8 �

� �F¡��À%�ÆcÇ¨% �50{!}%»'�% ~L2 ��I�1.O)2#�'O�;!}�i0}#&��.�) +¥ "� ÃO!$#\�c,O% ) ¦

MµWunS�N5ZFWU@A<=nS�A<w>V�A<wWunp�1>Vnwn=D6 AZFW�?iW ~L2 n=>V8�: �

~2L(L + 1), L = 0,1

2, 1, . . .

x�±i4em5m�~

¾�>V@i<O@i8L?i>V8A8AT�Á{ZFWunµ�N5Z6Wu@A<wnO�i<=>V�A<wWun��>Vn=nwDF AZ6Wun�?AW

Lz?AWUn�5WuBC:EWu@i<=nÈ�i<=>V�A<wWunOBu>V`-`É@A8in

7 ~L2 WU:Lz

nw>V8�: �~M, M = −L,−L + 1, . . . , L

x\±i4^m5�;~�Y>V8Ant:E<=@iBU:ED6>V8�?AWJZ�NÉ /N5nwW ��8É8A>5:=W |L,M〉 ZFW�VWUBU:=Wu@A<¨�A<w>V�A<wWYBu>5à`É@i8 7 ~L2 WU: Lz ?AWY�N5ZFWU@A<=nO�A<w>V�A<=WUn ~2L(L+1)WU:~M

4"¾�N5<w:EN58�:p?AW |L, L〉 Á/>V8�N

|L, L− n〉 ∝ Ln−|L, L〉x�±i4emV|V~

��>V<w` N�ZFDFn=N�:ED6>V8 � s @A�A��>Vn=>V8in |L,M〉 8A>V<w` N5Z6DFnwT54∥∥∥L−|L,M〉

∥∥∥2

= 〈L,M |L+L−|L,M〉 = ~2 (L(L + 1) −M(M − 1))x\±h4emV±V~

��8��Wu@i:p?A>58ABJBªRA>VDFnwDF<

L−|L,M〉 = ~√L(L + 1) −M(M − 1)|L,M − 1〉 x\±h4em��5~

� WH` �UàW�Á∥∥∥L+|L,M〉

∥∥∥2

= 〈L,M |L−L+|L,M〉 = ~2 (L(L+ 1) −M(M + 1))x\±h4emV¬V~

WU:p>V8©��Wu@i:�N5D68An=D�BªRi>VDFnwDF<

L+|L,M〉 = ~√L(L+ 1) −M(M + 1)|L,M + 1〉 x\±h4emVzV~

opWU` N�<=�;@AW � �YWUn_<=TUn=@AZ�:ªN�:=nYnw>V8�:_�N5Z�N5 iZFWunÀ��>V@A<�:E>V@i:Y>V��Tu<EN$:EWu@i< VWUBU:=>V<=D6WuZcn=N�:ED6nwv�N5D6nEN58�: ZFWUn<=kUjVZFWUnp?iWHBU>V`-`É@i:ªN$:EDF>58[Li, Lj

]= i~εijkLk

x\±h4^� �V~��8»:=WuZÈ>5�1TU<EN�:=Wu@A<�nu[^N5�A��WuZFZ6W5Á��/N5<�W � :EWU8An=D6>V8�Ác@i8 1 BO1 F^PRL @^Q PdY^LTQTSVUWF x �uN�8AjV@AZFN5<�`->V`-Wu8hq:E@A` �SWu8ºN�8AjVZ�N�DFnª~ Á5WC:YZ\[^>V��Tu<=N�:EWU@A< ~L = ~r∧ ~p nu[^N5�A��WuZ6ZFWgZ [ >V��Tu<=N�:EWU@A<�?AW 1 BO1 F^PRL�@^Q PdY^LTQTSVUWFBE] VQ L _`D 4h��8a�AN5<=Z6Wp�AN5<YN�DFZFZ6Wu@A<wn¨?AW � `->VàWU8�:�BuDF8iTU:ED6�;@AW�M �gZ6>V<=nw�;@�[^>58 Bu>58An=D6?Aku<wW�@A8a>V��Tu<EN}q:EWU@A<_?AW�à>V`-Wu8�: BuD68ATU:=DF�;@AW�<wWunt:E<=WUDF8�:SN�@ºn=>V@intq9WUn=�/N�BuWb?AWUn_TU:EN�:En �A<=>V�i<=WunY?AW ~L2 ?AWp5N�ZFWu@i<�A<=>5�A<=W

~2L(L + 1)4LWunw:��/N5<tv&>VDFn_N5�A��WuZFTbZFN � �AnwWu@A?A>$q98A>V<wàW �ª4

� � ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

� � � �� N��

¾O>V@A<�:E<EN�D6:EWU<QZFW+BuN5na�AN5<w:=DFBu@iZFDFWU<�?A@ à>5àWU8;:�BUDF8ATC:ED6� @iW�>5<= AD�:ªN5Z\Á�D6ZSWunw:-BU>Và`-> ?AW+?AW�/N5nwn=WU<�WU8�BU>h>V<w?A>V8A8iTuWungnw�ARATu<wDF�;@AWUn �

x = r sin θ cosϕ

y = r sin θ sinϕ

z = r cos θ

⇔

r2 = x2 + y2 + z2

tan2 θ = x2+y2

z2

tanϕ = yx

x�±i4^�i3}~

§¨8�@h:EDFZ6DFn=N58�:SZ�N�v&>V<w`É@AZ6W�?AWunp?ATU<=D65TuWUn�BU>Và��>VnwTuWun �∂f(r, θ, ϕ)

∂xi=∂f

∂r

∂r

∂xi+∂f

∂θ

∂θ

∂xi+∂f

∂ϕ

∂ϕ

∂xi

x�±i4^��KV~

��8�>5 i:ED6Wu8�:3V4 ∂r2

∂xi= 2r ∂r

∂xi

2r∂r

∂x= 2x⇔ ∂r

∂x=x

r= sin θ cosϕ

2r∂r

∂y= 2y ⇔ ∂r

∂y=y

r= sin θ sinϕ

2r∂r

∂z= 2z ⇔ ∂r

∂z=z

r= cos θ

x�±i4^�Vm�~

Kh4 ∂ tan2 θ∂xi

= 2 tan θ 1cos2 θ

∂θ∂xi

2 tan θ1

cos2 θ

∂θ

∂x=

2x

z2⇔ ∂θ

∂x=

cos θ cosϕ

r

2 tan θ1

cos2 θ

∂θ

∂y=

2y

z2⇔ ∂θ

∂y=

cos θ sinϕ

r

2 tan θ1

cos2 θ

∂θ

∂z= −2(x2 + y2)

z2⇔ ∂θ

∂z= −sin θ

r

x�±i4^�5�;~

mi4 ∂ tanϕ∂xi

= 1cos2 ϕ

∂ϕ∂xi

1

cos2 ϕ

∂ϕ

∂x= − y

x2⇔ ∂ϕ

∂x= − sinϕ

r sin θ1

cos2 ϕ

∂ϕ

∂y=

1

x⇔ ∂ϕ

∂y=

cosϕ

r sin θ1

cos2 ϕ

∂ϕ

∂z= 0 ⇔ ∂ϕ

∂z= 0

x�±i4^��|V~

� ��³ ¢£��¶Å% .10+��#�.O��0}#��,�%·¤�!}Ã�#&02�/¥ ��

�YWI�;@ADµ8i>V@Ang�1WU<=`-WU:g?�[ TuBU<=D6<=WJZ [ >V��Tu<EN$:EWu@i<~pWu8�Bu> >V<=?A>58A8ATuWUn�nw�ARATU<=DF�;@AWUn �

px =~

i

∂

∂x=

~

i

(sin θ cosϕ

∂

∂r+

cos θ cosϕ

r

∂

∂θ− sinϕ

r sin θ

∂

∂ϕ

)

py =~

i

∂

∂y=

~

i

(sin θ sinϕ

∂

∂r+

cos θ sinϕ

r

∂

∂θ+

cosϕ

r sin θ

∂

∂ϕ

)

pz =~

i

∂

∂z=

~

i

(cos θ

∂

∂r− sin θ

r

∂

∂θ

)

x\±h4^��±V~

N5DF8in=D��;@AWHZ [ >V��Tu<EN$:EWu@i< ~L �

Lx = ypz − zpy

=~

i

[(r sin θ sinϕ cos θ − r cos θ sin θ sinϕ)

∂

∂r

+

(−r sin θ sinϕ sin θ

r− r cos θ cos θ sinϕ

r

)∂

∂θ

+(−r cos θ

cosϕ

r sin θ

) ∂

∂ϕ

]

⇒ Lx =~

i

[− sinϕ

∂

∂θ− cosϕ

tan θ

∂

∂ϕ

]

x\±h4^�;�5~

Ly = zpx − xpz

=~

i

[(r cos θ sin θ cosϕ− r sin θ cosϕ cos θ)

∂

∂r

+

(r cos θ cos θ cosϕ

r+r sin θ cosϕ sin θ

r

)∂

∂θ

+

(−r cos θ

sinϕ

r sin θ

)∂

∂ϕ

]

⇒ Ly =~

i

[cosϕ

∂

∂θ− sinϕ

tan θ

∂

∂ϕ

]

x\±h4^��¬V~

�"� ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

Lz = xpy − ypx

=~

i

[(r sin θ cosϕ sin θ sinϕ− r sin θ sinϕ sin θ cosϕ)

∂

∂r

+

(sin θ cosϕ

cos θ sinϕ

r− r sin θ sinϕ

cos θ cosϕ

r

)∂

∂θ

+

(r sin θ cosϕ

cosϕ

r sin θ+ r sin θ sinϕ

sinϕ

r sin θ

)∂

∂ϕ

]

⇒ Lz =~

i

∂

∂ϕ

x�±i4^�Vz�~

L+ = Lx + iLy = ~eiϕ(∂

∂θ+ i cot θ

∂

∂ϕ

)

L− = Lx − iLy = ~e−iϕ(− ∂

∂θ+ i cot θ

∂

∂ϕ

)

~L2 = L−L+ + L2z + ~Lz

L−L+ = ~2e−iϕ[− ∂

∂θ+ i cot θ

∂

∂ϕ

]eiϕ[∂

∂θ+ i cot θ

∂

∂ϕ

]

= ~2e−iϕ[−eiϕ ∂

2

∂θ2− eiϕ

∂

∂θ

(i cot θ

∂

∂ϕ

)+ i cot θ

∂

∂ϕ

(eiϕ

∂

∂θ

)

+ i cot θ∂

∂ϕ

(eiϕi cot θ

∂

∂ϕ

)]

= ~2e−iϕ[−eiϕ ∂

2

∂θ2− eiϕ

−isin2 θ

∂

∂ϕ− eiϕi cot θ

∂2

∂ϕ∂θ+ i cot θieiϕ

∂

∂θ

+ i cot θeiϕ∂2

∂ϕ∂θ+ i cot θieiϕi cot θ

∂

∂ϕ+ i cot θeiϕi cot θ

∂2

∂ϕ2

]

= ~2

[− ∂2

∂θ2+ i

1

sin2 θ

∂

∂ϕ− cot θ

∂

∂θ− i cot2 θ

∂

∂ϕ− cot2 θ

∂2

∂ϕ2

]

= ~2

[− ∂2

∂θ2+ i

∂

∂ϕ− cot θ

∂

∂θ− cot2 θ

∂2

∂ϕ2

]

⇒ ~L2 = ~2

[− ∂2

∂θ2+ i

∂

∂ϕ− cot θ

∂

∂θ− cot2 θ

∂2

∂ϕ2− ∂2

∂ϕ2− i

∂

∂ϕ

]

⇒ ~L2 = −~2

(− ∂2

∂θ2+

1

tan θ

∂

∂θ+

1

sin2 θ

∂2

∂ϕ2

)

x�±i4L| ��~

� �F��À% ) ��A!$¶Å��.O#\��,�% )aÆ�Ç�� !$#��,�%h) ��¡

�Y>V8AnwDF?ATU<=>V8Ang@A8iWJv&>58ABU:=DF>V8©�A<w>V�A<=WJ?AWLz �

~

i

∂ψ(r, θ, ϕ)

∂ϕ= ~mψ(r, θ, ϕ) ⇒ ψ(r, θ, ϕ) = χ(r, θ)eimϕ

x\±h4L|h3{~�(N5D6nS�1>V@i<p�;@AW

ψ(r, θ, ϕ)nw>VD6:g@i8AWbv&>V8ABU:=DF>V8©@A8iD6V> �;@AWJ?AWunpBu> >V<w?A>V8A8ATUWunUÁ/DFZcv�N5@h:g� @iW

ψ(r, θ, ϕ+ 2π) = ψ(r, θ, ϕ) ⇒ ei2πm = 1

⇒ mWu8�:ED6Wu<

x\±h4L|VK5~

MµWbn=��WuBU:=<=WH?A@ 1 BO1 F^PRL�@^Q PdY^LTQTSVURF2BE] VQ L _`D Wunw:g�/8AN5ZFWUàWU8;:g?i>V8A8ATJ�/N5< �{ ¿ÀN5Z6Wu@A<wnS�A<=>5�A<=WUnp?iW ~L2 : ~2l(l + 1), l

Wu8�:=DFWu<¿ÀN5Z6Wu@A<wnS�A<=>5�A<=WUnp?iW

Lz : ~m, m = −l, . . . , lx\±h4L|5mV~

opWU` N�<=�;@AW � PO>V@An BuWUnY>V��Tu<=N�:EWU@A<=n 8AW�?iTu��Wu8A?AWU8�:S�;@AW�?AW θ WC: ϕ 4i��[ TU:EN5D6: �A<wTU DFnwDF AZ6Wg @�;@AW�Z6Wun�Bu>V`-��>VnEN58�:=WunÀ?A@ `->V`-Wu8�:�BuD68ATU:=DF�;@AWgBu>V`-`É@i:=Wu8�:YN{5WuB�ZFWS�A<w>h?A@iD6:À�/N5<�8�[ DF`-�1>5<w:EW�;@AWuZ6ZFWÉv&>V8ABC:ED6>V8 ?AW

r4 � @yBu>V@A�cÁ�>V8·�1WU@i:JBªRAWU<=BªRAWU<�ZFWUn�v&>V8ABC:ED6>V8Anb�A<w>V�A<=WUnJBu>V`-`É@A8AWUn[7

~L2 WC:Lz

BU>V`-àW-v&>V8ABU:=DF>V8inÉ?AWθWC:ϕ

@A8AD6�;@AWu`-Wu8�:24�M�W �A<=> ?A@AD�:É?�[ @A8AW+:EWUZFZ6W-v&>V8iBU:ED6>V8�A<=>5�A<=WJ�/N5<S8c[^D6à��>V<w:=W��;@AWuZ6ZFW�v&>V8iBU:ED6>V8©?AW

rn=WU<EN�Wu8ABU>V<=WJv&>V8ABC:ED6>V8+�A<=>5�A<=WJ?AW ~L2 WC:

Lz4

� � � � � � � �[��

�YRAWu<wBªRA>V8AnÈ?ATunw>V<=àN5DFn�Z6Wun�v&>V8ABU:=DF>V8in��A<w>V�A<wWunÈBU>V`-`É@A8AWUn�7 ~L2 WU: Lz ?iW �N5ZFWU@A<=nO�A<w>V�A<=WUn~2l(l + 1)

WU:~m

4 � [^N5�A<wkunpBuWJ�;@ADµ�A<=TUBukU?AW5Á">V8��1WU@i:pZFWUnSBªRAWu<wBªRAWu<�n=>V@AngZFN�v&>V<=`-W �Y ml (θ, ϕ) = Fm

l (θ)eimϕx\±h4L|��~

M�NQv&>58ABU:=DF>V8Y ll (θ, ϕ)

?A>VD6:gn=N�:ED6nwv�N5D6<=W �L+Y

ll = 0

⇒(∂

∂θ− l cot θ

)F ll (θ) = 0

x\±h4L|V|5~

�(N5D6nSZ [ Tu�;@/N�:=DF>V8©?AD �cTU<=Wu8�:=DFWuZ6ZFWdf

dθ= l

cos θ

sin θf

x\±h4L|5±V~NQ�1>V@i<pnw>VZF@h:EDF>58

f(θ) = (sin θ)l4"§�8�W��1WU:uÁ

df

dθ= l cos θ(sin θ)l−1 =

l cos θ

sin θf

⇒ Y ll (θ, ϕ) ∝ (sin θ)leilϕ

x\±h4L|V�5~

�A³ ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

��>5<=àN5ZFD6nEN�:=DF>V8 �shDY ll (θ, ϕ) = cl(sin θ)

leilϕÁ"N5ZF>5<=n

|cl|2∫ π

0

sin θdθ

∫ 2π

0

dϕ(sin θ)2l = 1

n=>VD�: |cl|2 × 2π × Il = 1N25WuB

Il =

∫ π

0

sin θdθ(sin θ)2l

x�±i4L|�¬�~

��8�v�N�D6:SZ6WbBªR/N58AjVWUàWU8�:�?iWJ�N5<wD�N5 iZFWun

u = cos θ, du = − sin θdθ, (sin θ)2 = 1 − u2

⇒ Il =

∫ 1

−1

du(1 − u2)lx�±i4L|�z�~

WU:Il

n=WHB2N�ZFBu@iZFW��/N�<p<wTuBU@A<=<wWu8ABUW �I0 = 2

Il =

∫ 1

−1

du(1 − u2)l−1(1 − u2)

= Il−1 −∫ 1

−1

du u× u(1 − u2)l−1

= Il−1 −[−[u

(1 − u2)l

2l

]1

−1

−∫ 1

−1

du(1 − u2)l

−2l

]

= Il−1 −1

2lIl

⇒ Il

(1 +

1

2l

)= Il−1

⇒ Il =2l

2l + 1Il−1 =

(2l)(2l − 2) · · ·2(2l + 1)(2l − 1) · · ·3 × I0

nw>VD6:Il =

[(2l)(2l − 2) · · · 2]2

(2l + 1)!× I0

>V@�Wu8ABU>V<=WIl =

22l+1(l!)2

(2l + 1)!

x�±i4e±��~

¯ÈD68/N5ZFWUàWU8�:2Ái>58�?i>VD6:pN2V>5DF<

|cl|2 =(2l + 1)!

4π

1

22l(l!)2

x�±i4e±h3}~MµWJBªRA>VD �ºBu>V8�5Wu8�:ED6>V8A8AWUZ�BU>V8An=D6nw:=W�7Q�A<wWu8A?i<=W

cl =(−1)l

2ll!

√(2l + 1)!

4π

x�±i4e±VKV~

� �F��À% ) ��A!$¶Å��.O#\��,�% )aÆ�Ç�� !$#��,�%h) ��

MµWun N5@i:E<wWun¨v&>V8ABC:ED6>V8An¨nu[ Wu8a?ATu?i@ADFnwWu8�:Y�AN5<ÀN5�A�AZ6DFB2N$:EDF>58Énwf nw:=Tu`aN�:ED6�;@AWg?AWL−

Á�n=N5BªR/N58�: �;@AW

Y m−1l (θ, ϕ) =

1

~√l(l + 1) −m(m− 1)

L−Yml (θ, ϕ)

N{5WuBL− = ~e−iϕ

[− ∂

∂θ+ i cot θ

∂

∂ϕ

] x\±h4e±VmV~

§ �iWU`a�iZFWun �3V4l = 0

Y 00 (θ, ϕ) =

1√4π

x\±h4e±5��~

Kh4l = 1

Y 11 (θ, ϕ) = −1

2

√6

4πsin θeiϕ

⇒ Y 11 (θ, ϕ) = −

√3

8πsin θeiϕ

Y 01 (θ, ϕ) =

1√2e−iϕ

(−√

3

8π

)[− cos θeiϕ + i

cos θ

sin θsin θieiϕ

]

=1√2

√3

8π2 cos θ

⇒ Y 01 (θ, ϕ) =

√3

4πcos θ

⇒ Y −11 (θ, ϕ) =

√3

8πsin θe−iϕ

x\±h4e±�|5~

mi4l = 2

Y ±22 (θ, ϕ) =

√15

32πsin2 θe±2iϕ

Y ±12 (θ, ϕ) = ∓

√15

8πsin θ cos θe±iϕ

Y 02 (θ, ϕ) =

√5

16π

(3 cos2 θ − 1

)

x\±h4e±V±V~

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:=DF�;@AWunU4"§¨ZFZFWUnSn=>V8�:p?A>V8iBH>5<w:ERi>VjV>V8/N�ZFWun �∫ 2π

0

dϕ

∫ π

0

dθ sin θ (Y ml )† Y m′

l′ = δl,l′δm,m′

x�±i4e±V�V~

Kh4pM�N�?ATu��Wu8i?/N58ABUW�WU8θ?AWUn�RAN5<=`->V8AD6� @iWun+n=�iRATu<wDF�;@AWun�Wunt:�<=WuZ6DFTUW 7�?AWun©v&>V8ABC:EDF>58An

�/N�<w:ED6Bu@AZ6DFkU<=Wun_N5�A��WuZFTUWun >RBEDKJEP��O1 FCc2F^L - BEPd@^LTQhBEPkca[FIb F <F^P [`]>F 4mi4��8©�1WU@i:pTU:EN5 AZFD6<_Z [ W �h�A<wWunwn=DF>58�jVTU8ATu<=N5ZFWbnw@AD6�N58�:=W@� �

Y ml (θ, ϕ) =

(−1)l

2ll!

√2l + 1

4π

(l +m)!

(l −m)!eimϕ(sin θ)−m

dl−m

d(cos θ)l−m(sin θ)2l

x�±i4e±5¬�~

�A4��8�N-WU8��AN5<w:=DFBu@iZFDFWU<G! �

Y −ml (θ, ϕ) = (−1)m (Y m

l (θ, ϕ))∗x�±i4e±5z�~

� � � � � ��0� � � � � � �� 6��

¾O>V@A< @i8AW(�N5ZFWU@A<�?A>V8A8iTuW»?AWlÁ >V8 �1WU@i:º?i>V8AB(BªRiWu<=BªRiWu<ºZ6Wun TC:ªN�:=nant:ªN�:=DF>V8i8/N5DF<wWun-?AW

Z [ Tu�;@/N$:EDF>58+?AWIshBªRA<=« ?AD68AjVWu<g?�[ @A8AWJ�/N�<w:ED6Bu@AZ6Wb?/N58Ang@A8��>5:EWU8;:=DFWUZ�BuWU8;:=<EN5Zµnw>V@AnpZ�N�v&>V<wàW �

ψ(r, θ, ϕ) = R(r)Y ml (θ, ϕ)

x�±i4L� ��~

>��R(r)

Wunw:Ànw>VZF@i:=DF>V8I?AWSZ [ Tu�;@/N$:EDF>58Q� @iDAnu[ >V i:=DFWu8�:¨WU8�<=Wu`-�AZFNM�2N58�: ~L2 �/N5< ~2l(l+1)Á�n=>VD�: �

[− ~2

2m

∂2

∂r2− ~2

mr

∂

∂r+

~2l(l + 1)

2mr2+ V (r)

]R(r) = ER(r)

x�±i4L� 3}~

#%$'&�(E)D5@>W)D3,Z43,LN5.CE3 � &�7.3dI� � >WI@&�-@1�� ( � �X(E- � �R>WCE&��^5.5bO�� .OP � �,LN36)HKÙ�Kd)?3dI@+`3 �45bO�� O

� � ��À% ¢¼��,��1% ¶Å% .10a'��/.O)a,O.�Â��/0{% .10}#&% ¥S�I��,�¥&��¶ ÃÈ#&% . �c¦

79ZµWunt:�BU>V`-à> ?AW�?AWJBªR/N�8AjVWu<p?AWbv&>V8ABC:ED6>V8+DF8ABU>V8A8;@AWJWU:p?AWJ��>Vn=WU<

R(r) =1

ru(r)

dR

dr= − 1

r2u(r) +

1

r

du

drd2R

dr2=

2

r3u(r) − 2

r2

du

dr+

1

r

d2u

dr2

⇒ − ~2

2m

[2

r3u(r) − 2

r2

du

dr+

1

r

d2u

dr2− 2

r3u(r) +

2

r2

du

dr

]

+

[~2l(l + 1)

2mr2+ V (r)

]u(r)

r= E

u(r)

r

nw>VD6: [− ~2

2m

d2

dr2+

~2l(l + 1)

2mr2+ V (r)

]u(r) = Eu(r)

x\±h4L�VK5~

� N58AnHZFWQBuN5nH�/N5<w:=DFBU@AZFD6Wu<�?A@��1>�:EWu8�:=DFWuZ¨BU>V@AZ6>V`É AD6Wu8·?�[ DF8�:EWU<EN5BC:EDF>58yWU8�:E<=W-@A8�8A>{f�N5@�?AWBªR/N5<wjVW

+ZeWC:�@A8-TuZFWUBU:=<=>V8�?AWgBªR/N5<wjVW −e Á V (r) = −Ze2

r

4V¾�>Vnw>V8Anα = +Ze2 ⇒ V (r) =

−αr

4�shDl 6= 0

Ái�1>V@i<r�1WC:ED6:uÁi>V8��Wu@i:p8ATUjVZFD6jVWu< −α

r

?AWC�N58�: ~2l(l+1)2mr2

ÁA?�[ >��

d2

dr2u(r) ' l(l + 1)

r2u(r)

x\±h4L�5mV~

sh@A�A��>Vnw>V8Anp� @iWu(r)

nw>VD6:g?iWHZFN�v&>V<=`-Wrs �

⇒ s(s− 1)rs−2 = l(l + 1)rs−2

⇒ s(s− 1) = l(l + 1)

⇒ s = l + 1>V@

s = −l

x\±h4L��~

�(N5D6nYZ�NJBu>58A?AD6:=DF>V8-?AW�8A>V<=àN5Z6DFnEN$:EDF>58Qn=@A<R(r)

D6à��>Vn=Wg�;@AW ∫ +∞0

r2R2(r)drn=>5D6:ÀBu>V8�5Wu<tq

jVWu8�:=W5Á/B5[ Wunt:wq 7$q9?AD6<=W��;@AW ∫ +∞0

u2(r)drn=>VD�:SBu>V8�5Wu<=j5Wu8�:EW�4�79Z1v�N5@i:g?A>V8iBbBªRA>VDFnwDF<_ZFNÉn=>VZ6@i:ED6>V8

�1>5n=D6:=D65Ws = l + 1

4shDl = 0

ÁA>V8�>V i:=DFWU8;:g�i<=kung?AWJZ\[^>V<wDFjVD68AW(r ' 0) �

d2

dr2u(r) ∝ −u(r)

rs Du(r) = a0 + a1r + a2r

2 + · · ·

⇒ d2

dr2u(r) = 2a2 + · · · , BUWI�;@ADµD6à��>Vn=W

a0 = 0

x\±h4L�V|5~

�Y>V8ABUZF@AnwDF>V8 � � N58ing:=>V@AngZFWUngB2N5nUÁ/>V8©?A>VD6:pN25>VDF<S��>V@A<r��WU:=D6: �

u(r) ∝ rl+1 x\±h4L�5±V~

�� ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 5 10 15 20

U(r

)

r

l = 0l = 1

� �� ±i463 � �J<=N5�ARAWb?iWHZFN�v&>V8ABC:EDF>58U(r) = l(l+1)

r2− 1

r

�1>V@i<l = 0

WU:l = 1

4

¾ÈN�<JN5DFZ6ZFWu@i<=nuÁ"Z6WÉ�1>�:EWu8�:=DFWuZ �UW��1WuBC:ED�v �U(r) = ~

2l(l+1)2mr2

− αr

NºZ [^N5Z6ZF@A<wWI<wWu�A<wTun=WU8�:ETuW-n=@i<bZ�N¯ÈD6jV@A<=WH±i4�3V4 � N58inp:=>V@An�Z6WungB2N�nuÁ�>V8(nU[eN�:w:EWU8A?»?A>V8iB@7�:E<=>5@iVWU<�?AWUn�TU:EN�:EnpZFD6Tung@A8AD6�;@AWu`-Wu8�:��>V@A<

E < 04A¾�ZFNM�u>58Antq 8A>V@Ang?/N58inpBUWJB2N5nUÁ/WU:gv�N5DFnw>V8AnSZ6WbBªR/N58AjVWU`aWU8�:�?iWJ�N5<wD�N5 iZFWun

λ =

√− E

EI, ρ =

r

a0

N2VWUBEI =

mα2

2~2, a0 =

~2

mα

x�±i4L�5�V~

� � ��À% ¢¼��,��1% ¶Å% .10a'��/.O)a,O.�Â��/0{% .10}#&% ¥S�I��,�¥&��¶ ÃÈ#&% . �

M¨[^TU� @AN�:ED6>V8©?AWU DFWU8�:

[− ~2

2m

1

a20

d2

dρ2+

~2l(l + 1)

2ma20ρ

2− α

a0ρ

]u(ρ) = −EIλ2u(ρ)

⇒[d2

dρ2+

~2l(l + 1)

2ma20ρ

2

−2ma20

~2+

2ma20α

~2a0ρ

]u(ρ) =

2mEIa20

~2λ2u(ρ)

⇒[d2

dρ2− l(l + 1)

ρ2+

2

ρ

]u(ρ) =

2m2α2

2~4

~4

m2α2λ2u(ρ)

nw>VD6: [d2

dρ2− l(l + 1)

ρ2+

2

ρ− λ2

]u(ρ) = 0

x\±h4L�5¬V~

��@/N58A?ρ→ +∞ ÁA>V8�N�N5�A�A<w> �iD6` N$:ED65Wu`-Wu8�:

d2u

dρ2' λ2u⇒ u(ρ) ' e±λρ

x\±h4L�5zV~

M�N�BU>V8A?AD�:ED6>V8+?AWH8A>V<=àN5Z6DFnEN$:EDF>58aD6à��>Vn=Wb?iWH<wW �wWC:EWu<gZFNÉn=>VZ6@i:ED6>V8eλρ

4�YRAWu<wBªRA>V8Anp?A>V8iBHZFNQn=>VZ6@i:ED6>V8+n=>V@inpZFN�v&>V<=`-W

u(ρ) = e−λρy(ρ)4

du

dρ= −λe−λρ + e−λρy′

d2u

dρ2= λ2e−λρy − 2λe−λρy′ + e−λρy′′

⇒[d2

dρ2− 2λ

d

dρ+

2

ρ− l(l + 1)

ρ2

]y(ρ) = 0

x\±h4e¬��V~

�YRAWu<wBªRA>V8An�Z�NÉnw>VZF@i:=DF>V8+nw>V@AnpZ�N�v&>V<wàWb?c[^@A8�?ATCVWuZ6>V�A��Wu`-Wu8�: �

y(ρ) = ρl+1

+∞∑

q=0

cqρq =

+∞∑

q=0

cqρq+l+1 x\±h4e¬i3{~

� ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

MµWJ�A<=TCv�N5BU:=Wu@A<S DFWU8;:p?AWJZFNÉBu>V8A?iD6:ED6>V8u(r) ∝ rl+1 �1>V@i<

r�1WC:ED�:24

dy

dρ=

+∞∑

q=0

cq(q + l + 1)ρl+q

d2y

dρ2=

+∞∑

q=0

cq(q + l + 1)(q + l)ρl+q−1

⇒+∞∑

q=0

[cq(q + l + 1)(q + l)ρl+q−1 − 2λcq(q + l + 1)ρl+q + 2cqρ

l+q

−cql(l + 1)ρl+q−1]

= 0

⇒{c0(l + 1)lρl−1 − c0(l + 1)lρl−1 = 0

cq(q + l + 1)(q + l) − 2λcq−1(q + l) + 2cq−1 − cql(l + 1) = 0

n=>5D6:cq [(q + l + 1)(q + l) − l(l + 1)] = 2cq−1 (λ(q + l) − 1)

x�±i4e¬VKV~

�;@AWHZ [ >V8+�1WU@i:p<=TUTuBu<wDF<wW

cqq(q + 2l + 1) = 2cq−1(λ(q + l) − 1)

⇒ cq = cq−12 (λ(q + l) − 1)

q(q + 2l + 1)

x�±i4e¬5m�~

]£�A<=D6>V<=D\Áh�;@/N�8A?q → +∞ Á

cq ∝ cq−12λ

q

⇒ cq ∝1

q!2λ

⇒ y(ρ) ' ρl+1e2λρ

⇒ u(ρ) ∝ eλρ

x�±i4e¬��;~

�;@AD"8�[ Wunw: N5@h:E<=Wg�;@AWpZ�Nbn=>5ZF@i:=DF>V8Q<wW �wWU:=TuWp�A<=TUBuTU?AWu`-`aWU8�:24;79ZAv�N5@i:À?A>V8iBp�;@AWpZFWS8;@A`-Tu<=N�:EWU@A<nu[^N58A8;@AZ6W5Á"N5@h:E<=WU`aWU8�:�?iD6:g�;@�[ DFZ�W �iD6nw:=WH@i8�WU8;:=DFWU<

k ≥ 1:=WuZµ�;@AW

λ(k + l) − 1 = 0

⇒ λ =1

k + l

x�±i4e¬V|V~

MµWungTu8iWu<=j5DFWung��>Vn=nwDF AZ6Wungn=>58;:p?A>58A8ATuWUn��AN5<

Ek,l = − EI(k + l)2

k = 1, 2, . . .x�±i4e¬5±�~

� � ��À% ¢¼��,��1% ¶Å% .10a'��/.O)a,O.�Â��/0{% .10}#&% ¥S�I��,�¥&��¶ ÃÈ#&% . ��

MµWungn=>VZ6@i:ED6>V8Anyk,l(ρ)

nU[^>5 i:ED6Wu8A8AWU8�:H7��/N5<t:ED6<_?AWJZ�NÉ<wWuZ�N$:EDF>58+?AWJ<=TuBU@A<=<wWu8ABUWInw@A<gZFWUncq4

§ �iWUà�iZFWun �3V4

k = 1, l = 0

λ = 1, c1 = 0

⇒ u1,0(ρ) = c0ρe−ρ

x\±h4e¬��5~

Kh4k = 1, l = 1

λ =1

2, c1 = 0

⇒ u1,1(ρ) = c0ρ2e−

ρ2

x\±h4e¬V¬V~

mi4k = 2, l = 0

λ =1

2, c1 = c0

2(

12× 1 − 1

)

1(1 + 1)= −1

2c0

⇒ u2,0(ρ) = c0ρ(1 − ρ

2

)e−

ρ2

x\±h4e¬VzV~

¯ÈDF8AN5ZFWUàWU8;:uÁ�Z6WunÈv&>58ABU:=DF>V8AnÈ<EN5?ADFN5ZFWUnÈnU[^>V h:EDFWU8A8AWU8;:ÀWU8-<wWu`-�AZ�N;�2N58�:ρ�/N5< r

a0

ÁVWU8-?iD6 DFn=N58�:�/N5<

rÁAWU:pWu8�8A>V<w` N�ZFDFn=N58�:_Z6Wb<=Tunw@AZ6:EN�:g�/N5<SZFNQBu>V8A?iD6:ED6>V8

∫ +∞

0

R2r2dr = 1x\±h4ez��V~

opTUB2N5�iD6:E@iZ�N�:=DF>V8 � M�WUnpv&>58ABU:=DF>V8AnS<=N5?ADFN5ZFWUn_n=>V8�:p?A>V8i8ATuWUn��/N�<

Rk,l(r) =1

re−λ r

a0

(r

a0

)l+1[c0 + c1

(r

a0

)+ ...+ ck−1

(r

a0

)k−1]

x\±h4ezi3{~

N2VWuBcq = cq−1

2 (λ(q + l) − 1)

q(q + 2l + 1)

WC:λ =

1

k + l

x\±h4ez�K5~

c0Wunt:p?ATU:=Wu<=`-DF8iTH�AN5<SZ�NÉBU>V8A?AD�:EDF>58+?AWJ8A>V<=àN5Z6DFnEN$:EDF>58 �

∫ +∞

0

r2(Rk,l(r))2dr = 1

x�±i4^zVm�~

"� ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

§ �hWu`-�AZFWUn �

R1,0(r) = c0e− r

a0

c0WUnw:�?A>V8A8iTH�AN5< �

∫ +∞

0

c20e− 2r

a0 r2dr = 1

�(N�DFn ∫ +∞

0

x2e−axdx =d2

da2

[∫ +∞

0

e−axdx

]=

d2

da2

(1

a

)=

2

a3

⇒∫ +∞

0

c20e− 2r

a0 r2dr = 2 ×(a0

3

)3

=a3

0

4

⇒ c0 = 2a− 3

20

⇒ R1,0(r) = 2a− 3

20 e

− ra0

x�±i4ez��;~

��8�:=<=>V@hVWJ?AWJ` �UàW�Á/N2VWUB ∫ +∞0

rne−ardr = n! a−1−n

R1,1(r) = (2a0)− 3

2r√3a0

e− r

2a0

R2,0(r) = 2(2a0)− 3

2

(1 − r

2a0

)e− r

2a0

x�±i4ezV|V~

]��i�AZFD6B2N�:=DF>V8 � Z\[^]_:=>V`-W�?�[^dpf ?A<=>5jVku8AW

V (r) = −e2

r

⇒{EI = me4

2~2 = 13.6 [eV ]

a0 = ~2

me2= 0.52 [ � ]

x�±i4ez5±�~

79Z_WUnw:Q:E<=N5?AD�:EDF>58A8AWuZÀ?AW©<=WU�A<=TUn=WU8;:=Wu<�ZFWUnQ8AD65W2N5@ ��?�[ Tu8AWU<=jVD6W+Wu8�v&>V8ABC:ED6>V8�?AWlBU>V`-àWºZ6W

`->V8�:E<=WbZFNÉ�/jV@A<wWJ±i4LK 4M¨[ Tu8AWU<=jVD6W�8AW�?ATU�1WU8A?��;@AW�?AW

k + l4 � @�Bu>5@A��Á�h@ Z [ DF`-�1>5<w:ªN�8ABuWº?iW�Z\[^TU8AWu<wjVDFW©��>V@A<

ZFWUn �A<w>V�A<wDFTU:=Tun-�AR�f n=DF�;@AWUnuÁ_D6ZgWunw:ºR/N5 AD�:E@AWUZp?iW(<=WUà�iZ�N5BUWu<-ZFW�8A>V`É A<wW(�;@/N58�:=DF�;@AWk�/N5<

n = k + lÁ�WU:É?iWº<=WU�1TU<=Wu<IZFWUn�v&>V8ABC:EDF>58AnI?c[^>V8i?AWunQ�/N�<

(n, l,m)4nnu[^N5�A��WuZFZ6W DGF PdBO1 V]>FSVUk_`PRL QTSVURF?> ]gQ Pd@^Q >k_`D 4

XYD6Wu8 �;@AWQZ\[^TU8AWu<wjVDFWI8AWÉ?ATU�1WU8A?AWQ�;@AWQ?AWnÁcWC:b�;@AWQZ�N-BuZ�N�n=n=D��/B2N$:EDF>58»�/N5<

(n, l,m)n=>5D6:

Bu>5à�AZ6kU:=W5Á/DFZµ��Wu<wn=D6nw:EWH?AWUn�?ATUjVTu8iTu<=WUn=BUWu8ABUWunu4�§¨8�W��1WU:2Á"@A8iWH�N5Z6Wu@A<g?AWn��Wu@h:�WU8(jVTU8ATu<=N5Z

�U:=<=W©>V i:=Wu8;@AW��1>5@A<Q�AZF@in=DFWU@A<=nQ�N5Z6Wu@A<wnQ?AWlÁ¨�iZF@An��A<wTuBuD6n=TUàWU8;:�:=>V@i:EWUn�ZFWunQ�N5Z6Wu@A<wnQ?AW

lDF8hv&Tu<=D6Wu@A<wWung>V@�Tuj�N5Z6Wun 7n− 1

4 � WI�iZF@AnUÁ/�1>V@i<p@i8AWI�N5Z6Wu@A<S?AWlÁ"DFZ�f�N

2l + 1�N5Z6Wu@A<wnp?iW

� � ��À% ¢¼��,��1% ¶Å% .10a'��/.O)a,O.�Â��/0{% .10}#&% ¥S�I��,�¥&��¶ ÃÈ#&% . �¡

4s 4p 4d 4f

3s 3p 3d

2s 2p

1s

0

−13.616

−13.69

−13.64

−13.6

§ \ W ¿H]

l = 0 l = 1 l = 2 l = 3

�� ±i4eK � ogWu�i<=TunwWu8�:ªN$:EDF>58�?A@�n=��WuBC:E<wWI?AWJZ\[eN�:=>V`-W�?�[ R;f ?A<w>VjVkU8AW54

m��>Vn=nwDF AZ6WunU4A]�@©:E>5:EN5Z ÁhZ�NQ?ATUjVTu8ATU<=WUn=BuWU8ABuWI5N�@i: �

1 + 3 + . . .+ (2n− 1) = (2 − 1) + (4 − 1) + . . .+ (2n− 1)

= 2(1 + 2 + . . .+ n) − n

= 2n(n+ 1)

2− n = n2

x\±h4ez��5~

M�N�?iTujVTU8ATu<wWun=BUWu8ABUWÉ?A@�8AD65W2N5@nWunt:

n2 4opWU` N�<=�;@AWun �

3V4pM�N�?ATUjVTu8ATU<=WUn=BuWU8ABuWp?AWUn�8AD65W2N5@ �QTuZ6WuBC:E<=>58ADF�;@AWUnÈ?AW_Z\[eN$:E>V`-W ?c[^R�f ?A<=>5jVku8AWSWUnw:¨WU8Év�N5D6:2n2 ?A@©v�N5D�:S?A@�n=�AD68Åx�V>VD6<S�AZF@inpZ6>VDF8/~C4

Kh4g79ZifÉNJTUj�N5Z6Wu`-Wu8�:¨?AWUnÀnw>VZF@h:EDF>58AnÈ?ATUBu<wD6�N58�:�Z�N�?AD �c@in=DF>58Q?�[^TUZFWUBU:E<w>V8An¨�/N�<¨ZFWunÈ�A<=>5:=>V8AnU4§¨ZFZ6Wungn=WU<=>V8�:�TC:E@A?iDFTuWUn��iZF@Ang:ªN5<w?�?AN58An�BUWIBU>V@A<wnuÁ/ <EN5D6n=Wu`� AZ�N5 iZFWu`-Wu8�:gWU8 ¾¨R;f nwDF�;@AW��@/N58�:ED6�;@AWJ7=7=7ª4

A³ ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

� �� N��Y>V`-`-Wu8�:��Wu@i:tq9>V8�>V AnwWu<w5Wu<bZFN-?ATUjVTu8iTu<=WUn=BUWu8ABUW-?iWun�8AD�VWuN5@��

n > 1� � N58An�BUW��/N5<=N$q

jV<=N5�ARAW�Á$8i>V@An�N�ZFZF>58An�V>VD6<�� @c[^D6Z nw@ ��:�?AWÀ�AZ6>V8AjVWU<�Z6WÀntf nw:EkUàW ?/N58An�@A8�BªRAN5à�I` N�jV8ATU:=DF�;@AW�4§¨8(TUZFWUBU:E<w>h?hfh8AN5àD6�;@AW�BuZFN5n=nwDF�;@AW�Á/ZF>V<wn=�;@AWJ?AWung�/N�<w:ED6Bu@AZ6WunSBªR/N�<=jVTUWunpn=WH?ATu�AZFN5BuWU8�:�?AN58An

@A8�BuD6<=Bu@iD6:_v&WU<=`-T5Á/WuZ6ZFWUn_Bu<wTuWu8�:�@A8�à>V`-Wu8�:gàN5jV8ATC:ED6� @iW

~M = I ~Sx�±i4ez5¬�~

>��IWUnw:�Z [ DF8�:EWU8An=D�:ETH?A@»Bu>5@A<EN58�:�WU: ~S =

∮~r ∧ d~r WUnw:�ZFWJVWuBC:EWU@A<bn=@A<tv�N5BuWI?A@»BUDF<=BU@AD6:u41�YW

`->VàWU8�:J` N5j58ATU:=DF�;@AW�n=W-Bu>V@i�AZFW�7©@A8ÅBªR/N�à�y` N�jV8ATU:=DF�;@AW�W � :=Tu<=D6Wu@A< ~B Wu8Å�i<=> ?A@ADFn=N58�:@A8AW©BU>V8�:E<=D6 A@i:=DF>V8 − ~M · ~B 7·Z [ Tu8AWU<=jVD6W54��b<UÁ¨�1>5@A<Q@A8 TuZFWUBU:=<=>V8 :E>V@i<=8/N58�:-N5@h:E>V@i<-?c[^@A88A>{f�N�@�Á"Z [ DF8�:EWU8An=D�:ET

I = eT

Á/>��T

Wunt:�ZFNÉ�1TU<=D6>h?AW �

~M =e

2T

∮~r ∧ d~r =

e

2Tm

∮~r ∧md~r

dtdt =

e

2Tm

∮~Ldt

x�±i4ez5z�~

�(N5D6nS�1>5@A<S@A8�à>V@hVWu`-Wu8�:%7Év&>V<=BUWJBuWu8�:=<EN5Z6W5Á ~L WUnw:pBu>V8inw:ªN�8;:

~M =e

2m~L

1

T

∮dt =

e

2m~L

x\±h463��~

MµWbà>5àWU8;:YBuD68ATU:=DF�;@AWbWunt:g?A>V8ABbBU>V@A�AZ6TbN5@+BªR/N�à�©àN5jV8ATC:EDF�;@AW��/N5<_@A8+:=Wu<=`-W�?�[ Tu8iWu<=j5DFW− e

2m~L. ~B

4�YWUBuD <=Wunt:EWÀh<=N5D;Wu8IàTUB2N58AD6�;@AW �;@/N�8;:=DF�;@AW�4 � TUà>58;:=<=>V8intq9Z6W�W �h�AZFD6BuD�:EWu`-Wu8�:u4}MONg�A<wWu`-DFku<wW

TU:EN5��W+Bu>V8AnwDFnt:EW�7 ?ATU:=Wu<wàD68AWu<ÉZFN(v&>V<=`-Wº?AW+Z\[ed�N5`-DFZ6:=>V8AD6Wu8y?�[^@i8AW+�/N5<t:EDFBU@AZFW BªR/N5<=j5TuW©Wu8�A<wTun=WU8ABuWI?�[ @A8�BªRAN5à�©àN5jV8ATC:ED6� @iW54M�N��/N5<w:=DFBU@AZFW�BªRAN5<=jVTUWI?AN58Ang@A8�BªR/N5`-��àN5jV8iTU:ED6�;@AW ��8Qnu[ DF8�:=Tu<=WUn=nwWgN5@É�A<w>V AZFkUàWÀ?�[ @A8AW_�AN5<w:=DFBu@iZFWÀ?AW_BªR/N5<wjVW

e?/N�8An�@A8QBªR/N�à�ÉàN5jV8iTU:ED6�;@AW

~B4"§¨8�àTUB2N58AD6�;@AWbBuZFN5n=nwDF�;@AW5Á/ZFNÉ�/N5<t:EDFBU@AZFWbWUnw:pn=>5@AàD6n=W[7Q@A8iWHv&>V<wBuW

~F = e~v × ~B

WU:pZ\[^TU� @AN�:ED6>V8+?A@�à>V@hVWu`-Wu8�:pnu[ TuBU<=D�: �

md2

dt2~r = e

d~r

dt× ~B

¾�>5@A<��;@/N�8;:=D6�/WU<gZFWJ�A<=>5 AZFkUàW�Á/Z�NQ�A<=WUàD6ku<wWHTC:ªN5��WJWunt:�?�[ TU:EN5 AZ6DF<SZ�NÉv&>V<w`É@AZFN�:ED6>V8+R/N5`-DFZ�:E>�q8AD6Wu8A8AWJ?A@��A<w>V AZFkUàW�4A¾O>V@A<gBuWUZ�NiÁ/TC:ªN5 AZ6DFnwn=>V8AnS?c[eN5 �>V<w?�ZFNÉv&>V<=`-W�?A@�Z�N5j5<EN58Aj5DFWu8c4

¾¨<=>5�1>VnwD6:=DF>V8 �

� ��¦·¢£��¶Å% .10a¢¼� �.��;0{#\��,�% ��

MµWbZ�N58Aj5<EN58Aj5DFWu8L =

1

2m~r2 +

e

c~r. ~A(~r)

Bu>V8i?A@AD6: 7�Z [ Tu�;@/N$:EDF>58+?A@�`->V@i5Wu`-Wu8�:pBªRAWu<wBªRATuWIn=D ~B = ~<w>5: ~A� Tu`->V8Ant:E<=N�:ED6>V8 �M¨[^TU� @AN�:ED6>V8©?�[e§¨@AZ6Wu<rqrM�N5jV<=N58AjVWbnU[^TUBu<=D�: �

d

dt

(∂L∂~r

)− ∂L∂~r

= ~0

∂L∂~r

= m~r +e

c~A(~r)

⇒ d

dt

(∂L∂~r

)= m

d2~r

dt2+e

c

(~r.~∇)~A

∂L∂~r

=e

c~∇(~r. ~A(~r)

)

` N�DFn~∇(~a.~b) = ~a× ~<w>5: ~b + (~a.~∇)~b +~b× ~<=>5: ~a+ (~b.~∇)~a

¾ÈN5<�N5D6ZFZFWU@A<=nUÁY?/N�8An�ZFN v&>V<=`É@iZ�N�:=DF>V8�ZFN5jV<=N58AjVD6Wu8A8AW ?iWyZFN�`aTUB2N58AD6�;@AW5Á~rWC:~rnw>V8�:(?iWun

�N5<=DFN5 AZFWUn_DF8i?ATu��Wu8A?AN58�:EWunU4�M�WUn�?iTu<=D�VTUWunp?AW~r�/N5<S<=N5�A��>V<w: 7

~rnw>V8�:p?A>V8ABJ8;@AZFZ6WunU4

]�D68An=D\Á~∇(~r. ~A(~r)

)= ~r × ~<=>�: ~A+

(~r.~∇)~A

M¨[^TU� @AN�:ED6>V8©?A@�`->V@i5Wu`-Wu8�:pnu[ TuBU<=D6:p?A>58AB �

md2~r

dt2+e

c

(~r.~∇)~A− e

c~r × ~<=>5: ~A− e

c

(~r.~∇)~A = 0

md2~rdt2

= ec~r × ~B

¾�>V@i<É�/N5nwn=Wu< 7(Z�N�v&>5<=`É@AZFN�:ED6>V8·R/N5`-DFZ�:E>V8AD6Wu8A8iW5ÁµDFZ¨v�N5@h:�v�N5D6<=W @i8AW :E<=N58Antv&>V<=`aN�:ED6>V8y?AWMµWujVWU8A?A<=W �

~p =∂L∂~r

= m~r +e

c~A(~r)

H(~r, ~p) = ~p.~r − L(~r, ~r)�(N5D6n

~r =1

m

(~p− e

c~A(~r)

)

⇒ H =−1

2m

(~p− e

c~A(~r)

)2

− e

c

1

m

(~p− ~A(~r)

). ~A(~r) + ~p.

1

m

(~p− ~A(~r)

)

� ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

⇒ H = 12m

(~p− e

c~A(~r)

)2

M�NI�;@/N�8;:=D6�/BuN�:ED6>V8�n=Wgv�N5D6:ÀN�ZF>V<wnÀnwDF`-�AZFWUàWU8�:�Wu8 <wWu`-�AZ�N;�2N58�:~rWC:~p�/N5<�?AWUnY>V��Tu<=N�:EWU@A<=n

nEN$:EDFntv�N5DFn=N58�:gZFWUnS<=kuj5ZFWunS?iWHBU>V`-`É@i:ªN$:EDF>58+R/N5 AD�:E@AWUZFZFWUnu4�Y>V8in=DF?iTu<=>58AnJ?ATunw>V<=àN5DFn�ZFWQ`->V@iVWUàWU8�:J?�[^@i8�TUZFWuBC:E<w>V8·?/N58Anb@A8y��>5:EWU8;:=DFWUZ

VWu8·�A<wTCq

n=WU8ABuWH?�[ @A8�BªR/N5`-��àN5jV8ATC:ED6� @iWb@A8AD6v&>5<=`-W ~B 4"M¨[^RAN5àD6Z6:=>V8ADFWU8 nU[^TUBu<=D�:p?A>V8AB �

H =1

2m

(~p− e

c~A(~r)

)2

+ V (~r)x\±h463��h3}~

��8�NhÁ�Bu>V`-`-W+?�[^RAN5 AD6:=@A?AW�Á�ZFW+BªRA>5D � WU8;:=<=W©�AZ6@An=D6Wu@A<wn-��>5:=Wu8�:ED6WuZFnÉ5WuBC:EWu@i<u4 � N58An-ZFW+BuN5n�A<wTun=WU8�:2Á"DFZ�WUnw:pBu>5à`->h?iW�?AWJBªRA>VD6n=DF<

~A = −1

2~r ∧ ~B

x\±h463��VKV~

��8(NQ AD6Wu8 ~<=>�: ~A = ~BÁ/WC:p?AWJ�AZF@AnS?iD6 ~A = 0

4� TUà>58Anw:=<EN�:=DF>V8 �3V4

~<=>5: (~r ∧ ~B) = ~r?iD6 ~B − (~r · ~∇) ~B + ( ~B · ~∇)~r − ~B

?iD6(~r)

x\±h463��5m�~

MµWung?AWU@��A<wWu`-DFWu<wn_:EWU<=`-Wungn=>V8�:p8;@AZFng�A@iDFn=�;@AW ~B Wunt:�@i8AD6v&>V<wàW�4

( ~B · ~∇)~r =

(Bx

∂

∂x,By

∂

∂y, Bz

∂

∂z

)xyz

= ~B

?AD6(~r) =

∂

∂xx +

∂

∂yy +

∂

∂zz = 3

⇒ ~<=>5: (~r ∧ ~B) = −2 ~B

⇒ ~<=>5: ~A = ~B

x\±h463��;~

Kh4

?AD�(~r ∧ ~B) = ~B · ~<=>5: (~r) − ~r · ~<w>5: ~B x\±h463��V|V~

MµWQ?AWU@��hDFku`-WÉ:EWU<=`-WQWunt:I8;@AZ¨BuN5< ~B Wunt:I@i8AD6v&>V<wàW�4 � W��AZ6@AnuÁ ~<=>5: (~r) = ~0B2N�<b8�[^D68hq

:=Wu<w D6Wu8A8AWU8�:g� @iWb?AWunS?ATU<=D�VTuWUng?�[^@i8AWbBu>V`-��>VnEN58�:=W��/N5<_<=N5�A��>V<w:SN5@��ºN5@i:E<wWunU4A]�D68An=D\Á?AD� ~A = 0

4

� ��¦·¢£��¶Å% .10a¢¼� �.��;0{#\��,�% c¦

��<uÁ/n=Dµ?iD6 ~A = 0Á~pWC: ~A BU>Và`�@i:EWU8;:u4/§¨8�W �cWC:2Á

[~p, ~A

]= [px, Ax] + [py, Ay] + [pz, Az]

��N5DFn[px, Ax(x, y, z)]ϕ(x, y, z) =

~

i

[∂

∂x(Axϕ) − Ax

∂

∂xϕ

]

=~

i

(∂Ax∂x

ϕ+ Ax∂ϕ

∂x− Ax

∂ϕ

∂x

)

=~

i

∂Ax∂x

ϕ

]�DF8in=D Á [~p, ~A

]ϕ =

~

i

(∂Ax∂x

+∂Ay∂y

+∂Az∂z

)ϕ

=~

i

?AD6 ~Aϕ

x\±i4�3��V±�~

��8��Wu@i:p?A>58ABJTuBu<wDF<wW �

H =~p 2

2m− e

mc~A · ~p +

e2

2mc2~A2 + V (~r)

nw>VD6:H =

~p 2

2m+

e

2mc(~r ∧ ~B) · ~p+

e2

8mc2(~r ∧ ~B)2 + V

(~r ∧ ~B) · ~p = −( ~B ∧ ~r) · ~p = − ~B · (~r ∧ ~p) = − ~B · ~L

x\±i4�3��V~

?�[ >��A8/N5ZFWUàWU8�:

H =~p 2

2m− e

2mc~L · ~B +

e2

8mc2(~r ∧ ~B)2 + V

x\±i4�3��V¬�~79Z¨N5�A�AN5<EN��6:�?A>58ABQ@A8 :EWU<=`-WQZFDF8iT2N5D6<=WH�/N5<�<=N5�A��>V<w:JN�@yBªRAN5à� àN5jV8iTU:ED6�;@AW54cM�[ >V��Tu<=N�:EWU@A<1 BO1 F^PRL(1�_=<`PdY^L QTSVURF2BE] VQ L _`D Wunt:�?iTU�/8ADµ�AN5<

~M =e

2mc~L

x\±i4�3��Vz�~

�Y>V`-àWºZ6WunÉ�N5Z6Wu@A<wnQ�A<=>V�i<=Wun��>Vn=nwDF AZ6WunQ?AWLz

n=>V8�:-?AWun�`É@iZ6:ED6�AZFWUn�WU8;:=DFWU<=n�?AW~Á�D6Z_Wunt:

@An=@iWuZ�?�[ DF8�:E<w>h?A@iDF<=W�Z�NQ�;@/N58�:ED�:ETbn=@iD6�N58�:EW

µB =e~

2mc

x\±i4�3V3��~N5�A��WuZ6TuWJZFW � _=<`PdY^L BEP [F �IBfO] 4"�YWH� @iDµWu8�:E<=N��8AW

~M =µB~~L

x\±i4�3V3V3}~

� N58An�@A8�TU:ªN$:s (l = 0)

Á/ZFWJà>5àWU8;:pàN5jV8ATC:ED6� @iWbWunw:p:E>5@��w>V@A<wn�8;@AZ\Á/?/N58Anp@A8(TC:ªN�:pÁ�Z6Wun

�N5ZFWU@A<=ng�A<w>V�A<=WUng�1>Vnwn=D6 AZFWUnS?AW ~Mznw>V8�: −µB, 0, µB Á/WU:�N5D68An=Dc?AWJn=@AD�:EW54

� ¢¼�1,��% ¶Å% .10a'O�/.O) ,O.�Â��/0{% .10}#&% ¥Y��% .102!$�A¥

ogWuàN5<=�;@AW � 79Zcf+N-@A8�N5@i:=<=W�:=Wu<=`-WJ�;@ADµ?ATu��Wu8i?�?AW ~B ?/N58AngZ\[^R/N�àD6Z6:E>58ADFWU8�4;79Zµnu[ TuBu<wD6:

(~r ∧ ~B)2 = ~r 2 ~B2 sin2 θ = ~r 2 ~B2 − (~r · ~B)2 = B2(x2 + y2)n=DB ‖z x\±h463V3{KV~

� N58inIZ6WunJN�:=>VàWUnuÁ 〈x2 + y2〉 ∝ a20

ÁµWU:HBUW�:=Wu<wàW�Wunt:H8ATujVZ6DFjVWuN5 AZ6WÉ?/N58AnJZ6WunJBªR/N5`-�AnJ` N}qjV8ATC:ED6� @iWunJN5BUBuWunwn=D6 AZFWUnbnEN5@iv n=DÈZFW��i<=Wu`-DFWU<�:EWU<=`-WÉWunt:J<=D6jV>V@A<wWu@AnwWu`-Wu8�:b8;@AZ Á1Bu>V`-`-WÉ�1>5@A<@A8·TU:EN�:

s4 ��>5@Anbf»<=WC DFWu8i?A<=>V8inH�iZF@An�:EN5<=? Z6>V<=nw� @iW-8i>V@An�:E<=N5D6:=Wu<w>V8An�?AWUnJ�A<=>V�i<=DFTC:ETUnb?AWun

N�:=>VàWUnu4 � N58An�Z6Wun�nw>VZFD6?AWunUÁc>��(ZFWun�v&>V8ABU:=DF>V8in�?�[ >V8A?AWÉnw>V8�:JTU:=Wu8A?A@iWunuÁµBUW�:EWU<=`-WÉ8�[ Wunt:J�/N5n8ATUjVZFD6jVW2N5 iZFW54

� � ��

� ��D�

� �� L��6��]�Z6>V<=nb�;@AW-Z�N+nw:=<=@ABC:E@A<wWa?iW- AN5n=W-?A@yn=��WuBU:=<=Wa?AW�Z [^N�:E>V`-WÉ?�[ R�f ?A<=>Vj5ku8AWaWunw:IWu8y�/N5<tv�N5D6:

N5BuBU>V<=? N2VWuB�Z�Nb<=TUn=>VZ6@i:ED6>V8a?iW�Z\[^TU� @AN�:ED6>V8a?iW�shBªRi<=« ?ADF8Aj5Wu<À?AW�Z\[^TUZFWUBU:E<w>V8-?/N58AnÀZFWp�1>�:EWu8�:=DFWuZBu>V@iZF>V`É iDFWu8+?i@��A<w>5:E>58�ÁA@A8�BuWU<w:ªN�DF8©8A>V`É i<=Wb?AWJ?A>V8i8ATuWUn�W �i��Tu<wDF`-Wu8�:ªN�ZFWunSBU>V8ABUWu<=8AN58�:�ZFNnw:=<=@ABC:E@A<wWÀ?i@Inw�1WUBU:E<wW�Wu8H�A<=TUn=Wu8iBuWÀ?�[^@i8HBªRAN5à�HàN5jV8ATC:EDF�;@AWÈn=W�n=>V8�:ON2VTU<=TUWunÈWU8J?ATun=N5BuBU>V<=?n=TU<=DFWU@�� N2VWuB�BuWpà> ?AkUZFW54;¾O>V@A<�v�N5DF<wWgn=DF`-�AZ6WHx&V>5DF<�Z [^N5<w:=DFBUZFW_?AW �b>V@A?AnwàD�:��1>V@i< ZFN � h<=N5DFW �jVWu8ikun=W2~CÁV@i8QBªR/N5`-�É` N�jV8ATU:=DF�;@AW W � :ETU<=DFWU@A<È8iW_?AWU <=N5D6:È�/N5n¨N2V>5DF<È?c[^D68 �/@AWU8ABuW_nw@A<�Z6WunÈTC:ªN�:=ns?AWHZ [ TuZFWUBU:=<=>V8�Á/WC:pDFZµ?AWCh<=N5D6:pZ6WUVWU<�ZFN-?iTujVTU8ATu<wWun=BUWu8ABUWQ?AWun�N5@i:=<=WUn�8AD�VW2N�@��© D�N�ZFWb:EWU<=`-W

−µB

~

~L · ~B ?AWJZ\[^R/N�àD6Z6:E>58ADFWU8 �

s

l = 0

~B = 0 ~B 6= 0

l

l = 1

~B = 0

µBB

µBB

~B 6= 0

l

l = 2

~B = 0

µBB

µBB

µBB

µBB

~B 6= 0

� �� h4�3 � MµWU5TuWQ?AWI?ATuj5Tu8ATU<=WunwBuWU8ABuWaN�:=:=Wu8A?i@AWÉZF>5<=n=�;@�[ >V8 N��A�AZFD6�;@AWI@i8 BªR/N5`-�»` N�jV8ATCq:ED6� @iW ~B 4

��<uÁcZ6WunbZ6WUVTUWunJ?AW�?ATUjVTu8ATU<=WUn=BuWU8ABuWa8AW�Bu>V<w<=WUn=��>V8A?AWU8;:H�/N�n[7©BUW�n=BªRATU` Nyx\¯�DFjV@A<wW��h4�3}~C4§¨8��/N�<w:ED6Bu@AZ6DFWU<uÁiZ6Wb8AD65W2N5@

1s?AWJZ [^N�:E>5àW�?�[^R�f ?A<w>VjVku8iW�n=WJnwBuDF8i?AWJWu8(KQWU8��i<=TunwWu8ABUW�?�[ @A8

�� ÆcÇÈ#�.

BªR/N5`-��` N5j58ATU:=DF�;@AW �

1s

~B = 0

' 2µBB

~B 6= 0

� �� h4eK � M�WCVTUWa?AWQ?iTujVTU8ATu<wWun=BUWu8ABUW >V in=Wu<tVTUWa?A@·8AD�VWuN5@1s

?AW�Z [^N�:E>V`-W�?c[^R�f ?A<=>5jVku8AWWu8��A<wTun=WU8ABuWI?�[ @A8�BªRAN5à�©àN5jV8ATC:ED6� @iW ~B 4

M�N�ZFWU5TuWJ?AWJ?ATUjVTu8ATU<=WUn=BuWU8ABuWQ?iWung8AD65W2N5@ �l 6= 0

?A>V8i8AWJZFDFWU@�7-?iWungn=��WuBC:E<=WUn� �W2N5@iBu>V@A��AZ6@AnbBu>V`-�AZ6DF�;@ATunb�;@AW�Z6WunJ�A<=TU?ADFBC:ED6>V8An��A<wTuBUTu?AWU8;:=WunU4�§¨8 32z�KV|hÁ ��RAZ6Wu8; 1WUB�� WU: �b>V@A?AnwàD�:>V8�:ºTUàD6n Z\[^R�f �1>�:ERAkUn=W»�;@AW»Z\[^TUZFWUBU:E<w>V8��>Vn=nwTu?/N�D6:º@i8 1 BO1 F^PRL�1�_=<`PdY^LTQTSVUWF Q PRLT]gQ Pkc �\SVUWF?A>V8i8ATJ�/N5<

~µs = gµB~S

~

x\�h4�3}~

>��gWUnw:H@A8AW�BU>V8Anw:EN58�:EW 7+�1WU@��A<wkunJTujVN5ZFW^7�KhÁ�WC:H>�� ~S

Wunt:H@A8�>5�1TU<EN�:=Wu@A<b?AW�`->V`-Wu8�:BuD68ATU:=DF�;@AW-N2f�N58�:H@A8

l = 12

4�M�N+�N5Z6Wu@A<1/2

N+TU:=T-BªRi>VDFnwDFWQ��>V@A<JW �h�AZ6DF�;@AWu<bZFN+n=Tu�AN5<EN�:=DF>V8Wu8�K�8AD�VWuN5@��yx

m = −1/2WC:m = 1/2

~ ÁiWC:SZFW�v�N�BU:EWU@A<gWunt:gZ 7��>V@A<_W �h�AZFD6� @iWu<_ZFNI�N5ZFWU@A<

?AW�Z\[^TUB2N5<t:YWU8�:E<=W�ZFWun 8AD65W2N5@ �º?�[ Tu8iWu<=j5DFWQx2µB

WU:Y8A>V8µB

nwD1Z6W�`->V`-Wu8�:YàN5jV8iTU:ED6�;@AW�TC:ªN5D�:n=D6à�iZFWu`-Wu8�:p?A>V8i8ATÉ�/N5<

~µ = µB~S~

~C4c�YWU:w:EWIR;f ��>5:ERikun=W�NaTU:ETI8A>5:EN5à`-Wu8�:�Bu>V8h�/<=`-TuWI�/N5<Z [ W �h��Tu<wDFWu8iBuWJ?AWIs :EWU<=8�WU: �bWu<=ZFN5BªR �

]�j

~B = Bz8A>58�@A8iD6v&>V<wàW

⇒ ~F = ~∇( ~M · ~B) 'Mz∂B∂zz

KÉ:ªN5BªRAWUn]�j � \ �b<N] 4d105s1

� �� h4^m � shBªRATuàN�n=DF`-�AZ6D6�/T�?AWJZ [ W �h�1TU<=D6Wu8ABUWH?iW�s;:EWu<w8hq��bWu<w8AZ�N�BªR

MµW·:EWu<wàW �Unw�ADF8 �» DFWU8;:�?AWyZ [ DF?ATUW �;@AWÅZ\[^>V8Ënu[ Wunt:(v�N5D�:EW 7 Z [ Tu��>h�;@AWy?AWyZ [ >V<=D6jVDF8iW»?AWBuWy`->VàWU8�:+` N�jV8ATU:=DF�;@AW�4gM�[ TuZ6WuBC:E<=>58G8�[ TU:EN58�:��/N5n©ZF> B2N5Z6DFnwT5ÁYnU[^D6Z�:E>V@i<=8AW·n=@i<�Z6@AD�q ` �UàWx&:=>V@A<w8AWu<�� C:E>Qnw�ADF8 ��WU8�N�8AjVZ�N�DFnª~ ÁiZFN�?AWu8in=D6:=TJ?AWbBªR/N5<wjVWungWu8�`->V@iVWUàWU8�:g?A>VD6:_Bu<=TUWu<g@A8

¦��F��À% ´µ�"!$¶Å�/¥�#\)u¶Å%�'�,ËÆcÇ�#\. 12

��O�

à>5àWU8;:�` N�jV8ATU:=DF�;@AW�4c§¨8»v�N5D�:2Á�BuWÉ`->V`-Wu8�:�àN5jV8ATC:ED6� @iW�8�[^N >k_Ec [�� _`P _`DGBF<ÙRF @^D _EcCc^QTSVUWF 479Z_nu[^N5jVD�:É?�[ @A8 ?AWUjV<=Tº?iW+ZFDF �Wu<t:ETan=@A�A�iZFTu`-Wu8�:EN5DF<wW �;@AW©�1>5n=n=kU?AWu8�:�@A8 jV<=N58A?�8A>5`É A<=W ?AW�/N5<t:EDFBU@AZFWUnu4M�Na:ERATU>V<=D6WÉ<=WUZ�N�:=D6 DFnt:EWH?AWQZ [ TuZ6WuBC:E<=>58�Á1�;@ADÈBu>V8AnwDFnt:EW27º<wWu`-�AZ�N5BUWu<�Z [ Tu�;@/N�:=DF>V8»?iW�shBªRA<w«�q

?ADF8ijVWu<��/N5<�n=NIjVTU8ATu<=N5ZFD6nEN�:=DF>V8É<wWuZFN�:ED�hD6nw:=WS?A@AW%7 � D6<EN5BgWC: BU>V8A8;@AW�n=>V@AnÀZFWg8A>5` [�� Y\SVU�_`LTQhBEP[F 9 Q ]<_@ Á�?ATUà>V8�:=<=W��;@AWaBUW�?AWujV<wTa?iW-Z6DF �Wu<t:ETQDF8�:=Wu<=8iWaN+@A8iWa>V<wDFjVD68AWÉ8/N$:E@A<wWuZFZ6WQ?/N58AnJZ6WB2N5?i<=Wp?AWgZ�N�:ERiTu>V<wDFWg�;@/N58�:ED6� @iWp<wWuZFN�:ED�hD6nw:=W5ÁVWC: BU>V8A?A@iD6:#7JZ�N��A<wTu?AD6BU:ED6>V8

g = 2.002319305Á

VTU<=D6�ATuWHW �h�1TU<=D6àWU8;:EN5ZFWUàWU8�: �w@An=�;@�[ 7-ZFNÉR;@AD6:=DFku`-Wb?ATUBuDFàN5Z6W �

� � � � � � �� 6� � � �L� � � � � 12

��8aà>V`-Wu8�: BuD68ATU:=DF�;@AW 12

Wunw:Y@A8ºBuN5nY�/N�<w:ED6Bu@AZ6DFWU<À?iW�`->VàWU8�: BUDF8ATC:ED6� @iW5Á WU:À:E>V@i:YBuW�� @iDN+TU:ETQ @ 7©Z\[^> BuBuN5n=D6>V8 ?A@·`->VàWU8�:JBuDF8iTU:ED6�;@AWQ>V<= iD6:ªN�ZÈ<=WUnw:=W��N5ZFN5 AZ6W54�MµWUnHBU>V`-�1>Vn=N58�:EWUn��nEN�:=DFntv&>V8�:gZFWung<wkujVZ6WunS?AWJBu>5à`É@h:ªN�:=DF>V8

[Si, Sj

]= i~εijkSk[

~S2, Si]

= 0 ; i, j, k = x, y, z

x\�h4LK5~

M¨[^WUn=�/N5BUW�?iW�dpDFZF �Wu<t:�N5nwn=> BuDFTpWunt:À?AW�?iDF`-Wu8AnwDF>V8aKhÁ�WC: >58a��Wu@i: BªRA>VD6n=D6<¨Z�Nb /N5nwW�?AWUn TC:ªN�:=n�A<=>5�A<=WUn�BU>Và`�@A8An 7

SzWC: ~S2 �Sz|m〉 = ~m|m〉

~S2|m〉 = ~2 1

2

(1

2+ 1

)|m〉 = ~2 3

4|m〉

x\�h4emV~

>��m = 1

2

>5@m = −1

2

!24�sh@AD6�N58�:�ZFW�BU>V8�:EW �h:=W5Áµ?AD �cTU<=WU8;:=Wunb?AWUn=BU<=DF�h:EDF>58An�?AWQBUWU:bWUn=�/N�BuW?AWHd�D6ZF �Wu<w:_n=>V8�:pBu>V@i<EN5`-àWU8�:g@i:ED6ZFDFnwTuWUn �3V4��Y>V`-àW-Z [ Wunw�/N5BUW Wunt:É?AWa?ADF`-Wu8in=DF>58�KhÁ�>V8 ��Wu@h:�?ATUBu<wDF<=WaBªR/N5�;@AWºTUZFTUàWU8;:��/N5<H@A8

VWUBU:=Wu@A<pBu>VZ6>V8A8AW7QBu> W��-BUDFWu8�:=npBU>V`-�AZFW �iWUn �(αβ

)= α

(10

)+ β

(01

)

(10

)⇔ |1

2〉(

01

)⇔ |− 1

2〉

x\�h4^��~

�YWU:w:EW�?AWUn=Bu<wDF�i:=DF>V8bWUnw:µ:=<=kun�@i:=DFZ6WÈB2N�<µWuZFZ6WÈ��Wu<wàWC:�?�[^W �h�A<=D6àWU<�ZFWUnµ>V�1TU<EN�:=Wu@A<wnSx, Sy, SzBu>5à`-Wb?AWungàN�:E<wDFBUWun

2 × 24

# �A>WI@/'+,3�+07@>W5@( =H)?3 ��(*C.3G/ = 3dI"=?3,I_14-F;�-@3~S ≡ ~S

3d/:= -@IF&�5BK,)0>�=?3d-.) 3`= &�IN/?325_3d)?LN3,= 1432UV>W(E)?32CYS KG+`&�I.&�LN(E314-c/?89L�]B&�CE3�� OP � Ch3G/ =D=H)H>�14( =H(*&�I.I.3dCb1hS &�LN3`=:=H)?3l = 1

2

1.>WI_/2CE>NI.&�=H>W=?(E&�I|l,m〉

O

�� ÆcÇÈ#�.

Kh4pMµ>V<wn=�;@�[ >V8+nu[ DF8�:ETU<=WUn=n=W7�@i8+Wu8AnwWu`É AZ6Wb?AW�nw�ADF8An 12

ÁiDFZ�Wunt:S�AZ6@An_Bu>5à`->h?iW�?AW�j�N5<=?iWu<ZFNÉ8A>5:ªN$:EDF>58 �U A<EN �bWC: � �5WC: ��4"¾O>V@A<J3Jn=�AD68 1

2

Á/ZFNÉ8A>5:ªN$:EDF>58+nw:ªN�8A?/N5<w?�WUnw:�N5ZF>5<=n

|12〉 → |↑〉 x�nw�ADF8��U@A� �2~

|− 1

2〉 → |↓〉 x�nw�ADF8��U?A> �g8 �u~

x\�h4e|V~

MµWun �/kUBªRAWun�<EN5�i�1WUZFZFWU8�:SZ [ >V<=D6Wu8�:ªN$:EDF>58º?A@�à>V`-Wu8�:gàN5jV8ATC:ED6� @iW54mi4g79Z�WUnw:pn=>5@iVWU8;:p@i:=DFZ6W5ÁiWU8©�/N5<w:=DFBU@AZFD6Wu<YZ6>V<=nw�;@�[^>58�?iDFn=BU@i:EWbZ6WunS�A<w>V�A<wDFTU:=TunS?AWb�iZF@AnwDFWu@i<=n

�/N�<w:ED6Bu@AZ6WunSD6?AWu8�:ED6�;@AWunUÁ�?iWJ:=<EN5D�:EWU<SZFWung?AWUjV<=TUn�?iWHZ6DF �Wu<t:ETb?AWJZ [ Wunw�/N5BUW(x, y, z)

WU:p?AWnw�ADF8�n=@A<p@A8��ADFWU?�?�[ TujVN5ZFD�:ET54AM�[ Wunw�/N5BUWI?AWId�D6ZF �Wu<t:g?A@(nw�ADF8�Wunt:�N5Z6>V<=nS?iTuBu<wD6:p�/N5<gZ6Wunv&>V8iBU:ED6>V8An 7É�N5ZFWU@A<SBu>5à�AZ6W �hWb?�[ @A8AWJ�N5<=DFN5 AZFW

σ�i<=Wu8AN58�:�Z6WunS�N5ZFWU@A<=n ±1

2�

χ :

{−1

2,1

2

}→ C

x �h4^±�~

MµWungTC:ªN�:=np?iWH AN5n=WUn�BU>V<=<wWun=��>V8A?iWu8�:�N�@��+v&>V8ABC:ED6>V8An �

|12〉 ↔ χ 1

2: χ 1

2

(1

2

)= 1 , χ 1

2(−1

2) = 0

|− 1

2〉 ↔ χ− 1

2: χ− 1

2

(1

2

)= 0 , χ− 1

2(−1

2) = 1

x\�h4e�V~

�YWUn�<wWuZ�N$:EDF>58An_n=WJ<wTun=@iàWU8;:pnw>V@AnpZ�N�v&>V<wàW

χσ(σ′) = δσσ′

x\�h4^¬�~

�p>V@AnYnwWu<w>V8AngN5`-Wu8ATUn 7�@h:EDFZ6DFnwWu<�BuWUn_?AD �1Tu<wWu8�:EWUn_?AWUn=Bu<wDF�i:=DF>V8inu4i�Y>Và`-Wu8"�u>V8AnY�/N5<ÀZ�N�A<wWu`-DFkU<=W5Ái� @iDµBu>V8A?i@AD6: 7Q?AWUn�BU>V8AnwDF?ATU<EN�:=DF>V8An :E<=kUngDF8�:ETU<=WunwnEN58�:=Wunpn=@A<gZFWUnS<=>5:EN�:ED6>V8AnU4

�� c� �A�6�� D��

�Y>V`-`-Wu8��U>V8An��/N5<¨TC:ªN5 AZ6DF<�Z�N�v&>V<wàWS?AWUn�>V�1TU<EN�:=Wu@A<wnSz, S+, S−, Sx

WU:Sy

?AN58An¨Z�Nb /N�n=W?ATC�/8ADFWJ�/N�<SZ [ Tu�;@/N�:=DF>V8��h4^¬i4

Sz|12〉 =

~

2|12〉

Sz|− 1

2〉 = −~

2|− 1

2〉

⇒ Sz =~

2

(1 00 −1

) x\�h4^z�~

¦��F¡��À% )-¢£�h0{!$#\�V% )Q'�%»Â��/,O¥\# ��1¡

S+|12〉 = 0

S+|− 1

2〉 = ~

√1

2

(1

2+ 1

)−(−1

2

)(−1

2+ 1

)|12〉 = ~|1

2〉

⇒ S+ = ~

(0 10 0

)

S−|12〉 = ~

√1

2

(1

2+ 1

)− 1

2

(1

2+ 1

)|12〉 = ~|− 1

2〉

S−|− 1

2〉 = 0

⇒ S− = ~

(0 01 0

)

x � 463��V~

{S+ = Sx + iSy

S− = Sx − iSy⇒

{Sx = 1

2(S+ + S−)

Sy = 12i

(S+ − S−)

⇒ Sx =~

2

(0 11 0

)Sy =

~

2

(0 −ii 0

)

x � 463V3{~

��8�5>VD6:g�;@AWSx

ÁSy

WC:Sz

nu[ W �h�A<wDF`-Wu8�:pn=DF`-�AZ6Wu`-Wu8�: 7�Z [^N5DF?AW�?AWun 1�_`LT]^Qh@CFCc [F#H�_ÙDKQ �σx =

(0 11 0

)σy =

(0 −ii 0

)σz =

(1 00 −1

) x � 463}K5~

�A@AD6n=�;@AWSi = ~

2σi (i = x, y, z)

4

¾¨<=>V�A<wDFTC:ETung?AWUng` N�:=<=D6BuWunS?iWI¾�N5@AZ6D �� TunwDFjV8i>V8Ang�/N5<

σ0Z�NQ` N$:E<=D6BuW�D6?AWu8�:=D6:ET�4

3V4σ2i = σ0Kh4gPO<(σi) = 0mi4

det(σi) = −1�A4[σi, σj] = 2iεijkσk|h4σiσj = −σjσi

��>V@A<i 6= jMµWungàN�:E<wDFBuWUnS?AWH¾ÈN5@iZFDµN58�:=DFBu>5à`É@h:EWu8�:u4

��/³ ÆcÇÈ#�.

� TUà>58Anw:=<EN�:=DF>V8 �(

0 11 0

)(0 −ii 0

)=

(i 00 −i

)

(0 −ii 0

)(0 11 0

)=

(−i 00 i

) x\�h4632m�~

±i4Ix&�;~YWU:�x |5~ ⇒ σiσj = −σjσj = iεijkσk��>V@A<

i 6= j�h4pMµWung�A<w>V�A<wDFTU:=TunIxt3{~YWU:�x\±�~YnwWJ<=Tunw@A`-Wu8�:�WU8

σiσj = δijσ0 + iεijkσkx\�h463u�;~

¬i4p§¨8�?ATU�/8iDFn=n=N58�:~σ ≡ (σx, σy, σz)

Á/>V8©WU8�?ATU?A@AD�:�N�DFn=TUàWU8�: �(~σ · ~a)(~σ ·~b) = σ0(~a ·~b) + i~σ · (~a ∧~b)

⇒[~σ · ~a, ~σ ·~b

]= 2i~σ · (~a ∧~b)

x\�h463{|V~

zi4 {σ0, σx, σy, σz}Bu>58Anw:=D6:E@iWu8�:g@A8AW� AN5n=W�?iWunSàN�:E<wDFBuWUn

2× 24i��[^WUnw:gBUZ�N5D6< �A@iDFn=�;@AW�ZFWun

àN�:E<wDFBUWun(

1 00 0

)=

1

2(σ0 + σz)

(0 00 1

)=

1

2(σ0 − σz)

(0 10 0

)=

1

2(σx + iσy)

(0 01 0

)=

1

2(σx − iσy)

x\�h4632±�~

BU>V8Anw:=D6:=@AWu8�:pWuZ6ZFWunrq9`��u`-Wun_@i8AWJ /N5n=WHTU D6?AWu8�:EW�4

�� c� �A�6�� D�� N� �� c�� @��MµWun¨àN�:=<=DFBUWun�?AWS¾ÈN�@AZFDhn=>V8�:�8A>5:EN5`-àWU8;:O:E<wkun¨@i:=DFZ6WunÈ��>V@A<�?ATuBU<=D6<=W_Z6Wun�<w>5:ªN�:=DF>V8inu45�Y>V8An=D q

?ATU<=>V8AnYWu8ºW �cWC:_ZFWp�A<w>V AZFkUàWpn=@iD6�N58�: � n=>VD6Wu8�: n WU:m

?AWu@ � 5WuBU:=Wu@A<wnS@A8AD�:ªN5D6<=Wun�<wWuZFD6TunÀ�/N5<@A8AWJ<w>5:ªN�:=DF>V8

R~α?�[^N58Aj5ZFW

α = ‖~α‖ N5@i:E>5@A<S?AWJZ�NQ?AD6<=WuBC:ED6>V8α = ~α

‖~α‖ÁiB5[ Wunw:tq*7$q ?ADF<wW �

n = R~α(m)x\�h463{�V~

�Y>V`-àWU8�:ÉZFWUn�TC:ªN�:=n��A<w>V�A<=WUn |n ↑〉 WC: |n ↓〉 ?AW ~S · n n=>V8�:wq DFZ6nI<wWuZFD6TunIN5@��ÅTU:EN�:EnÉ�i<=>V�A<wWun|m↑〉 WC: |m↓〉 ?AW ~S · m �¾¨<=>V��>VnwD6:ED6>V8 �

~S · n = e−i~S·~α/~

(~S · m

)ei~S·~α/~ x\�h4632¬�~

¦��³ ¢£�h0{!$#\�V% )Q'�%»Â��/,O¥\#À%�0 � �"02�i0}#&�1.O) ��1�

n

mα

�

α

� �� 4^� � op>�:ªN�:=DF>V8+?�[ @A8�N58AjVZ6WαN5@i:=>V@A<S?AWJZ [^N �hW

~αx~α�1>VD68�:EWbRA>V<wng?AWJZ�N�v&Wu@iDFZFZ6W{~ 4

� Tu`->V8Ant:E<=N�:ED6>V8 �§¨8�BU>V8An=@iZ6:ªN�8;:gZFNÉ�/jV@A<wW5Á/>V8+V>5D6:pN5D6n=Tu`-Wu8�:g�;@AW �

n = cosα m+ sinα α ∧ m x � 463{zV~

¾�>V@i<dα

D68i�/8AD�:ETUn=DF`aN5Z\Á >58(NQ?A>V8iB �n(α + dα) = n(α) + dα α ∧ n(α) +O(dα2)

⇒ ~S · n(α + dα) = ~S · n(α) + dα~S · (α ∧ n(α)) +O(dα2)

x � 4LK��V~

�(N5D6nSZ�NQ<=WUZ�N�:=DF>V8·x\�h4�3{m�~YDF`-�AZ6DF�;@AW[~S · ~a, ~S ·~b

]= i~~S · (~a ∧~b)

⇒ ~S · n(α + dα) = ~S · n(α) − i

~dα[~S · α, ~S · n(α)

]+O(dα2)

x � 4LKh3{~

shDµ>V8©Bu>58An=D6?Aku<wW ~S · n Bu>V`-`-W�v&>V8ABU:=DF>V8+?iWαÁA>V8�NQ?A>V8AB �

d

dα

(~S · n

)= − i

~

[~S · α, ~S · n

]+O(dα2)

x � 4LKVK5~

N2VWuBHZ�NQBU>V8A?AD�:EDF>58+DF8AD�:ED�N�ZFW (~S · n)

(α = 0) = ~S · m 4� Tu`->V8�:E<w>V8AnI� @iW

A ≡ e−i~S·~α/~

(~S · m

)ei~S·~α/~ Wunt:Én=>5ZF@i:=DF>V8 x�WU8 n=Wa<EN5�i�1WUZ�N58�:I?AW ZFN

?ATU�A8AD6:=DF>V8α = ~α

‖~α‖ ↔ ~α = α×‖α‖ ≡ α× α~ 4

A(α = 0) = ~S · m :E<=D� D�N5ZdA

dα= − i

~S · α~

A+ Ai~S · α

~= − i

~

[~S · α, A

] �S�b¯ �x � 4LK5mV~

�� ÆcÇÈ#�.

ogWU5Wu8A>V8in^7(Z�N©�;@AWunt:EDF>58�8A>V@AnI8A>V@AnIn=>V`-àWUn��>Vn=TUWºN5@Å?ATu�/N�<w:2Á 7(n=N{5>VDF<IZFW-<EN5�i�1>V<t:Wu8�:=<=WHZFWunSTC:ªN�:=ng�A<=>V�i<=Wung?AW ~S · n WU:gBuWU@��?AW ~S · m 4��Y>58An=D6?ATu<w>V8AngZ [ TU:EN�:

e−i~S·~α/~|m↑〉 4

~S · ne−i~S·~α/~|m↑〉 = e−i~S·~α/~

(~S · m

)ei~S·~α/~e−i

~S·~α/~|m↑〉

= e−i~S·~α/~~S · m|m↑〉

= e−i~S·~α/~ ~

2|m↑〉

⇒ |n ↑〉 = e−i~S·~α/~|m↑〉

x\�h4LK$�;~

� WJ`��u`-W5Á |n↓〉 = e−i~S·~α/~|m↓〉 4

ogWuàN5<=�;@AW �M�N�?ATu`->V8Ant:E<EN$:EDF>58+<=Wu��>VnwWHnw@A<gZ [ DF?AWU8�:ED6:=T �[~S · ~a, ~S ·~b

]= i~~S · (~a ∧~b) x\�h4eKV|V~

��<uÁ;BuWC:=:=W�DF?AWU8�:ED6:=TgWunw:Yh<=N5DFWp��>V@A<À8�[^D6à��>V<t:EWg�;@AWuZ�à>5àWU8;: BuD68ATU:=DF�;@AW�4h§¨8ºW �cWC:2Á 7I�/N�<w:ED6<?AWUn�<wWuZ�N$:EDF>58An_?AWJBu>5à`É@h:ªN�:=DF>V8

[Lx, Ly] = i~Lz>V8+:E<w>V@i5W

[Lxax + Lyay + Lzaz, Lxbx + Lyby + Lzbz]

= axby[Lx, Ly] + axbz[Lx, Lz]

+ aybx[Ly, Lx] + aybz[Ly, Lz]

+ azbx[Lz, Lx] + azby[Lz, Ly]

= i~Lz(axby − aybx) + i~Lx(aybz − azby) + i~Ly(azbx − axbz)

= i~~L · (~a ∧~b). x\�h4eK5±�~]�D68An=D\ÁA>V8�NQ?AWbv�NM�u>58�:=>V@i: 7Év�N5D6:_jVTu8ATU<EN5Z6W �

~L · n = e−i~L·~α/~

(~L · m

)ei~L·~α/~ x\�h4LK5�V~

WU:|nLM〉 = e−i

~L·~α/~|mLM〉. x �h4eK5¬�~� N�8An�Z6WQB2N5nH?A@Ånw�ADF8 1

2

ÁµBuWC:�>5�1TU<EN�:=Wu@A<HN©@A8AW�v&>V<=`-WQ:E<=kUnJn=DF`-�AZ6W54µ§¨8ÅW �cWC:2Á�>V8y�1WU@i:TuBU<=D6<=W

e−i~S·~α/~ = e−i~σ·~α/2

x\�h4LK�z�~

¦��³ ¢£�h0{!$#\�V% )Q'�%»Â��/,O¥\#À%�0 � �"02�i0}#&�1.O) ��¦

>��~σWunt:�Z6W�VWUBU:EWU@A<g?ATU�A8AD6:g�A<wTuBUTu?AWUà`-Wu8�:p?A>V8�:pZFWUnSBu>V`-�1>5nEN58�:EWUnSn=>V8�:pZFWUnS` N$:E<=D6BuWUn_?AW

¾ÈN5@iZFD 4/�(N5DFnUÁ

e−i~σ·~α/2 =

+∞∑

n=0

(−i~σ · ~α2

)n1

n!= sum+∞

n=0

(−iα~σ · α2

)n1

n!

x � 4em��V~

WU:p�/N5<Hx\�h4�3{m�~

(~σ · α)2

= σ0α · α + i~σ · (α ∧ α) = σ0

x � 4emi3{~?�[ >��

e−i~σ·~α/2 = σ0

∑

n=0,2,4,...

(−iα2

)n1

n!+ σ · α

∑

n=1,3,5,...

(−iα2

)n1

n!

= cosα

2σ0 − iσ · α sin

α

2

⇒ e−i~σ·~α/2 = cosα

2σ0 − i sin

α

2σ · α

x � 4em�K5~

§ �iWUà�iZFW � �Y>V8AnwDF?ATU<=>V8Ana@A8AW�<=>5:EN�:ED6>V8�?�[^N58AjVZ6W −π2

N5@i:=>V@A<�?AWx

7·�/N5<w:=DF<Q?AWz4�]O5WuB

ZFWUn_8A>5:EN�:ED6>V8AnY�i<=TuBUTu?AWU8�:EWunUÁ ~S · m = SzWU: ~S · n = Sy

4Ash>VD6: (10

) Z [ TU:ªN$:_�A<w>V�A<=W�?AWSz

?AW�N5ZFWU@A<S�A<=>5�A<=W

~

2

4 � [eN5�i<=kunpZ�N�v&>5<=`É@AZ6W��;@ADµ�A<wTuBukU?AW5Á�DFZ�v�N5@i:pN5�i�AZFD6� @iWu<SZ [ >V��Tu<=N�:EWU@A<√

2

2σ0 + i

√2

2σx =

√2

2

(1 ii 1

)

⇒ |n↑〉 =︸︷︷︸(7.22)

√2

2

(1 ii 1

)(10

)=

√2

2

(1i

) x � 4emVmV~

��<uÁ

σy

(1i

)=

(0 −ii 0

)(1i

)=

(1i

)

⇒ Sy

(√2

2

(1i

))=

~

2

(√2

2

(1i

)) �uN-àN5<=BªRiW �x � 4em5��~

opWU` N�<=�;@AWun �

�� ÆcÇÈ#�.

3V4�shD1>V8 v�N5D�:Y@i8AW�<=>�:ªN�:=DF>V8a?AW2π

ÁhZ\[^>V��Tu<=N�:EWU@A<_N�n=n=> BuD6T��N5@i: −σ04i]�@i:E<wWu`-Wu8�:S?AD�:2Á;>V8

N�:=<EN58Antv&>V<=`-T�@A8©TC:ªN�:SWU8�nw>V8+>V�A��>VnwT ��YWuBUDµWunw:S <=N5D��>V@A<_Z6Wun_nw�ADF8AnS?iWu`-D�q9WU8�:EDFWU<=nUÁàN5DFnc�/N5nµ��>V@A<µZFWUnµn=�AD68An�Wu8�:ED6Wu<wnu4$�YWU:w:EW��A<w>V�A<=D6TU:=T�N_TU:ET¨5Tu<=D��/TuW��/N�<µ?AWunOW �h��Tu<=D6Wu8ABUWun?�[ DF8�:=Wu<wv&TU<=WU8ABuW�n=@A<À?AWun 8AWu@h:E<=>58AnuÁ �;@AD Á;Bu>5à`-WSZFWUn TUZFWuBC:E<w>V8AnuÁ;��>Vn=nwku?AWU8�:S@A8 nw�ADF8 1

2

479Z¨v�N5@i:Jv�N5D6<=WQ@A8iWa<w>5:ªN�:=DF>V8·?AW

4π��>V@A<Jn=W-<=WC:E<=>5@iVWU<�?/N58inIZ6W-`��u`-W�TU:EN�:IN{5WuBaZFW

`��u`-Wbn=D6jV8AW-x s�4^]I4 �ÅWU<=8AWU<pWC:�N5Z 4�Á/¾¨R�fhnwDFBuN5Z�opWC DFW ��MµWU:=:=Wu<wn ¡"� Áµ3��|�miÁ�xt32z��V|V~w~C4��[ Wunt: @i8aW �iWUà�iZFWS?�[ @A8AWp�A<=>5�A<=D6TU:ETSj5Tu8ATU<EN5Z6WS?AWun�<=>�:ªN�:=DF>V8An � BuWg8�[ Wunw:À�;@�[^N5�A<wkunÀ@A8AW<w>5:ªN�:=DF>V8J?AW

4π�;@�[ @A8�>V �wWU: F^P�]>F^D _`LTQhBEP N{5WuB n=>V8IWu8� DF<=>58A8AWu`-Wu8�:�n=WÀ<=WC:E<=>5@iVW ?/N58An

nw>V8aTC:ªN�:ÀD68AD6:=D�N5Zcx\N5<wjV@A`-Wu8�:À? � 7 � D6<EN5B�Á5V>5DF<�ZFWSZ6D6 <=WS?AW �5W2N�8hqt�(N�<=BpMµTU;f�qrMµWu AZ6>V8A?WC:�¯/<=N58��u>5DFn=WJXYN5ZFD6 /N5< � � Uk_`PRLTQTSVUWF ~C4

Kh4�shDÈ>V8yN5�A�AZ6DF�;@AWÉZFNº` �UàWÉ<w>5:ªN$:EDF>58 7+Z [ TU:ªN$: |m ↓〉 Ác>V8·>V h:EDFWU8�:J ADFWU8 n �A< |n ↓〉 4µ§¨8W �cWC:2Á |m↓〉 = |−m↑〉 → |− n↑〉 = |n↓〉 4;shD/@A8�TU:EN�:�Wunt:À?ATuBU<=D6:��AN5<¨ZFW_5WuBU:=Wu@A< (a

b

)

<wWuZ�N$:ED6vZ7ºZ�N �;@/N58�:=D6�/BuN�:ED6>V8»?/N58inbZ�N ?AD6<=WUBU:ED6>V8mÁ1Z [ >V�1TU<EN�:=Wu@A<�<=>5:EN�:ED6>V8

e−i~σ·~α/2Z6W

:=<EN58Antv&>V<=`-WbWu8�Z [ TU:EN�: (ab

) <=WUZ�N�:=D6v 7�Z�NÉ�;@/N58�:=D6�/BuN�:ED6>V8©?/N58AngZ�NÉ?iDF<=WUBU:=DF>V8n4

mi4 �p8AW�� @iWunw:=DF>V8JV>5DFn=D68AW�` N�DFn�?AD �cTU<=WU8;:=W�Wunw:O?AWÀnEN25>VDF<µBU>Và`-Wu8�:�W �i�i<=DF`-Wu<µ@i8H5WuBU:=Wu@A<(ab

) <wWuZFN�:ED�v#7©ZFN �;@/N58�:ED��/B2N$:EDF>58 ?/N58AnbZFNº?ADF<wWuBU:=DF>V8m

?/N�8AnbZ�Nº /N�n=WQ<=WUZ�N�:=D65W27ºZ�N�;@/N58�:=D6�/BuN�:ED6>V8©?/N58AngZ�NÉ?iDF<=WUBU:=DF>V8

n4

M¨[ TU:ªN$: (ab

) Bu>V<w<=Wunw�1>58A? 7 |ψ〉 = a|m ↑〉 + b|m ↓〉 4ÈM�WUn�Bu> >V<=?A>58A8ATuWUn (a′b′

) ?A@5WuBU:=Wu@A<p?/N58inpZFNQ /N5n=W |nσ〉 nU[^TUBu<=D�VWU8;: �

{a′ = 〈n↑|ψ〉b′ = 〈n↓|ψ〉

nw>VD6:{a′ = a〈n↑|m↑〉 + b〈n↑|m↓〉b′ = a〈n↓|m↑〉 + b〈n↓|m↓〉

��<uÁ |n↑〉 = e−i~S·~α/~|m↑〉 ⇒ 〈n↑| = 〈m↑|ei~S·~α/~

⇒

a′ = a(1 0

)ei~σ·~α/2

(1

0

)+ b(1 0

)ei~σ·~α/2

(0

1

)

b′ = a(0 1

)ei~σ·~α/2

(1

0

)+ b(0 1

)ei~σ·~α/2

(0

1

)

⇒(a′

b′

)= ei~σ·~α/2

(ab

)

x\�h4emV|V~

¦��F�� ¥�% �50{!}��.�.��.�!}%h¥&�i0}# ��#�)U0{% ��

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Z�NJ�1>5n=D6:=DF>V8Q?AN58AnYZ\[^WUn=�/N�BuWpWU: @A8�Á;?ADFnwBu<wWU:2Á;��>V@A<�ZFWgnw�ADF8�4;]�fVN58�:YBªRA>5DFn=D"@A8iW�?AD6<=WUBU:ED6>V8-�1>V@i<�;@/N58�:ED��/Wu<gZFWbnw�ADF8cÁ/>V8©�1WU@i:p?A>V8ABJZFWb?iTuBu<wDF<wWb�/N5<g@A8AWbv&>V8ABC:ED6>V8©?�[^>58A?AW54

ψ(x, y, z; σ) ∈ C x, y, z ∈ R σ ∈ {−1

2,1

2} x\�h4em5±�~

M�NÉ�A<=>V AN5 ADFZ6D6:=T�?AWb:=<=>V@i5Wu<SZ\[^TUZFWuBC:E<w>V8©N5@©�1>5DF8�:x, y, z

N2VWUBJZ�NÉ�A<w>��wWuBC:EDF>58σnw@AD6�N58�:

z�1>5@A<¨ZFWSnw�ADF8�Wunt:À?A>V8A8ATUW��AN5< |ψ(x, y, z; σ)|2 4;�YWUZFZFWS?iWg:=<=>V@hVWu<�Z\[^TUZFWuBC:E<w>V8�N5@��1>5DF8�:x, y, zN2VWuB @A8 n=�iDF8 �;@AWUZFBu>58A�;@AWºWunt: ∑

σ |ψ(x, y, z; σ)|2 WU:ÉBUWuZ6ZFW-?AW Z6W-:=<=>V@i5Wu<I8�[^D6à��>V<t:EW->��` N�DFnSN2VWUBI@i8�nw�ADF8�?ATU:=Wu<wàD68AT��N5@i: ∫

d3~r |ψ(x, y, z; σ)|2 4§¨8i�/8�Á1Bu>V`-àWJZFNa�A<w>V /N5 AD6ZFD�:ETb?AWÉZ6WH:=<=>V@hVWu<�N{5WuBÉ8c[^D6à��>V<w:=WI�;@AWUZ�nw�ADF8(8�[^D6à��>V<t:EWH>��N5@i:H3VÁA>V8©?A>5D6:gDF`-��>Vn=WU<_Z�NQ8A>V<w` N�ZFDFn=N�:ED6>V8 �

∫d3~r

∑

σ

|ψ(x, y, z; σ)|2 = 1x\�h4emV�V~

��@AWUnw:=DF>V8 � sh@i�A�1>5n=>V8An��;@�[ @A8 TuZFWUBU:=<=>V8�nw>VD6:Q?iTuBu<wD6:��/N�<É@A8AW+v&>V8ABC:ED6>V8�?�[ >V8A?AW+>�� ZFN�N5<=DFN5 AZFW�n=�AD68�WUnw:�<=WUZ�N�:=D6VW87-@A8iWH?iDF<=WUBU:=DF>V8

m �ψ(x, y, z; mσ)

x\�h4^mV¬�~�Y>V`-àWU8;:gnU[^TUBu<wD6:pZ�N�v&>V8iBU:ED6>V8©?�[ >V8A?AWHn=DµZ\[^>V8©BªRA>5DFn=D�:g@A8AWJ?AD6<=WuBC:ED6>V8

n�1>V@i<S� @AN58�:ED6�AWu<

ZFWbnw�ADF8 �opTU�1>58An=W � Z�N�v&>58ABU:=DF>V8©?�[/>V8i?AW

ψ(x, y, z; mσ)Bu>V<w<=Wunw�1>58A?(N�@ ��WU:

|ψ〉 =∑

σ

∫d3~r ψ(x, y, z; mσ)|~r〉 ⊗ |mσ〉 x\�h4em5z�~

¾ÈN5<g?ATC�/8AD�:EDF>58�ÁiZFN�v&>V8ABU:=DF>V8©?�[/>58A?AWψ(x, y, z; nσ′)

WUnw:p?A>V8AB �

(〈~r| ⊗ 〈nσ′|)|ψ〉 =∑

σ

ψ(x, y, z; mσ)〈nσ′|mσ〉

= 〈nσ′|(ψ(x, y, z; m ↑)|m ↑〉 + ψ(x, y, z; m ↓)|m ↓〉) x � 4^� �V~��<uÁA?�[^N5�A<wkunpZFWbBuN5ZFBU@AZ�n=@i<pZ6WunS<w>5:ªN�:=DF>V8inuÁABUWungBu>V`-��>VnEN58�:=Wungn=>V8�:�?A>V8A8iTuWunp�/N5< �

(ψ(x, y, z; n ↑)ψ(x, y, z; n ↓)

)= ei~σ·~α/2

(ψ(x, y, z; m ↑)ψ(x, y, z; m ↓)

) x � 4^�A3{~�YWU:w:EWJ�A<=>5�A<=D6TU:ETJN�àkU8AW[7Q�A<=TCv&Tu<=WU<g?/N58AngBuWU<w:EN5DF8AngBuN5ng<=Wu�i<=TunwWu8�:EWU<�ZFN�v&>V8ABC:EDF>58�?c[^>V8i?AW

?�[ @A8�TUZFWUBU:E<w>V8��/N5<S@A8©VWUBU:EWU@A<pBu>VZ6>V8A8AW

�"�� ÆcÇÈ#�.

(ψ(x, y, z; ↑)ψ(x, y, z; ↓)

) x\�h4^��KV~N5�A��WuZ6T c > Q PdFÛ6] 4]��i�AZFD6B2N�:=DF>V8 � nw@A�A��>Vn=>58An+�;@�[^@i8¹TuZFWUBU:=<=>V8¹nw>VD6:a?/N58An @A8¹TU:EN�:

ψ(x, y, z; σ)Á >��

σWunw:

<=WUZ�N�:=D6v 7QZ�NQ?AD6<=WuBC:ED6>V8m

4��@AWUZFZFWbWUnw:�Z�NÉ�A<w>V /N5 iDFZFD�:ET�?iWHZ6Wb:E<=>5@iVWU<�N�@��>VDF8�:x, y, z

N2VWUB@A8AWJ�A<w>��wWuBC:ED6>V8

~

2

?AN58AngZ�NQ?AD6<=WuBC:ED6>V8n�

ogTu��>V8AnwW � n=D ~α ?ATUn=D6jV8AW Z�Ny<=>5:EN�:ED6>V8��>V@A<a�/N5n=nwWu<+?AWm

7nÁYWUZFZFW�Wunt:©?A>V8A8ATUW»�/N5<

|ψ(x, y, z; n ↑)|2 Á��;@AD¨nu[ >V i:=DFWu8�:@7��AN5<w:=DF<�?AWψ(x, y, z; σ)

7+Z [^N5DF?AWQ?iWaZ\[^TU� @AN�:ED6>V8·�A<=TUBuTCq?AWU8;:=W54

�� X�� #R� &'*��&R '�� X(��8' R !(I��&R )TY8'*��Y>V`-`-W�ZFWHn=�iDF8»WC:bZFWUn�Bu> >V<w?A>V8A8ATUWun�nw>V8�:�?AWUnb?AWuj5<=Tun�?AW�Z6DF �Wu<w:=T5Á"DFZ6n�BU>V<=<wWunw�1>V8i?AWu8�:7

?AWUn�>V in=Wu<t�N5 AZFWUng� @iDµBu>V`-`É@i:=Wu8�:2Á

[~S,~r] = [~S, ~p] = [~S, ~L] = 0x\�h4 ��m�~

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(Aspin ⊗ Aorb)|ψ〉 =∑

σ

∫d3~r |ψ(x, y, z)| Aorb|~r〉 ⊗ Aspin|σ〉

x\�h4^�5�;~

¾�N5<gW �hWu`-�AZFW�Á/Z [ >V��Tu<EN$:EWu@i<S<=>5:EN�:ED6>V8+nu[ TuBU<=D6:

e−i~S·~α/~ ⊗ e−i

~L·~α/~ x\�h4^��|V~�;@AWHZ [ >V8©TuBU<=D6:gnw>V@i5Wu8�:

e−i~J ·~α/~ >�� ~J = ~S + ~L

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(1

2m2c21

r

dU

dr) ~S · ~L x\�h4^�V±�~

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?/N58in @A8�BªRAN5à��TuZFWUBU:=<=DF�;@AW ~E7yZ�N·hD�:EWUn=n=W

~vn=WU8;:º@A8�BªR/N5`-��àN5jV8iTU:ED6�;@AW ~v∧ ~E

c

4Y�YWBªR/N5`-��n=WJBu>V@i�AZFWJN5@�n=�iDF8

¦��F�� ¥�% �50{!}��.�.��.�!}%h¥&�i0}# ��#�)U0{% �"��

→ |e|mc

ρ~v ∧~E

c

x � 4^�;�5~�(N5D6n

e ~E = ~∇V = ~r 1rdVdr

→ Hs.o =|e|mc

~S ~v ∧ ~r1r

dV

dr

1

ec= (

1

m2c21

r

dU

dr) ~S · ~L x � 4^��¬V~

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�"� � ÆcÇÈ#�.

� � ��

� ��

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M�>@/2B {~J1

2|j1m1〉 = ~2j1(j1 + 1)|j1m1〉Jz1 |j1m1〉 = ~m1|j1m1〉

�� `L�}�

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[Jx1 + Jx2 + . . . , Jy1 + Jy2 + . . . ] = [Jx1 , Jy1 ] + [Jx2 , J

y2 ] + . . .

= i~Jz1 + i~Jz2 + . . .�� `s�=�

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`� efM@WI6K8�1x[�efEk37<IG?/�g=E�-IE26KM@[P/wGI/�1�V&;=V&/2-A371dB2JL-IE�3KJP_]<I/�12q�[P/�1d>OM@[L/2<I691dWI6K;=W?6K/21xWY;@1K1KJLiI[P/�1dGI/

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W�;=<I6 ~J2/�3

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�� 0��. �� .10��.��;0��,��

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xj + Jyi J

yj + Jyi J

yj

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Jzi + Jzi`

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yi ]J

yj

= i~Jyi Jxj − i~Jxi J

yj

[Jzj ,~Ji · ~Jj] = Jxi [Jzj , J

xj ] + Jyi [J

zj , J

yj ]

= i~Jxi Jyi − i~Jyi J

xj

⇒ [Jzi + Jzj ,~Ji · ~Jj] = 0

�� `��]�⇒ [Jαi + Jαj ,

~Ji · ~Jj] = 0�� `! =�

⇒ [∑

i

Jαi ,~Ji · ~Jj] = 0

�� `�"A�

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q ~J2

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|j1j1m1m2〉 = |j1m1〉 ⊗ |j2m2〉�� `!0=�

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JzGI/�>OM@[P/�<I6K1(WI69;=WI69/21

~2j(j + 1)/�3

~m(m =−j,−j + 1, . . . , j)

`+�h;=V&V\/Jz = Jz1 + Jz2

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2j + 1E�37MO371l69/2[PJLE21l/�-]3K6K/w/2<U�ZWNM@6

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m = −j,−j + 1, . . . , j`#3x-X3K/2[/�-I1K/�V5iI[L/mGefEk3�MO3K1

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n(m) =∑

j≥mN(j) (m ≥ 0)

�� ` � �

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n(j1 + j2) = 1

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n(j1 − j2) = j1 + j2 − (j1 − j2) + 1 = 2j2 + 1

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· · ·n(0

;=< 12) = 2j2 + 1

�� `��=�

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qNGI;=-IB

N(j1 + j2 − 1) = 1

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...

N(j1 − j2) = 1

N(j1 − j2 − 1) = 0

...

�� `L�� =�

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`

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jGI;=-I-IEwW�;=<I6

| j1 − j2 |≤ j ≤ j1 + j2M

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j19;=-A3��|j1 − j2| , |j1 − j2 + 1| , . . . , j1 + j2

�� `L�}�=�� B �NM@_]<I/w>OM@[P/�<I6lGI/

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|j1〉 ⊗ |j2〉 = |j1 − j2| ⊕ (|j1 − j2| + 1) ⊕ · · · ⊕ (j1 + j2)�� `L��A�

�� ! "�#��$��%"��'&�(*)+�-,. "/0�

� E�1KJLg=-I;=-I1ZB�/21ZEk3�MO3K1XWNM@6 |j1j2jm〉 `�1v;=<I1XG?JP1KW�;=19;=-I1XGIE�1K;=69V�MOJP1�G?/�GI/2<c��iNM@19/21ZGI/[neo/219WNM@B�/QG?/�~vJP[LiY/�693�`.��/QW?6K;=iI[L82V&/m/2143xG?/Q[L/21l69/2[PJL/26l/�-]3K6K/w/2[L[P/�1m�

|j1j2jm〉 =∑

m1,m2 2 (m1+m2=m)

〈j1j2m1m2|j1j2jm〉|j1j2m1m2〉�� `L� �]�

��/21dB2;U/43&B�JP/2-A3K1 〈j1j2m1m2|j1j2jm〉 12efM@WIW�/2[L[P/2-A3d[L/21lB2;U/43&B�JP/2-A3K1xGI/��h[L/2iI19B �cp 5m;=6KGIM@- `6 /2[PMO37JL;=-ZGeo;=693 �I;=g=;=-IM@[PJz37E ��h;@V\V&/ 〈j1j2j ′m′|j1j2jm〉 = δjj′δmm′

qN;=-�MZ�∑

m1,m2 2 (m1+m2=m)

〈j1j2j ′m′|j1j2m1m2〉〈j1j2m1m2|j1j2jm〉 = δjj′δmm′

�� `L�� =�

�|;@<I6j ′ = j

/�3m′ = m

qNJL[Y>UJP/�-]3Q�∑

m1,m2 2 (m1+m2=m)

| 〈j1j2m1m2|j1j2jm〉 |2= 1�� `z��"A�

��

�h/�3937/Q6K/2[PMO37JL;=-X19/267M#<?3KJP[P/�WY;=<?6d�c�?/�6d[�M -I;=69V�M@[LJP1KMO37JL;=-`�|;=<?6QB�;=-I14376K<?JP6K/#[L/21wB2;U/43&B2JL/2-A371QGI/\�h[P/�iI1KB �Up 5m;=6KGI;@-q+[P/ WI[L<I1w1KJPV&WI[L/5/2143:GI/ C�M@JL6K/ [L/

B�M@[LB2<I[ GI/#WI6K;UB �I/#/2- WI6K;UB �I/#/2- <?3KJP[PJL17M@-A3�GI/21Q6K8�g=[P/�1mGI/#6KE2B�<I6K19JP;=-�3K6K821w19JPV&WI[P/�12`��ueoJPGIE�/GI/wiNM@19/�/�193vGeoE2B269JP69/&�

J+ = J+1 + J+

2

qJ− = J−

1 + J−2

M@JL1J+|j1j2jm〉 = ~

√j(j + 1) −m(m + 1)|j1j2jm+ 1〉/�3

〈j1j2m1m2|J+1 = (J−

1 |j1j2m1m2〉)†

= (~√j1(j1 + 1) −m1(m1 − 1)|j1j2m1 − 1m2〉)†

= ~√j1(j1 + 1) −m1(m1 − 1)〈j1j2m1 − 1m2|

�uefJLGI/2-A37Jz37E〈j1j2m1m2|J+|j1j2jm〉 = 〈j1j2m1m2|J+

1 + J+2 |j1j2jm〉B2;=-?GI<IJL3vF(�

√j(j + 1) −m(m+ 1) 〈j1j2m1m2|j1j2jm+ 1〉=√j1(j1 + 1) −m1(m1 − 1) 〈j1j2m1 − 1, m2|j1j2j,m〉

+√j2(j2 + 1) −m2(m2 − 1) 〈j1j2m1m2 − 1|j1j2j,m〉

�� `L��0=�� /QV�1�V\/

√j(j + 1) −m(m− 1) 〈j1j2m1m2|j1j2j,m− 1〉=√j1(j1 + 1) −m1(m1 + 1) 〈j1j2m1 + 1, m2|j1j2j,m〉

+√j2(j2 + 1) −m2(m2 + 1) 〈j1j2m1m2 + 1|j1j2j,m〉

�� `L� � �

�h;=-I19JPGIE�6K;=-I1j1qj2/�3jGI;=-I-?E212` ��/21.>OM@[L/2<I691mWY;@1K1KJLiI[P/�1mGI/

m19;=-A3w1KJPV&WI[L/21mMZ6K/�WI6KE p

1K/�-]3K/26v1K<I6d<I-�g=6KM@W#�IJL_U<?/&�

�h;=-I19JPGIE�6K;=-I1d<I-(3769J�M@-?g=[P/Oq

��

� efM@WI69821 [�M�W?6K/2V&JP8�6K/069/2[PMO37JL;=- GI/S69E2B2<?6K1KJL;=-q.19J5;=- B2;@-I-NM � 3�GI/2<U� B2;U/43&B2JL/2-A371�q.;=-B2;@-I-NM � 3d[P/

3 �� ` � /wV�12V&/@qI19J�;=-(B2;=-?1KJPG?826K/Q[P/.3K6KJPM@-Ig=[L/

Ge+MOWI6K8�1Q[PMXGI/2<U�?JL82V&/#6K/2[PMO37JL;=- GI/#69E2B2<?6K1KJL;=-q�JL[t1K<.3#3QGI/#B2;@-I-NM � 376K/:GI/2<c� G?/21wB2;U/43 pB2JL/2-A371wW�;=<I6wB2;@-I-NM � 376K/ [L/

3 �� ` �h;=V&V\/�37;=<I1w[L/21wB2;U/43&B2JL/2-A371QF([neo/k�U37E�6KJL/2<I6mGI/�1m>@MO[P/2<?6K1W�;=1K19JPiI[L/21 1�esMO-I-]<I[P/�-]32q@37;@<I1t[P/�1tB2;U/43&B�JP/2-A3K1uWY/�<?>=/�-]3r19/lGIE2G?<IJP69/^Geo<I- B2;U/43&B2JL/2-A3uGI< i�;=69GqWNM@6m/ �c/2V&WI[P/ MO< W�;=JL-]3��5`+�cJ ;=-�B �I;=JL1KJL3.[P/5WI69/2V&JP/�6.B2;U/43&B2JL/2-A3w6KE�/2[�qJP[L1�[P/:1K/269;=-A3w37;=<I1�` JL-NM@[P/�V\/�-A3�q?[L/mWI6K/�V\JL/26^B�;c/ 3&B2JP/�-A3x/�193v�?�cEw/2-�JPV&WY;=1KM@-A3l[�M 69/2[�M 37JP;@-

∑

m1,m2 2 (m1+m2=j)

|〈j1j2m1m2|j1j2jj〉|2 = 1�� `z��A�

�\�� & ��* )�� "+* � ! �� ,�� 'd) "�$ � ! %�� !-+ ' ! � " �� '

�ue��NM@V&JP[L3K;=-IJL/2-XGIE�B269JL>OM@-A3v<I-(E2[P/�B�3K6K;=-�M@<?3K;=<I6lGeo<I-�-I;}TAM@<�12eoE2B269JL3w�H = H0 + λ~L · ~S

�� ` � A�;=R

H0

/�193x[�M WNM@6437JP/mG?/�[�e��NMOV\JL[L37;@-IJP/�-�_]<IJ+-I/QGIE2W�/2-IG�WNM@1dGI<�1KWIJL-`��h;=V&V\/L+, L−, LzB2;@V\V5<c37/2-A3.M}>@/2B

H0

qλ~L · ~S B2;=V&V5<?3K/5M�>=/�B

H0

`�,.- W�/2<?3mG?;=-IB5[L/21�G?J�M@g=;@-NM@[PJL1K/�6vGIM@-I1<I-I/diNMO1K/vB2;=V&V5<I-I/O` � M@6�M@JL[P[L/2<I6912q ~L B2;=V&V5<?3K/dM}>@/2B

H0

/k3λ~L · ~S `U,.-\W�/2<c3�G?;=-IBdB2[PM@1K19/26[P/�1�Ek3�MO3K1�GefM@WI69821X[L/21\>OM@[L/2<I691�GI/ ~L2 ` �|M@6XB2;@-]3K6K/Oq Lz -I/HB�;=V&V5<?37/�WNM@1XM}>@/2B λ~L · ~S `,.- /�193QGI;=-?B\MOV\/�-IE#F�B �?/26KB �?/26Q[P/21wEk3�MO3K1QW?6K;=WI69/21QFZ[neoJP-A37E�6KJL/2<I6.Gjef<I- 1K;=<?14pb/�1KWNMOB2/&GI/

lGI;=-?-IE@qIV\M@JL1�/�-�B�;=<IWI[PM@-A3^GIJzyjE�6K/�-]3K/21^>OM@[L/2<I691^GI/m� iIJL/2-Z/�-]3K/2-IG?<�[PM:WNM@6437JL/�67MOGIJ�M@[L/�/�193

B2;@V\WI[L8�3K/2V&/2-A3QGIE2B�;=<IWI[LE2/}� `t�h;@V\V&/H0

/21435WI69;=WY;@6937JL;=-I-I/�[rF�[�efJLGI/2-A37Jz37E GNMO-I1:<I- 1K;=<?14p/219WNM@B�/�GI/

lG?;=-I-IEOqNJP[19<.3#3xG?/QG?J�M@g=;@-NM@[PJL1K/�6h[neo;=W�E26KMO37/�<I6

λ~L · ~S ` M@JL1

λ~L · ~S =1

2[(~L + ~S)2 − ~L2 − ~S2] =

1

2(~L+ ~S)2 − l(l + 1)

2− 3

8

�� ` �U� �,.- /�193�GI;=-IB M@V&/2-IE F�B �?/26KB �?/26H[L/21�E�37MO371�WI6K;@WI6K/�1�/k3H[L/21Z>OM@[P/�<I6K1�WI6K;@WI6K/�1�GI/

(~L +~S)2

` � esMOWI6K8�1\[L/XWNM@6KM@g=67MOW#�I/ZWI69E2B2E�GI/2-A32q�/�[P[L/21 1K;@-]3#G?/�[PMHCD;=6KV&/~2j(j + 1)

M}>@/2Bj =∣∣l − 1

2

∣∣ , . . . l + 12

`#�|;=<I6l = 0, j = 1

2 �WY;=<?6

l > 0qj = l ± 1

2

`�,.-(/2-�GIE2G?<IJL3Q�

~L · ~S|l, s, j,m〉 =~2

2(j(j + 1) − l(l + 1) − 3

4)|l, s, j,m〉

�� [��5��=� �(l +

1

2)(l +

3

2) − l(l + 1) − 3

4= l

��

⇒ ~L · ~S|l, s, l + 1

2, m〉 = ~2 l

2|l, s, l + 1

2, m〉 �� ` �=�@�

�� [ p��@�(l > 0)

(l − 1/2)(l + 1/2) − l(l + 1) − 3

4= −l − 1

⇒ ~L · ~S|l, s, l − 1

2, m〉 = ~2−l − 1

2|l, s, l − 1

2, m〉

�� ` �@�=�

� /2Bk37/2<?6K1vWI6K;=W?6K/21 ��,.-([P/�1d3K6K;=<c>=/w/2-�<?37JL[PJL17M@-A3l[L/21lB2;U/43&B�JP/2-A3K1xGI/��h[L/2iI19B �cp 5m;=6KGIM@-`�^M@1

l = 0qj = 1/2

�

〈l, s,m1m2|l, s, j,m〉 = δm2 ,m

WI<IJP19_]<I/m1 = 0 (l = 0)

�^M@1l 6= 0

��h;=-I1KJLGIE269;=-I1&[P/�B�M@1j = 1 + 1/2

`��h;=V&V\/(JL[v-jeoT M _U<jef<I-�E�3�M 3XM�>=/�Bl + 1/2 = m

qN;=-�G?;=JL3dM�>@;=JP6m�

〈l, s, l, 12|l, s, l + 1

2, l +

1

2〉 = 1

� /QV�1�V\/Oq〈l, s,−l,−1

2|l, s, l + 1

2,−l − 1

2〉 = 1

�|;=<?6 −l − 12< m < l + 1

2

q JP[r19<.3#35Geo<?37JL[PJL1K/26w[L/21:69/2[PMO37JL;=-I1wGI/�69E2B�<I6K69/2-IB�/Z<I-I/�19/2<I[L/CD;=JP1dW�;=<I6v6K/�[PJP/�6d[P/21vB2;U/43&B�JP/2-A3K1./2-A3769/:/�<c�Y` { - /kyj/k3�q�[�M#GI/2<U�?JL82V&/:69/2[�M 37JP;@-�MOWIWI[PJL_]<IE2/�Fm2 = 1

2

69/2[PJL/hGIJL6K/�B�37/�V\/�-A3 〈j1j2m1m2|j1j2j,m−1〉 F 〈j1j2m1 +1, m2|j1j2j,m〉 WI<IJL1K_]<I/〈j1j2m1m2 + 1|j1j2j,m〉 = 0

` � / V�1�V\/Oqj[PM�WI6K/�V\JL8269/:6K/�[�MO3KJP;=- M@WIWI[LJP_]<IE2/:Fm2 = −1

26K/�[PJP/mG?JP6K/�B�3K/2V&/2-A3 〈j1j2m1m2|j1j2j,m〉 F 〈j1j2m1 − 1, m2|j1j2j,m〉 `,.62qI19J

j1 = lqj = l + 1

2

/�3m = m1 + 3

2

qI;=-�M

j(j + 1) −m(m− 1) = l2 + 2l +3

4−m2 −m = (l +m+

1

2)(l −m+

3

2)

/�3

j1(j1 + 1) −m1(m1 + 1) = l2 + l −m2 − 3

4+ 2m = (l −m+

3

2)(l +m− 1

2)

��JL-I1KJ�[PM5GI/2<c�cJL82V&/Q69/2[PMO37JL;=-XB2;@-IGI<IJz3�FZ�

〈l, s,m− 3

2,−1

2|l, s, l + 1

2, m− 1〉 =

√l +m− 1/2

l +m + 1/2〈l, s,m− 1

2,1

2|l, s, l + 1

2, m〉

� ��

;=<(/2-?B2;=69/

〈l, s,m− 1

2,1

2|l, s, l + 1

2, m〉 =

√l +m + 1/2

l +m + 3/2〈l, s,m+

1

2,1

2|l, s, l + 1

2, m+ 1〉

�h/�3937/.69/2[PMO37JL;=-#WY/�<?3 1k376K/�6KE2<c37JP[LJP19E2/ � <I1K_]<efF5B2/x_]<I/.[P/vGIE�-I;=V&JP-NMO3K/2<I6r19;=JL3�E2gAMO[jFl+(l+

12) + 1

2

`

〈l, s,m− 1

2,1

2|l, s, l + 1

2, m〉

=

√l +m+ 1/2

l +m+ 3/2

√l +m + 3/2

l +m + 5/2. . .

√· · ·

2l + 1〈l, s, l, 1

2|l, s, l + 1

2,1

2〉

=

√l +m+ 1/2

2l + 1��/21d6K/�[�MO3KJP;=-?1^g=E2-?E267MO[P/21d19;=-A3��

〈l, s,m− 1

2,±1

2|l, s, l + 1

2, m〉 =

√l ±m+ 1/2

2l + 1

�� ` � �]�

〈l, s,m− 1

2,±1

2|l, s, l − 1

2, m〉 = ∓

√l ∓m + 1/2

2l + 1

�� ` � =�

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Lz/k3 ~L2

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A→ f(A)� �?`z� �

WY/�<?3v1K/wV&/�3K3K6K/w1K;@<I1d[�M:CD;=69V\/

f(A) = U−1f AUf Uf

�N;=W�E26KMO37/�<I6d[PJL-IE�M@JL6K/ � �?` �@�{ -�/kyj/k3�qN[P/�1l>@MO[P/2<?6K1lWI69;=WI6K/�1dGI/

f(A)1K;=-A3vGIE�3K/269V\JL-IE2/�1vWIM@6l[neoE2_]<NMO3KJP;=-

|f(A) − λ� [ | =

∣∣U−1f AUf − λ

� [ ∣∣= |A− λ

� [ | = 0� �?`f�A�

� � � ��

1KJλ/2193v>OM@[P/�<I6lWI69;=WI6K/mG?/

A`

{ -�V&E2B2M@-IJL_U<?/Z_U<IM@-A37JP_]<I/Oqr1K/2<?[P/21&[P/�15>OM@[P/�<I6K1#V&; T@/2-I-?/21&GI/21&;=iI19/269>OM@i?[P/21#GNMO-I1#GI/21E�37MO371 GI/r[neo/219WNM@B�/�GI/�~xJL[Pi�/2643+;=-A3|<?-I/�19JPg=-?JL�NB2MO37JL;=-` � /�V�12V&/r_U<?/�[�efEk>=;=[L<?37JL;=-.37/�V\W�;=6K/�[P[L/W�/2<?3 1�3K6K/QGIE2B�6KJL3K/�G?<HW�;=JP-A3vGI/w>c<?/:GI/�~x/�JP1K/�-]iY/�6Kg�B�;=V&V\/w[neoE�>@;=[P<c37JP;@-�G?/21�;@iI1K/�69>OM@iI[L/21;=<(GI<�WY;@JP-A3lGI/m>U<I/wGI/Q�cB �I6��cG?JP-Ig=/�6lB2;=V&V&/m<I-I/mE�>@;=[P<c37JP;@-�G?/21l>=/�B�37/�<I6K1vGeoE�37MO371�qNJP[/�193[P;@g=JP_]<I/.GjefJLV\W�;=1K/�6^_]<eo<I-I/m376KM@-I19CD;@6KV\MO37JL;=-XWI<IJL1K19/m1K/mC�M@JP69/.GI/.C�M�;=-(E2_]<IJz>@MO[P/2-A3K/m1K<I6l[L/21>=/�B�3K/2<I691xGeoE�37MO371�` � E�1KJPg@-I;=-I1dM@[L;=6K1�WNM@6

Uf[neo;=WYE�67MO3K/2<I6�B2;=696K/�1KW�;=-IGNM@-A3xF5[�MQ3767MO-I19CD;=69V�M p

37JL;=-f : |Ψ〉 → Uf |Ψ〉 � �?`��]�

ab[YC�MO<?3^_]<I/.[PM�>OM@[P/�<I6�V\;}T=/�-I-I/mGI/x37;=<?3K/.;=iI19/264>@MOiI[P/xGNM@-I1h3K;=<?3hE�3�M 3dM@JL3�[�MQV�1�V\/v>OM@[P/�<I6_]<I/2[+_]<I/w1K;@JL3d[P/mW�;=JL-]3^GI/Q>U<I/w_]<eo;=-�M@GI;=W?3K/@` �xJP-I19JnqI;=-(GI;@JL3dM�>=;@JP6

〈Ψ|U−1f AUf |Ψ〉 = 〈UfΨ|A|UfΨ〉

� �?`! =� M@JL1 〈UfΨ| = 〈Ψ|Uf

†

→ 〈Ψ|U−1f AUf |Ψ〉 = 〈Ψ|Uf

†AUf |Ψ〉

→ Uf = Uf/k3

Uf†= U−1

f

� �?`�"A�

�xJP-I19Ju[neo;=WYE�67MO3K/2<I6Uf

GI;=Jz3 1�3769/ 37/2[ _U<?/Uf

†= U−1

f

`��x<?3769/2V&/2-A3QGIJL32q<I-I/#;=W�E26KMO37JL;=-GI/w19TUV&E�3K6KJP/mW�/2<c3 1k3769/QG?E2B269JL3K/Q/�-�V&E2B2M@-IJL_U<?/m_U<IM@-A37JP_]<I/QWNM@6l<?- �� !��

U`

{ [P[L/.WY/�<?3 1�3K6K/wGIE2B�6KJz37/wGI/wGI/2<U�(C�M2;=-?1Q��=`

{A→ U †AU

�NWY;@JP-A3dGI/m>U<I/wWNM@1K19JLC|Ψ〉

� �?`!0=��c`

{|Ψ〉 → UΨ

�NW�;=JP-A3dGI/m>U<I/QM@B�3KJLCAJP-IB �IM@-Ig=E � �?` � �

�cJw<I- ;=W�E26KMO37/�<I6S/2143 �I/�6KV&JL3KJP_]<I/@qh[neo;=WYE�67MO3K/2<I6

U = eiS/2193�<I-IJz3�M@JL6K/O` { - /�yY/�32q

U † = e−iS/�3

UU † = eiSe−iS =� [�` � / 3K6K821�-I;=V:iI6K/�<I1K/�1H;=W�E26KMO37JL;=-I1ZGI/ 19TUV&E�3769JP/

W�/2<?>@/2-A3 1k3769/Q69/2WI69E219/2-A37E�/21.WNMO6l<I-�;@WYE�67MO3K/2<I6dGI/w[�M:CD;@6KV&/eiS`

1v;=<I1tM�>@;=-I1 G?;=-IBhG?E2V&;=-A376KE�_]<eo<I-5;=W�E26KMO37/�<I6|[LJP-IE2M@JP69/u/�3|<I-IJz3�M@JL6K/uW�;=<?>OM@JL3 69/2WI69E21K/�-A37/26<I-I/�;=W�E26KMO37JL;=-wGI/�14TUV\Ek376KJL/@`}ab-A>=/�6K19/2V&/2-A3�qO;=-�W�/2<c3|[LE2g=Jz37JLV\/�V\/�-]31K/�GI/2V\M@-IGI/�6+_U<?/2[A3bTUWY/Geo;=W�E267M 37/2<?6�/�193�1K<I19B2/2Wc37JPi?[P/vGI/x69/2WI69E219/2-A37/�6h<I-I/v;=W�E26KMO37JL;=-#GI/x14TUV\Ek376KJL/@` � /vB2/vW�;=JP-A3�GI/>U<I/@q@[P/�B26KJz378�6K/lM@G?;=W?37E�/2143r/2- g=E�-IE26KM@[c_]<I/l[L/21 � �� l19;=JP/�-A3tB2;@-I1K/�69>=E�/21�` �x<?3769/2V&/2-A3GIJz3�qI19J�<I-I/w;=W�E267M 37JP;@-Z/2193vGIE�B269JL37/wWNMO6Q�

|Ψ〉 → U |Ψ〉 � �?` �A�

��

JP[jC�MO<?3d_]<I/|〈UΨ|UΦ〉|2 = |〈Ψ|Φ〉|2

� �c`L�� =�� JPg=-?/26+MlGIE2V&;=-A3769Er<I-�69E219<I[L37MO3+g=E2-?E267MO[ � B2;@-I-]<:1K;@<I1+[P/u-I;=V GI/u3%�IE�;=6K8�V\/rGI/ � JPg@-I/267�

1K/�[P;=-�[P/�_U<?/2[�[P/�1d;=WYE�67MO3KJP;=-?1lGI/�14TUV\Ek376KJL/m-I/QWY;=<c>@MOJP/2-A3 1k376K/Q6K/�WI6KE�1K/�-]3KE2/�1._]<I/�WIM@6lGI/2<U�3bTUWY/�1\Geo;=W�E26KMO37/�<I6K1��u[P/�1 ;=WYE�67MO3K/2<I6#<I-?JL3�MOJP6K/�1 � >@;=JP6 B�JzpbG?/21K19<I1��5/k3\[L/21#;=W�E26KMO37/�<I6&M@-A37Jzp<I-IJz3�M@JL6K/21�qIB@eo/2143xF&GIJL6K/m[L/21l;=W�E26KMO37/�<I6K1

K1KMO37JL19C�M@JL17M@-A3dMO<c�(GI/�<c��WI6K;@WI6KJLE�37E�1d1K<IJz>@MO-]3K/21w�

�=`�� /�193�M@-A37JL[PJP-?E�M@JL6K/Q�

K(|Ψ〉 + |Φ〉) = K|Ψ〉 +K|Φ〉/�3K(c|Ψ〉) = c∗K(|Ψ〉)

� �?`L�@� ��c`

〈KΨ|KΦ〉 = 〈Ψ|Φ〉∗ (= 〈φ|Ψ〉)� �?`z� �=�

� M@69V\Ju[L/21w19TUV&E�3K6KJP/�1�B�;=<I6KM@V\V&/2-A3w69/2-IB�;=-A376KE�/215/�- W#�ATU1KJL_]<I/@q 1K/�<I[P/#[neo;=W�E26KMO37JL;=- 6K/2-Up>=/�6K1K/�V\/�-A3mGI<�3K/2V&WI1v/2193x6K/�WI6KE�1K/�-]3KE2/ WNMO6v<?-H;=W�E26KMO37/�<I6�MO-]3KJzp <I-IJL37M@JP69/ � >@;=JP6dB�JzpbG?/21K19<I1�� `� ;@<?37/�1x[L/21dM@<?3K6K/�1v;@WYE�67MO3KJP;=-I1l19;=-A3v6K/2W?6KE219/2-A37E�/21.WIM@6l<I-�;=WYE�67MO3K/2<I6l<?-IJL37M@JP69/@`

�� $ � ' + + � ,^"�$|* ' +�� ! %�� ' ! " � � ' +� M@-I1hB2/�3937/m19/2B�3KJP;=-jq?-I;=<?1lM@[P[L;=-I1�WNM@191K/26h/2-(6K/k>c<?/.[P/�1h376KM@-I19CD;@6KV\MO37JL;=-I1�Gef/�1KWNMOB2/m/�3^GI/

37/�V\WI1^CD;@-IGNM@V&/2-A3�MO[P/21h/2-(V\E�B�M@-?JP_]<I/._]<NM@-A3KJP_]<I/m-I;=-(69/2[�M 37JL>UJL1937/Oqc/k3l-I;=<?1dM@[P[L;=-I1hEk3�M@i?[PJP6[�M5CD;=69V\/.G?/Q[�ef;@WYE�67MO3K/2<I6l_]<IJ�[L/2<I6l/�193�M@1K19;cB�JPE@`

�� ,./0�"�� 0,�� / �.��0�"� �� /0��

{ - V&E2B2M@-IJP_]<I/ B�[�M@191KJL_U<?/@q�<I-I/ 3K67M@-I14CD;=6KV\MO3KJP;=-�Geo/219WNM@B�/\B�;=6K69/21KW�;=-IG FZ<I-I/ CD;=-IBk37JL;=-GI/21vB2;U;=69GI;=-I-IE�/21w�

~r → f(~r)� �?`z�}�A�

� efM@WI6K8�1h[L/vW?6KJP-?B2JPW�/lGI/vB2;=696K/�1KW�;=-IGNM@-?B2/@qU[�MmV�12V&/l3767MO-I19CD;=69V�M 37JP;@- /2-�V&E2B2M@-IJL_U<?/v_]<NMO-cp37JL_U<?/QG?;=JL3 1k3769/QG?E2B269JL3K/QWIM@6

~r → f(~r)� �?`L� �]�

M@JL1lB2/�3937/w376KM@-I19CD;@6KV\MO37JL;=-XGI;=Jz3dWY;@<?>=;=JL6^1K/QGIE�B26KJL6K/mWNMO6v<?-I/m3767MO-I19CD;=69V�M 37JP;@-X<I-IJz3�M@JL6K/5�

f(~r) = U †f ~rUf

� �c`L� @��cJ�[neo;=-�M@GI;@W?37/m[L/mWY;=JL-A3dGI/m>c<?/��M@B�3KJLC��OqI[L/21��O/�371v1K;@-]3d3K67M@-I14CD;=6KV&E21l19/2[L;=- �

|Ψ〉 → Uf |Ψ〉 � �?`L� "A�

� �#� ��

�u6K;@WY;=19JL3KJP;=- �cGNM@-I1h<I-I/.376KM@-I14CD;=6KV\MO37JL;=-&GIE��N-?JP/mWNM@6~r → f(~r)

qc[�M�CD;@-IB�3KJP;=-XGjef;=-?GI/.1K/376KM@-I14CD;=6KV&/m1K/�[P;=-

Ψ(~r) → Ψ(f−1(~r))� �?`z�0=�

� E�V\;@-I193K67MO3KJP;=- �NWNM@6lG?E��N-IJz37JL;=-(GI/21l>=/�B�3K/2<I691�WI69;=WI6K/�1dGI/~r;=-�MZ�

~r|f(~r0)〉 = f(~r0)|f(~r0)〉� �?`z� � �

��/2V&V&/ �Uf |~r0〉 = |f(~r0)〉

� �?`z��A�� E�V\;@-I193K67MO3KJP;=- �

f(~r) = U †f ~rUf

⇒ ~rUf = Uff(~r)

⇒ ~rUf |~r0〉 = Uff(~r)|~r0〉= Uff(~r0)|~r0〉= f(~r0)Uf |~r0〉

� �?` � A�

Uf (~ro)/2143 GI;=-?BX>=/2Bk37/�<I6 WI6K;@WI6K/XG?/

~rG?/�>OM@[P/�<I6:WI69;=WI69/

f(~r0)q JL/@q

Uf |~r0〉 = |f(~r0〉� M�>@/2BQ<I-�B �?;=Jz�XGI/�W �NM@1K/QF [�M B�[PE}� `6 /�>@/2-I;=-?1.F [�M5WI69;=W�;=1KJz37JP;@-`

|Ψ〉 → Uf |Ψ〉Ψ(~r) = 〈~r|Ψ〉 → 〈~r|Uf |Ψ〉 = (U †

f |~r〉)†|Ψ〉 M@JL1

f−1(~r) = Uf ~rU†f{ -�/kyj/k3�qNWNMO6vG?E��N-IJz37JL;=-q

f−1(f(~r)) = ~r/�3Uff(~r)U †

f = UfU†f ~rUfU

†f = ~r

��JL-I1KJ�qIGefM@WI69821d[P/m[L/2V&V\/Oq

U †f |~r〉 = |f−1(~r)〉

→ 〈~r|UfΨ〉 = 〈f−1(~r)|Ψ〉 = Ψ(f−1(~r)).� �?`s�c� �

�h;=-?B2[P<?1KJP;@- �UGNM@-I1�<I-I/x3K67M@-?19CD;=69V�MO3KJP;=-~r → f(~r)

qU[�MmCD;=-IBk37JP;@-\Geo;=-IG?/.1K/v3767MO-I19CD;=69V\/1K<?JL>OM@-A3

Ψ(~r) → Ψ(f−1(~r))`

��

� � � ��

�h;=-I19JPGIE�6K;=-I1d<I-?/w3K67M@-?1K[�M 37JP;@-ZWNM@6aGjef<I-�19TU143782V&/�F#<?-�GI/�g=6KEwGI/w[LJPi�/2693KE5�x→ x+ a

,.-�B �I/�6KB �I/�<I-(;=W�E267M 37/2<?6T (a)

37/2[�_]<I/T †(a)xT (a) = x + a

� �c` �=�@�{ -�V5<?[L37JLWI[PJPM@-A3^F gAMO<IB �I/QWNM@6

T (a)qNJP[Y>UJL/2-A3Q�

[x, T (a)] = aT (a)� �?`s�@�A�

�^M@[PB�<I[P;@-I1T (a)|x〉 `�,.-�M(�

(xT (a) − T (a)x)|x〉 = aT (a)|x〉xT (a)|x〉 − xT (a)|x〉 = aT (a)|x〉

x(T (a)|x〉) = (x+ a)T (a)|x〉⇒ T (a)|x〉 = |x+ a〉 � �?`s� �]�

V\;UGI<I[L;([P/&B �I;=Jz� Gjef<I- C�M@B�3K/2<I6QGI/\W#�NM@19/@` ,m- 6K/k376K;@<?>=/\[P/#6KE�1K<I[z3�MO3�g=E2-?E267MO[�W?6KE2B�E2GI/�-A3�`,.-�/�-�GIE�GI<IJz3d-I;@3�MOV\V&/2-A3d_]<I/

Ψ(x) → 〈x|T (a)|Ψ〉 = (T (a)†|x〉)†|Ψ〉= 〈x− a|Ψ〉 = Ψ(x− a)

� �c` � @�1x;@3K;=-I1&M@<�WNM@1K1KM@g=/�_]<I/H[L/21#;=W�E267M 37/2<?6K1

T (a)CD;=6KV&/2-A3&<I-�g@6K;=<IW�/@` { -�WIM@693KJPB2<?[PJP/�62q

T (a)T (b) = T (b)T (a) = T (a+ b)`

{ 1K17M�T@;=-I1ZV\M@JP-A37/�-NM@-A3&GI/�3K6K;=<c>=/26\<I-I/�CD;=6KV&/(/k�cWI[PJLB2Jz37/(GI/H[�ef;@WYE�67MO3K/2<I6T (a)

` � ;=<I6B2/�[�M?q B2;=-?1KJPG?E26K;@-I1:<I-?/�376KM@-I1K[PMO37JL;=- JP-?�N-?JL37E�1KJLV�M@[L/

T (δ)q+/�3�C�M@JP19;=-I1Q<I-SGIEk>=/2[L;=WIW�/2V&/2-A3

[PJLV\Jz37E�G?/T (δ)

�T (δ) =

� [ − iδK +O(δ2)� �c` � "=�

��/#C�MOB�37/�<I6 −1/�193�B2;=-A>@/2-A37JL;=-I-I/�[nq�V\M@JP1.[L/ B �I;=J ��Geo<I- C�MOB�37/�<I6

i/2143�JL-A37E269/21917M@-A3�B�M@6mJL[

B2;=-?GI<IJL3vF#<I-�;=W�E267M 37/2<?6K

�I/�6KV&JL37JL_]<I/@` { -�/�yY/�32qT (δ)

E�37M@-A3x<?-IJL37M@JP69/@q?;@-HMZ�T †(δ)T (δ) =

� [1K;=Jz3

� [ − iδK + iδK† +O(δ2) =� [

⇒ K† = K� �?` � 0=�

6 /�WY;@6937;@-I1dB2/wGIE�>@/2[L;=WIW�/2V&/2-A3v[PJPV&JL3KE�GNMO-I1d[�M56K/�[�MO3KJP;=-[x, T (a)] = aT (a)

`Nab[>UJP/�-]3w�[x,

� [ − iδK] = δ� [

+O(δ2)

→ [x, K] = i� [ � �?`s� � �

� ��

,.-�/�-�G?E2GI<IJz3d[P/m69E21K<?[L3�M 3l376K8�1dJPV&WY;=643�M@-A3l19<IJL>OM@-A3w�

K =p

~

� �?` � �A�

{ -?�N-jq|;=- WY/�<?3QiIJP/�- 1��I6�G?E2B2;@V\W�;=1K/�6�<?-I/&3K67M@-?1K[�M 37JP;@- �I-IJP/#/2- WI6K;UGI<IJz3QGI/&3K67M@-?1K[�M p37JL;=-I1m�

T (a) = T (a

N)N

= [� [ − i

a

nK +O(

1

N2)]N

M@JL1limN→∞

[1 +x

N+O(

1

N2)]N = ex

� �?`s� A�{ -�/kyj/k3�q

limN→∞

ln[(1 +x

N+O(

1

N2))N ] = lim

N→∞N ln[1 +

x

NO(

1

N2)]

= limN→∞

N(x

NO(

1

N2))

= x

��JL-I1KJ�q

T (a) = exp(−ipa/~)� �?`f�?� �

,.-�GIJL3d_]<I/w[neo;=W�E26KMO37/�<I6p/2143�[L/�� ! �� ! � � � � �� ! �k`

�+M�g=E�-IE26KM@[PJL17MO3KJP;=-�FX�\GIJPV&/2-I19JP;=-?1v/�193mJLV\V&E2G?J�MO3K/@`��+M\3K67M@-I19[�MO3KJP;=-�WNM@6v<I-H>@/2Bk37/2<?6~a/2143vGIE��N-?JP/wWNM@6

T (~a)†~rT (~a) = ~r + ~a� �?`f�A�=�

�uef;@WYE�67MO3K/2<I6T (~a)

>@E269JL�N/T (~a)|~r〉 = |~r + ~a〉

� �?`f�=�A�� /mC�M�;=-�/ �cWI[PJLB2JL3K/@qcJP[W�/2<?3v1�efE�B26KJL6K/#�

T (~a) = exp(−ip~a/~)� �?`f� �]�

� � � � ��

�h;=-?1KJPG?E26K;@-I1�G?E21K;@6KV\M@JP1w<I-I/&6K;O3�MO3KJP;=-R~α

GefM@-Ig=[L/α = ‖~α‖ ` �uef;=W�E26KMO37/�<I6wGIE2B�6KJz>@MO-]3B2/k3K3K/Q69;@3�M 37JP;@-Z/2-�V\E�B�M@-IJL_]<I/w_U<IM@-A37JP_]<I/w/�193vGIE��I-IJ+WNM@6

R†(~α)~rR(~α) = R~α(~r)� �?`f�� =�

��

� efM@WI6K8�1d[P/21�CD;=6KV5<?[P/21�g=E2-?E267MO[P/21hGI/.376KM@-I19CD;@6KV\MO37JL;=-I1�Gef/�1KWNMOB2/@qNB�/�3d;=W�E26KMO37/�<I6lM@g@JL3^19/2[L;=-

R(~α)|~r〉 = |R~α(~r)〉� �?`f� "A�

�|;=<?6x3769;=<?>@/26.1KM\CD;=6KV&/w/k�cWI[LJPB2Jz37/Oq�B�;=-I19JPGIE�6K;=-I1�F\-I;@<?>=/2M@<�<I-I/�6K;@37MO37JL;=-�JL-?�N-IJz37E�1KJPV\M@[L/@`� efM@WI6K8�1xB�/w_U<jef;=-�M >c<�1K<?6v[PM56K;@37MO37JL;=-ZGI/�1x19WIJP-I1�qN;=-�MZ�

Rδ~n = ~r + δ~n ∧ ~r +O(δ2)� �c`s��0@�

;=R~n/2143�<I-(>=/�B�3K/2<I6v<I-IJz3�M@JL6K/@`.1x;=<?1vB �?/26KB �?;=-I1�[�ef;@WYE�67MO3K/2<I6 R(δ~n)

1K;=<?1v[PM:CD;=6KV&/

R(δ~n) =� [ − iδK +O(δ2)

� �c`s� � �

��/mB2;=V&V5<?37MO37/�<I6dM�>=/2BQ<I-I/wB�;=V\W�;=1KM@-A37/xiGI/~r12eoE2B�6KJz3��

[xi,R(δ~n)] = δ(R(~n ∧ ~r))i +O(δ2)Geo;=R − i[xi, K] = (~n ∧ ~r)i1K;=Jz3[xi, K] = iεijknjxk

� �c`s� �=�,.6:-I;=<?1 M�>=;=-I1�>c<S[L;=6K1QGI/X[�efEk37<IG?/�GI<0V&;=<?>@/2V&/2-A3 B�/2-A376KM@[h_]<I/X[L/\V&;=V&/2-A35B�JP-IEk37JL_U<?/17MO3KJP14C�M@JP1KM@JL3h[P/21d6982g=[L/21lGI/wB2;@V\V5<c3�MO3KJP;=-

[Li, xj] = i~εijkxk

[~L · ~n, xj] = i~εijknixk

;=<�/2-IB�;=6K/

[xi, ~L · ~n] = i~εijknjxk

K =~L · ~n

~

� �c`�� =�

{ -�E�B26KJz>OM@-A3 R(~α) = R( ~αN

)NqN/�3v/2-�WI6K/�-NM@-A3v[�M [PJLV\Jz37/

N → +∞ qNJP[Y>cJL/2-A3w�

R(~α) = exp(−i ~α · ~L~

)� �c`��I�}�

,.- 69/�3769;=<?>@/&[L/ 6KE�1K<I[z3�MO3wEk3�M@iI[LJ [L;=6K1mGI/#[�efEk37<IGI/ G?/&[PMX6K;@37MO37JL;=-�GI< 1KWIJL-`+,.- GIJL3w_]<I/#[L/V\;@V\/�-]3dB�JP-IEk37JL_U<?/m/2193x[P/ � ��! �� ! �k`

� � � ��

� � � ��

��eo/2143�[�ef;=W�E26KMO37JL;=-Z_]<IJ�B2;@-I1KJL1937/QF JP-A>@/26K19/26v[P/m19JPg=-I/mG?/21vB2;U;=6KGI;@-I-IE2/�1w�~r → −~r

� �?`��]�=��ueo;=WYE�67MO3K/2<I6vM@191K;UB2JLE@qN/2-�g=E�-IE26KM@[�-I;@3KE

ΠqN/2143vGIE��N-?J WIM@6

Π|~r〉 = | − ~r〉� �c`��A�=�

,.-�/2-�GIE2GI<?JL3d_]<I/Π2 =

� [�`��/21dCD;=-IB�3KJP;=-?1lGef;@-IGI/w1K/m376KM@-I14CD;=6KV&/2-A3v1K<IJz>OM@-A3w�Ψ(x, y, z) → Ψ(−x,−y,−z) � �?`�� ]�

�+M 3767MO-I19CD;=69V�M 37JP;@-ΠB2;=-I19/264>=/Q[P/mWI69;cGI<?JL3d1KB2M@[�MOJP6K/O` { -�/�yY/�3�q

∫Ψ∗(−~r)Φ(−~r)d~r =

∫ ∣∣∣∣d(−~r)d~r

∣∣∣∣Ψ∗(~r)Φ(~r)d~r =

∫Φ†(~r)φ(~r)d~r

� �c`�� @�

Π/2193XGI;=-IB�<I-IJL37M@JP69/ �

Π† = Π−1`h�h;=V&V\/

Π2 =� [�q�;@-�/2- G?E2GI<IJz3

Π† = Π`��/21

;=W�E26KMO37/�<I6K1d1K/m3K67M@-I14CD;=6KV&/2-A3vGI;=-?BQ19<IJL>OM@-A3w�A→ ΠAΠ

� �?`��"A��|;@<I6

~rq?;=-�1�esM 3K37/�-IG�F 3K6K;=<c>=/26Q�

Π~rΠ = −~r� �?`�� 0=�

{ 3v/2-�/�yY/�3�qN;@-�M(�Π~rΠ|~r〉 = Π~r| − ~r〉 = −~rΠ| − ~r〉 = −~r|~r〉

Π~rΠ = −~r� �?`��

� /wV�12V&/ qΠ~pΠ = −~p ` � E2V&;=-A376K;@-I1d[P/mW�;=<I6l<I-I/QB2;@V\W�;=17MO-]3K/ �

[x, p] = i~

Π[x, p]Π = i~

ΠxΠΠpΠ − ΠpΠΠxΠ = i~

[x,ΠpΠ] = −i~ΠpΠ = −p

� �?`�� A�� MO6vB�;=-A3769/@qN[P/mW?6K;UGI<IJL3

xipj-I/wB �NMO-Ig=/QWNM@1dGI/w1KJLg=-I/@` { -�/�yY/�32q

ΠxipjΠ = ΠxiΠΠpjΠ = (−xi)(−pj) = xipi� �?` A�

,.-�/�-�G?E2GI<IJz3d_]<I/�[L/.V&;=V\/�-A3dB2JP-?E�37JL_]<I/w/2193vJL-]>OM@69J�M@-A3m�

Π~LΠ = ~L� �?` U� �

��/21+>=/�B�37/�<I6K1 _U<?JUB �IM@-Ig=/�-]3 GI/�1KJPg@-I/rGNM@-I1|<I-I/�;=WYE�67MO3KJP;=-wGI/�WNM@69JL3KE�19;=-A3tMOWIWY/�[PE�1�� k`��h/2<U��_U<?J�-I/wB �NM@-Ig=/�-A3�WNMO1vG?/Q19JPg=-?/m1K;=-A3xM@WIW�/2[LE21�� =`

��

� �� /� � ��

ab[�M@696KJL>@/w1K;=<?>@/2-A3x_U<?/�[L/m[�M@g=6KM@-Ig=JL/2-X19;=JL3dJL-IGIE�WY/�-IGNM@-A3vGI<�1KJPg@-I/mGI/q

Lcl(q, q) = Lcl(−q, q)� �?`! =�=�

;=<�_U<?/Q[L/w~�M@V&JP[z37;=-?JP/2-\1K;=Jz3dJP-A>OM@6KJPM@-A3lWNM@6dB �NMO-Ig=/2V&/2-A3vGI<�1KJLg=-I/mGI/~p�

H(~p,~r) = H(−~p,~r)� �?` O�A�

� M@-I1vB2/wB2M@12qN19J~r(t)

/2143�19;=[P<c37JP;@-q~r′(t) ≡ ~r(−t) /�193x1K;=[L<?37JL;=-`

� E2V&;=-A3769;=-I1m[L/5GNM@-I1m<?-�B�M@1m19JPV&WI[P/\�+�c;=Jz3q(t), p(t)

<I-I/ 19;=[P<?3KJP;=-HGI/21mE�_U<IMO37JL;=-I1mGI/~�M@V&JP[z37;=-H�

{q(t) = ∂2H(q(t), p(t))

p(t) = −∂1H(q(t), p(t))

� �c` �A�/�3v1K<?WIWY;@1K;=-I1v_]<I/

H(q,−p) = H(q, p)

→ −∂2H(q,−p) = ∂2H(q, p)� �?` =�

� E2V&;=-A3769;=-I1m_]<I/ {q(t) ≡ q(−t), p(t) ≡ −p(−t)} /2143w1K;=[L<?37JL;=-` �+/�1wE2_]<NMO3KJP;=-I1�GI/#~�M pV\JL[L3K;=-ZM@WIW?[PJP_]<IE�/21vM@<Z37/�V\W?1 −t GI;=-I-I/�-A3:�{q(−t) = ∂2H(q(−t), p(−t))p(−t) = −∂1H(q(−t), p(−t))

� �c` "=�

,.62q( ˙q(t) = −q(−t) q ˙p(t) = p(−t)) `Y,m-(/�-�GIE�GI<IJz3��

{− ˙q(t) = ∂2H(q(t),−p(t))˙p(t) = −∂1H(q(t),−p(t))

M@JL1

∂1H(q(t),−p(t)) = ∂1H(q(t), p(t))

∂2H(q(t),−p(t)) = −∂2H(q(t), p(t))

��JL-I1KJ�q{

˙q(t) = ∂2H(q(t), p(t))˙p(t) = −∂1H(q(t), p(t))

�� ,.-�/�193vGI;=-IBw-NM 37<I69/2[P[L/2V&/2-A3dM@V&/2-?E�F B�;=-I1KJLGIE269/26d[�M53K67M@-I14CD;=6KV\MO3KJP;=-XG?E��N-IJL/wWNM@6m�

{θ−1~rθ = ~r

θ−1~pθ = −~p

� ��

E2_]<IJz>OM@[P/�-]3vF(�{θ~rθ−1 = ~r

θ~pθ−1 = −~p�u6K;@WY;=19JL3KJP;=- ��+M5376KM@-I19CD;@6KV\MO37JL;=-

θ/2143�MO-]3KJP[LJP-IE2M@JP69/@`

� E�V\;@-I193K67MO3KJP;=- � � efM@WI69821d[P/�1d3K67M@-?19CD;=69V�MO3KJP;=-?1^GI/~r/k3~pq

θ[xi, pj]θ−1 = θxipjθ

−1 − θpjxiθ−1

= θxiθ−1θpjθ

−1 − θpjθ−1θxiθ

−1

= xi(−pj) − (−pj)xi= −[xi, pj]

� �?` 0=��^M@[LB2<I[L;=-I1 〈θΦ|θ[xi, pj]|Ψ〉 GI/��:C�M�;=-I12`

〈θΦ|θ[xi, pj]|Ψ〉 = 〈θΦ|θi~δij|Ψ〉 � �?`! � �V\M@JP1lMO<I1K19J

〈θΦ|θ[xi, pj]|Ψ〉 = 〈θΦ|θ[xi, pj]θ−1θ|Ψ〉= 〈θΦ| − [xi, pj]θ|Ψ〉= −〈θΦ|i~δijθ|Ψ〉

⇒ θi|Ψ〉 = −iθ|Ψ〉� �c` �=�

� efM@WI69821v[P/.3%�?E2;=6982V&/mGI/ � JPg=-I/�62qI[PM53767MO-I19CD;=69V�M 37JP;@-θ/2193vGI;@-IBQM@-A37Jzp <I-IJz3�M@JL6K/@`

� �I/�6KB �I;@-I1 V\M@JP-A37/�-NM@-A3 <I-I/ CD;=6KV&/ /k�cWI[LJPB2Jz37/ GI/ [�M 376KM@-I14CD;=6KV\MO37JL;=-θ` � efM@WI69821

θ−1~r θ = ~rqI;=-�M

~rθ|~r〉 = θ~r|~r〉 = θ~r|~r〉 = ~rθ|~r〉 � �?`�" A�� ;=-?B

θ|~r〉 /2143d>=/2Bk37/�<I6�WI69;=WI69/mGI/ ~r qNGI/m>OM@[P/�<I6lWI69;=WI6K/ ~r `��cJ�;@-(C�MOJL3l[L/mB �I;=Jz�ZGI/wW#�NM@19/

θ|~r〉 = |~r〉 � �?`�"?� �[�M 376KM@-I14CD;=6KV\MO37JL;=-(/2143mWNM@64C�M@JL3K/2V&/2-A3vGIE��N-?JP/@` { - /�yY/�32qYB�;=V&V\/w/2[L[P/Q/2143mM@-A37JL[PJL-IE�M@JL6K/Oq?;=-M

θc|~r〉 = c∗|~r〉� �?`�"A�=�

� <�B2;=<?WqNWY;@<I6d<I- �@/�3 |Ψ〉 GI;@-I-IEwWNM@6

|Ψ〉 =

∫d~rΨ(~r)|~r〉

� �?`!"@�A�JP[Y>cJL/2-A3Q�

θ|Ψ〉 =

∫d~rΨ∗(~r)|~r〉

�� /� � � � ��

1K;=Jz3θ|Ψ〉 = |Ψ∗〉

� �c`!" �A�{ -?�N-qN-I;O37;=-I1d[PM5[P;=JG?/w3K67M@-?19CD;=69V�MO3KJP;=-�GI<�V&;=V&/2-A3dB2JL-IE�3KJP_]<I/ �

θ−1 ~Lθ = −~L� �c`!"� @�

_]<IJ�GIE2B�;=<I[L/QG?/QB�/2[L[P/21lG?/~r/�3~p/�3vGI/w[�M5GIEk�N-IJz37JP;@- ~L = ~r ∧ ~p `

�� 0�0$ � ��0� �� -,'��$�$��

1x;=<?1mM�>=;=-I1�GIE � F\>U<�_]<I/ 19J|[�e��NMOV\JL[L37;@-IJP/�-�Ek3�M@Jz3�JL-IGIE�WY/�-IGNM@-A3mGI< 3K/2V&WI12q�[neoE�>@;=[P<c37JP;@-37/�V\W�;=6K/�[P[L/ � _U<?/m[neo;=-XW�/2<?3dB�;=-I1KJLGIE269/26lB�;=V\V&/�<?-I/�376KM@-I19[�MO3KJP;=-XGIM@-I1^[L/�37/�V\WI17��W�;=<?>OM@JL31�3K6K/w6K/�WI6KE�1K/2-A3KE2/:WIM@6d<I-�;@WYE�67MO3K/2<I6lGeoE�>@;=[P<c37JP;@-

U(t) = e−i~Ht � �c`!" "=�

� <�WY;=JL-A3lGI/m>U<I/wGI/w~x/�JP1K/�-]iY/�6Kg � ;=<(W�;=JP-A3lGI/m>U<I/wWNM@191KJzC�� qIB2/m19;=-A3x[L/21l;=i?1K/264>OM@iI[P/�1d_U<?J1K;=-A3d376KM@-I14CD;=6KV&E2/�1Q�

AH(t) = U †(t)AU(t)� �?`!" 0=�

� <�WY;=JL-A3dGI/m>c<?/�GI/��cB �?6 �UGIJP-Ig@/262q?B2/w1K;=-A3v[P/�1l>=/2Bk37/�<I6K1xGefEk3�MO3K1d_U<?J�1K;=-A3d376KM@-I19CD;@6KV&E21w�

|Ψ(t)〉 = U(t)|Ψ(0)〉� �c`!" � �

� efM@WI6K8�1w[�M�CD;=69V\/5G?/&[�ef;@WYE�67MO3K/2<I6.GjefEk>=;=[L<?37JL;=-qj;@- >@;=JL3.GIJP69/2B�3K/2V&/2-A3Q_]<I/ [ne��NM@V&JP[z37;=-IJL/2-/2143x[L/mg=E2-?E267M 37/2<?6xGI/�1l3767MO-I1K[PMO37JL;=-I1^GNMO-I1d[P/.37/�V\W?12qNB2/w_]<IJ+GIE�B2;=<I[L/wM@<I191KJ+GI/#�

dAHdt

=1

i~[AH , H]

� �c`!" �=�

�� !�� $|* � ! ) ' ' " ��*/+ %(' ) �:!-+ ' $� � "+*��!�+MxWI69/2V&JP8269/^M@W?WI[PJLB�MO3KJP;=-wGI/^B�/21tB�;=-I19JPGIE�67MO3KJP;=-I1 GI/^14TUV\Ek376KJL/@q /�3 WI6K;=iIM@iI[P/�V\/�-A3t[�MvWI[P<?1

JPV&WY;@693�MO-]3K/@q�/2143�B�/2[L[P/:;@R [�e��IM@V\JL[L3K;=-IJP/�-(/2193QJL-]>OM@69J�M@-A3�WNM@6.<I-?/ ;=WYE�67MO3KJP;=-�G?/ 19TUV\Ek3769JP/@`� M@6#/ �?/�V\W?[P/ZGNM@-?1&<?-�W�;@37/�-A37JP/�[^B2/�-A3767MO[nqu[ne��NM@V&JP[z37;=-?JP/2- /2143&V�M@-?JLCD/21437/�V\/�-]3#JL-A>@MO6KJ�MO-]3WNM@6�6K;@37MO37JL;=-` � /xC�M2;=-\g=E2-?E267MO[P/@q]B�/�3K3K/�JL-]>OM@69J�M@-?B2/v1K/x3K67M@G?<IJL3�WNM@6�[P/dC�M@JL3�_]<eoJP[�/k�cJL1937/x<I-;=W�E267M 37/2<?6

K� /�3�V�1�V\/d<I-?/vC�M@V&JP[L[P/�Gef;@WYE�67MO3K/2<I691hGNM@-?1�[L/vB2M@1hG?/21�19TUV\Ek3769JP/21�B2;=-A3KJP-]<I/217�

37/�[�_U<?/K−1HK = H ⇔ [H,K] = 0

� �?`!0 A�� /.C�M�;=-�37;@<?3dF�C�M@Jz3^g=E�-IE26KM@[P/OqcB�/�3937/m6K/�[�MO3KJP;=-\JPV&WI[PJL_]<I/d_U<?/m1KJ

Ψ/�193lEk3�MO3lW?6K;=WI69/.GI/

Hq

K|Φ〉 [neo/2143vM@<I191KJn` { -�/�yY/�32qN1KJ H|Φ〉 = E|Φ〉 q

HK|Φ〉 = KH|Φ〉 = KE|Φ〉 = EK|Φ〉 � �c` 0c�}�

� � � ��

��JL-I1KJ�q�1KJK|Φ〉 -eo/21435WNM@1QWI69;=WY;@6937JL;=-I-I/�[rF |Φ〉 q+[�MX>OM@[P/�<I6�W?6K;=WI69/ E /�193�GIE2g=E�-IE269E2/O`u�cJ

[neo;=W�E26KMO37/�<I6 � /2143dGIJ�M@g@;=-NM@[LJP17MOiI[P/lB�/2[�M5JLV\W?[PJP_]<I/d_]<eo;=-(WY/�<?3lGIJPM@g=;=-IM@[PJL1K/26H/k3KGIM@-I1

<I-I/wiNMO1K/wB2;=V&V5<I-?/@`{ -?�N-q?1KJ

K/2143 �I/269V\Jz37JP_]<I/Oq?B�/2[�M�JPV&WI[PJL_U<?/�_]<eoJP[jB�;=6K69/219WY;=-?G�F5<?-I/�� !�� ! � �� k`

� /#B2/&W�;=JP-A3wGI/ >U<I/@q+[L/21mGIJzyjE�6K/2-A3K/21Q19TUV\Ek3769JP/21wWI69E219/2-A37E�/21:GIM@-I1Q[�MX1K/�B�3KJP;=- WI6KE�B2E�GI/2-A37/� WIM@6KJz37E@qu69/2-A>=/�6K19/2V&/2-A3ZGI< 37/2V&WI1�qr3K67M@-?1K[�M 37JP;@-I12qt69;@3�M 37JP;@-��:1K/�1KE�WNM@6K/�-A3X-NMO3K<I6K/�[P[L/2V&/2-A3/2-�GI/�<c��B�[�M@191K/21 �^[L/21Z14TcV&E�3K6KJL/21ZGIJL1KB2698�3K/21(/�3([L/21Z19TUV&E�3K6KJP/�1ZB2;=-A3KJP-]<I/21�`��t37<IGIJL;=-I1Z[L/21B2;@-I1KE�_U<?/2-IB�/21.GI/w[�efJL-]>OM@69J�M@-?B2/.GI/w[�e��NMOV\JL[L37;@-IJP/�-\GNMO-I1dB2/21vGIJzyjE�6K/2-A3K1vB2M@12`

�� ,'�� "�� , ��.��

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� ��

� MO-I1XB2/HB�M@1�qh[neo;=W�E26KMO37/�<I6θ-eo/2193X-IJ �?/26KV&JL3KJP_]<I/Oq�-?Jx<I-?JL3�MOJP6K/O` � M@6\B2;=-A3769/@q�JP[dWY/�<?3

B2;@-IGI<IJL6K/hGIM@-I1tB�/26937M@JP-?1 B�MO1tF�<I-?/^GIE2g@E2-IE�6K/219B2/�-IB2/O`U�h/^1K/�67M�[L/hB�MO1 W�;=<I6 [L/21|E�3�M 371tWI69;=WI69/21GI/

H37/�[P1l_]<I/

θ|Φ〉 -?/�19;=JL3lWIM@1dWI6K;=W�;=6437JP;@-I-I/2[F |Φ〉 `�|;@<I6tWI69E2B2JL1K/�6 GNM@-?1t_]<I/2[L1tB�M@1|;=- W�/2<?3 12efMO3K3K/2-IGI69/lFx3K6K;=<?>@/26t<?-I/^GIE�g=E2-IE�6K/�1KB2/�-IB2/Oq]6K/2V\M@6 p_]<I;=-I1�37;=<?3hGefM@iY;@6KGX_]<I/

θ2-I/.B �NMO-Ig=/.WNMO1h[P/x19TU193K82V&/@`cab[jGI;=Jz3hG?;=-IB 1k376K/xWI6K;@WY;=6437JL;=-I-I/�[

F [neoJPGI/�-A37JL3KEmF#<I-ZC�M@B�3K/2<I6dGI/wW#�NMO1K/QWI6K8�1Q�

θ2 = eiα� [ � �?` 0@�=�

�u6K;@WY;=19JL3KJP;=- �eiα = ±1

� E�V\;@-I193K67MO3KJP;=- ��B2;@-I1KJLGIE269;=-I1:� �O/�3K1 |Φa〉/�3 |Φb〉

` � efM@WI6K8�1�[�esMO-]3KJzp <I-IJL37M@6KJz37EQGI/θq�JL[

>UJP/�-]3Q�

〈θΦa|Φb〉 = 〈θΦb|θ2Φa〉 = eiα〈θΦb|Φa〉= eiα〈θΦa|θ2Φb〉 = ei2α〈θΦa|Φb〉

e2iα = 1

eiα = ±1� �?`!0@�A�

�cJeiα = −1

qN;@-�M#M@[L;=6K1m�〈θΦa|Φb〉 = −〈θΦb|Φa〉

� �?` 0 �]�{ -�W?6K/2-IM@-A3 |Φa〉 = |Φb〉

qI;=-(/�-�GIE�GI<IJz3x_]<I/〈θΦ|Φ〉 = 0

� �?`!0 =�

�� /� � � � ��

�cJh[�esMOB�37JL;=- G?/X[�MH14TUV\Ek376KJL/�69/2-A>=/�6K1K/�V\/�-A3�GI<03K/2V&WI1:/�193 3K/2[P[L/\_]<I/θ2 = −1

q|[P/21�E�37MO371WI6K;@WI6K/�1vG?/m[ne��NM@V&JP[L3K;=-IJL/2-#1K;=-A3vGIE�g=E2-IE�6KE�12`��h/�3937/wWI69;=WI6KJLE�3KE./2143vM@WIW�/2[PE�/ � � � ��! �� !"�� k`� M@-I1:[P/&B�M@15G?<SV\;=<c>=/2V&/2-A3:Gef<?-I/ZWNM@6437JPB�<I[P/&17MO-I151KW?JP-q [P/\B �I;=Jz�

θ|Ψ〉 = |Ψ∗〉 JLV#pWI[PJL_]<I/miIJP/�-(1 �I6d_]<I/θ2 =

� [�`Iab[�-e T�M#G?;=-IBwWNM@1dGIE�g=E2-IE�6K/�1KB2/�-IB2/:GI/ �w6KM@V&/26K1�`�h;=-I19JPGIE�6K;=-I1�WNM@6�B2;=-A3K6K/�[L/xB2M@1hGjef<I-�V\;=V&/2-A3�B2JL-IE�3KJP_]<I/v_U<?/2[PB�;=-I_]<I/ ~J ` �|M@6�6K/2-A>@/26K19/kpV\/�-A3xG?<�3K/2V&WI12q ~J /�193d376KM@-I19CD;@6KV&Em/2- − ~J

`N,m-�M GI;=-IB#�

θJz = −JzθθJx = −JxθθJy = −Jyθ

θJ+ = θ(Jx + iJy) = θJx − iθJy = −Jxθ + iJyθ

θJ+ = −J−θ/�3

θJ− = −J+θ� �c` 0 "=�

�c;=Jz3 |j,m〉 <I-I/miIM@1K/.G?/�>=/�B�37/�<I6K1dWI69;=WI69/21lGI/ ~J2/�3Jz` �+>=/2Bm[PM:B2;@-]>@/2-A37JL;=-

~ = 1qU;=-(M

GI;=-IB#�Jz|j,m〉 = m|j,m〉 ` � esM@W?6K821v[P/�1l6K8�g=[P/�1lGI/Q3K67M@-?19CD;=69V�MO3KJP;=-jqcJL[j>UJP/2-A3w�

Jzθ|j,m〉 = −θJz|j,m〉 = −mθ|j,m〉� �?`!0 0=�

θ|j,m〉 GI;=JL3^GI;=-IB 1�3K6K/wWI69;=WY;@6937JL;=-I-I/�[�F |j,−m〉 `�� ;=1K;@-I1

θ|j,m〉 = eiα(j,m)|j,−m〉� �?` 0 � �

,.-HM#M@[L;=6K1m�

J+θ|j,m〉 = eiα(j,m)J+|j,−m〉= eiα(j,m)

√j(j + 1) −m(m− 1)|j,−m + 1〉

M@JL1J†θ = −θJ− q

J+θ|j,m〉 = −θJ−|j,m〉= −θ

√j(j + 1) −m(m− 1)|j,m− 1〉

= −eiα(j,m−1)√j(j + 1) −m(m− 1)|j,−m + 1〉

� �?`!0�A�,.-�GI;@JL3lGI;@-IBQM}>@;=JP6

eiα(j,m) = −eiα(j,m−1)� �c` � =�

B2/�1x69/2[�M 37JP;@-I1^1K;@-]3v/�-�WNMO6937JLB2<I[LJP/�6^17MO3KJP14C�M@JL3K/21^19J�;=-�B �?;=JP19JL3

eiα(j,m) = eiβ(j)(−1)j−m� �c` � �}�

� �#� ��

,.-HM GI;@-IB#�θ|j,m〉 = eiβ(j)(−1)j−m|j,−m〉θ2|j,m〉 = θ (θ|j,m〉)

= e−iβ(j)(−1)j−meiβ(j)(−1)j+m|j,m〉

θ2|j,m〉 = (−1)2j|j,m〉� �?` � �=�

�cJ � /�193�/2-A37JL/26�qθ2 =

� [�qA/k3h;@-�MwWNM@1�GI/vGIE2g=E�-IE269/219B2/2-?B2/mGI/ �m67M@V&/26912qA/2-�M@B2B�;=6KG�M�>=/2Bx-I;=1B2;@-IB2[L<I1KJL;=-I1 19<I6t[L/hV&;=V\/�-A3tB2JL-IE�3KJP_]<I/�;=6KiIJz3�M@[�`� M@6 B2;=-A3769/@qOWY;=<?6t[P/�1 V&;=V\/�-A371 B�JP-IEk37JL_U<?/21GI/�V\J pb/2-A3KJP/2691#/�3X/�-�WNM@6437JLB2<I[LJP/26 W�;=<I6&<I-�19WIJP- 1

2

q�JL[dGI;=JL3#T�M�>=;=JL6\<?-I/HGIE�g=E2-?E26K/�1KB�/2-IB�/Geo;=6KG?6K/�� 1KJl[ne��NM@V&JP[z37;=-?JP/2- /2143\JL-A>@MO6KJ�MO-]35WNM@6 6K/�-A>=/2691K/�V\/�-]3�GI<�37/2V&WI1�quGI;=-IBZGNMO-I1#[�M376KM@-I14CD;=6KV\MO37JL;=- ~J ⇒ − ~J

` �|;@<I6|[�efE�[P/�B�3769;=-QGNM@-I1 <I-:W�;@3K/2-A37JL/2[AB2;@<I[P;=V:iIJP/�-q�[ne��NM@V&JP[z37;=-?JP/2-/2143uJP-A>OM@69J�M@-A3|WNM@6 69/2-A>=/�6K19/2V&/2-A3hG?<:37/�V\WI1�q@/k3u[P/�1 -IJz>=/�MO<c�519;=-A3uGI/�<c�5CD;=JL1 GIE�g=E2-?E26KE�12`��|M@6B2;@-]3K6K/Oq�/�-�W?6KE219/2-IB�/:Geo<I-�B �NM@V&W�V\M@g=-?E�37JL_]<I/@qI[L/.37/�6KV&/ −gHB

~S · ~B iI6KJL1K/w[neoJP-A>OM@6KJPM@-IB�/WNM@6l69/2-A>=/�6K19/2V&/2-A3mGI<Z37/�V\W?12qN/�3v[PM5GIE2g=E�-IE269/219B2/2-?B2/5/�193x[P/�>@E2/O`��ef/�193 �� !N`

� � � ��

�ueo;=WYE�67MO3K/2<I6�WNM@69JL37EΠ/2143^<I-\;=WYE�67MO3K/2<I6�[PJL-IE�MOJP6K/l_]<eo;=-XW�/2<c3�C�M@B�JP[L/2V&/2-A3�GIJ�MOg=;=-NM@[LJP19/262`

�h;=V&V\/Π2 =

� [�qh19/21Z>OM@[P/�<I6K1\WI6K;=W?6K/21Z19;=-A3 ±1qh/k3(F0WNM@6437JP6&GI/�-eoJPV&W�;=693K/�_]<I/�[ �@/k3

GI/&[neo/219WNM@B2/&GI/\~xJL[Pi�/2693w_]<IJr-eo/21435WNMO1Q>@/2B�3K/2<I6�WI6K;@WI6K/&GI/Πq+;=- W�/2<c3�B�;=-I14376K<?JP69/\GI/�<c�

>=/�B�3K/2<I691�WI69;=WI6K/�1dGI/Π�

|Φ+〉 = |Φ〉 + Π|Φ〉� �?` � �A�

|Φ−〉 = |Φ〉 − Π|Φ〉� �?` � �]�

ab[P1^>@E269JL�N/�-]3x/2-�/�yY/�3Π|Φ+〉 = |Φ+〉

E�3�M 3dWNM@JP6Π|Φ−〉 = −|Φ−〉

Ek3�MO3vJPV&WNM@JL6 � �?` � =��cJt[�e��IM@V\JL[L3K;=-IJP/�-(/2193wJL-A>@MO6KJ�MO-]3�GNM@-I1�<I-I/ ;=W�E26KMO37JL;=- GI/ WNMO6KJL3KE@qY;=-�WY/�<?3mGI;@-IB B �I/269B �I/26[P/�1�E�37MO371+WI69;=WI6K/�1+1KE2WIM@6KE�V\/�-]3|GNM@-?1 [L/21�19;=<I14p /219WNM@B2/�1 GI/�1|Ek3�MO3K1�WNM@JL6K1�/�3 GI/�1|Ek3�MO3K1�JPV&WNM@JL6K12`{ �c/2V&WI[P/ ��V&;=<?>@/2V&/2-A3+Geo<I-I/rWNM@6437JLB2<I[L/tGNM@-I1+<I-QW�;@37/�-]3KJP/�[=B2/�-]3K67M@[�` �+M^3767MO-I19CD;=69V�M 37JP;@-

GI/lWNMO6KJL3KE~r → −~r [�MOJP1K19/^[ne��NM@V&JP[z37;=-?JP/2-QJP-A>OM@69J�M@-A3�` ��/21r1K;=[L<?37JL;=-I1|GI;=JL>@/2-A3�GI;=-IBdM�>@;=JP6u<I-?/WNM@69JL3KE&i?JP/2- GIE��I-IJP/O`t�h;=V&V&/&[PM�WIM@6KJz37E&[�M@JL1K1K/ ~L JP-A>OM@69J�M@-A3�q ~L B�;=V&V5<?37/\M�>=/�B

Πq|/k3:;=-

GI;=Jz3�WY;=<c>=;=JL6�[L/21�GIJ�M@g@;=-NM@[LJP1K/�6uGNM@-I1�<I-I/xiNM@1K/�B2;=V&V5<I-I/O`c��ef/�193^/kyj/�B�3KJL>=/�V\/�-A3l[P/vB�M@1�` { -B2;U;=69GI;=-I-IE�/21d1KW �IE269JP_]<I/21�qY[PM5WNM@69JL37EmB�;=6K69/219WY;=-?GHF [�M53K67M@-I14CD;=6KV\MO3KJP;=-

(r, θ, φ) ⇒ (r, π − θ, φ+ π)� �c` � "=�

/�3t;=-5W�/2<?3t>@E269JL�N/�6t_]<I/l[P/�1 �NM@69V\;@-IJP_]<I/�1t1KW#�?E26KJL_]<I/21u19;=-A3uWNM@JL6K/21 1KJl/�193rWNM@JL6|/k3uJPV&WNM@JL6K/�1

1KJl/2193vJLV\WNMOJP62`

�� /� � � � ��

�� ,'�� 0�� "��

� M@-I1�[P/#B�MO1:GI/�1�376KM@-I19[�MO3KJP;=-I1m;@< GI/�1:6K;O3�MO3KJP;=-I1�q�-I;=<I1:M�>=;=-I1Q>U<0_]<I/�[L/21Q;=WYE�67MO3K/2<I6911K/X3K67M@-I14CD;=6KV&/2-A3&F [�esM@JLGI/�Gjef;=W�E26KMO37/�<I6K1 <?-IJL37M@JP69/21:G?/Z[�M�CD;=69V\/

ei~~a·~p W�;=<I6 [L/21:3K67M@-I19[�M p37JL;=-I1v/�3

ei~~α·~L W�;=<I6v[P/21x6K;@37MO37JL;=-I1�` { -HC�MOJL3�q�;=-HWY/�<?3.GIE�V\;@-]3K6K/�6�_]<I/:W�;=<I6v37;=<c3.g=69;=<IW�/Geo;=WYE�67MO3K/2<I691u<I-IJz3�M@JL6K/21|GIE2W�/2-?GNM@-A3rGeo<I-I/^>OM@6KJPM@iI[L/�B�;=-A37JP-]<I/

α/�3r1KMO37JL19C�M@JL17M@-A3t[PMx69/2[�M 37JP;@-

U(α1 + α2) = U(α1)U(α2)� �c` � 0@�

JP[/ �cJP193K/w<I- �� !� � � � � �� 37/�[�_U<?/

U = e−iαT� �c` �=� �

�uef;=W�E26KMO37/�<I6T1�esMOWIWY/�[P[L/\[L/ � ��! �� JP-?�N-?JL37E�1KJLV�M@[��xGI<0g@6K;=<IW�/@` 1x;@<I1:69/�>UJP/�-IGI6K;@-I1

WI[P<?1d37M@6KG(19<I6d[neo<?37JL[PJP1KMO37JL;=-&19TU193KE2V\MO37JL_U<?/�GI/w[PM5193769<IB�3K<I6K/QGI/wg=69;=<IW�/@`�cJ[�e��NMOV\JL[L37;@-IJP/�-�/2143vJP-A>OM@6KJPM@-A3lWNM@6d[P/�1^376KM@-I19CD;@6KV\MO37JL;=-I1

U(α)qN;@-�MZ�

U †(α)HU(α) = HGeo;=R[H,U(α)] = 0

W�;=<I6l37;@<?3α

� �?` � �A��|;=<?6h_]<I/.[L/21r3767MO-I19CD;=69V�M 37JP;@-I1

U(α)CD;=69V\/�-]3�<I-Xg@6K;=<IW�/@q]/�[P[P/�1�G?;=JL>@/2-A3^B�;=-A37/2-?JP6

U(α =0) =

� [�` M@JP19;=-I1l<I-(GIEk>=/�[P;=WIW�/2V&/2-A3x[PJLV\Jz37EvGI/U(α)

M@<c37;=<I6lG?/α = 0

`Nab[j>UJL/2-A3Q�

U(α) =� [ − iαT +O(α2)

[H,U(α)] = −iα[H, T ] +O(α2) = 0

⇒ [H, T ] = 0� �?` � A�

�|;=<?6.[P/�1.19TUV&E�3769JP/�1�B2;@-]3KJP-]<I/�12qJL[|/ �?JL193K/ M@JP-?1KJ <I-�;=W�E267M 37/2<?6 �I/�6KV&JL37JL_]<I/@q�GI;=-IB:<I-I/5;=iUp1K/�69>OM@iI[L/@q?_]<IJjB2;=V&V5<?3K/.M�>=/�BQ[�e��IM@V\JL[L3K;=-IJP/�-` �x<?3769/2V&/2-A3lGIJz3�q?[L/�g=E�-IE26KMO37/�<I6lGI/.[PM�376KM@-I1 pCD;=6KV\MO3KJP;=-X/�193v<I-I/ � � � !�� ! � �� k`{ �?/�V\W?[P/21 �]1KJ�[ne��NM@V&JP[L3K;=-IJL/2- /2143^JL-]>OM@69J�M@-A3�WNM@6�376KM@-I1K[PMO37JL;=-q

[H,Pi] = 0`c�cJ�[ne��NM@V&JP[L3K;Op

-IJP/�-�/�193vJP-A>OM@69J�M@-A3lWNMO6l6K;@37MO37JL;=-q[H,Li] = 0

`�h;=V&V\/ <I- ;=W�E267M 37/2<?6 �I/�6KV&JL3KJP_]<I/ /2143:3K;=< � ;=<I691wGIJ�M@g@;=-NM@[LJP17MOiI[P/OqYJL[u/2- 69E219<I[L3K/\_]<eo;=-

WY/�<?3\GIJ�M@g@;=-NM@[LJP1K/�6H

/�3\[P/(g=E2-?E267M 37/2<?6\GI/(376KM@-I14CD;=6KV\MO37JL;=-I1 _]<IJd[�MOJP1K19/2-A3H

JL-A>@MO6KJ�MO-]3GNM@-I1\<I-I/�iNM@19/HB2;@V\V5<?-I/@`��YeoJP[^T M <I-�19/2<I[vg=E2-?E267M 37/2<?62q�B@eo/2193�iIJP/�-�1��I6�[L<IJl_]<eo;=-�>OM<?37JL[PJL1K/26�`�� M@6m/ �?/�V\W?[P/@qY/2-�GIJLV\/�-I1KJL;=-0�=qY;=-�W�/2<?3m<c37JP[LJP19/26

pB2;=V&V&/:g=E�-IE26KMO37/�<I62qB�/ _]<IJ

B2;=-?GI<IJL35M@<c� CD;=-IBk37JP;@-I1:WI69;=WI69/21eipx/~

q|BO`sF?`oG�M@<c� ;=-IG?/21 WI[PM@-I/21�` M@JP1�12eoJP[�T WI[L<I1KJL/2<I691g=E2-?E267M 37/2<?6K12qUJP[IC�MO<?3�GIJP1437JP-?g=<I/26uG?/2<c�\B�MO12q]JP[L[P<I14376KE�1uWNM@6r[P/�1�3K67M@-?1K[�M 37JP;@-I1u/�3�[P/�1�69;@3�M 37JP;@-I12`� 6KM@-I19[�MO3KJP;=-I1 ��JP[+T MX�\g=E2-IE�67MO3K/2<I691

pxqpyqpz`Yab[L1�B2;@V\V5<c37/2-A3�/2-A3769/ /2<c�Y` � <�B2;=<IWjq

1KJ�[�e��NMOV\JL[L37;@-IJP/�-\B�;=V\V:<?37/wM�>=/�B~pqI;=-�WY/�<?3d[P/mG?J�M@g=;@-NM@[PJL1K/�6hGNM@-?1v<?-I/wiNM@1K/wB�;=V\V:<I-I/wF

� ��

~pxq~pyq~pz`��h/w67M@JL1K;=-I-?/2V&/2-A3dB2;=-?GI<IJL3vMO<c�(;=-IG?/21vWI[�M@-?/21

ei~p·~r/~`

6 ;@3�M 37JP;@-I1 �UJP[�T\M M@<I1K19J�:g@E2-IE�67MO3K/2<I691LxqLyqLzqcV\M@JL1�JL[P1�-I/.B2;@V\V5<c37/2-A3hWNM@1^/�-]3K6K/

/2<U�j` � <0B�;=<IWq ;=- -?/�W�/2<?35FHWI6KJL;=6KJu<c37JP[LJP19/26w_]<ef<?-01K/�<I[hGI/\B2/�15g=E2-?E267M 37/2<?6K15F�[�MZCD;=JP1�`{ - 6KE�MO[PJL3KE@qY[neo<I-I/ GI/�1QB�;=V&WY;=1KM@-A37/�1mB2;=V&V5<?3K/#M@<I1K19JuM�>=/�B

Lzq/k3w;=- WY/�<?3wWNM@6m/ �?/�V\W?[P/

GIJPM@g=;=-NMO[PJP19/26HGNM@-I1x<I-I/�iNM@1K/�B2;=V&V5<I-?/:F

Lz/k3 ~L2

`Y��eo/2193.B2/:_]<I/�-I;=<I1vC�M@JP19;=-I1vWY;@<I6[P/�B2M@1ZGI/ [�M0WIM@693KJPB2<?[P/ GNMO-I1(<I-�W�;@3K/2-A37JL/2[.B�/2-A376KM@[n`��+MSWI69E21K/�-IB2/ GI/ g=E�-IE26KMO37/�<I6K1�_]<IJ-I/QB2;=V&V5<?3K/2-A3xWNM@1v/2-A3769/:/2<U�H/�193.F&[�ef;=69JPg=JL-I/.GI/�GIE2g@E2-IE�6K/219B2/�-IB2/ 1�efJL[P1dB2;=V&V5<?3K/2-A3�M}>@/2B[ne��NM@V&JP[z37;=-?JP/2-j` � M@-I1+[P/ B�M@1+GI/�1 69;@3�M 37JP;@-I12qk1KJA[�ef;=-wMl<?3KJP[LJP1KE

Lz/k3 ~L2

qL+

/k3L− /2-?g=/2-IG?6K/2-A3/2- /�yY/�3QGI/�1wE�3�M 371wGI/ V�12V&/ E2-I/�6Kg=JL/ 12eoJP[P1m-jesM@-?-U<?[P/2-A3QWNM@1w[L/:>=/�B�37/�<I6Q1K<I6w[P/�_]<I/2[u;=- [L/21

M@WIW?[PJP_]<I/O`

�\�� ,�� ' !#"�+ %(' "��(, �w$ * ' %(' + �w$� ' � ' +h�� ' �Z$, + ' !#" ��"+*��! +

�+M�1K<IB�B2/�1K1KJL;=-�G?/�G?/2<c��;@<�WI[L<I1KJL/2<I691 ;=W�E267M 37JP;@-I15GI/(19TUV&E�3K6KJP/Z/�193&/2-?B2;=69/�<?-I/(;=WYE p67M 37JP;@-SGI/�14TUV\Ek376KJL/@`-�ueo/2-I19/2V5i?[P/�GI/21#;=W�E26KMO37JL;=-I1 GI/(19TUV&E�3K6KJP/(_]<I/�[neo;=-S/2-?g=/2-IG?6K/�WNM@6M@WIW?[PJPB2MO37JL;=-X19<IB2B�/21K19JL>@/:GI/wB2/wWI69JP-IB�JPW�/mB2;=-?1937Jz37<I/w/�-�g=E�-IE26KM@[�<I- � � � � ��k`6 M@WIW�/2[ �N<I-�g=69;=<IW�/m/2193vGIEk�N-IJ+WNM@6l[L/21l6982g=[L/21d1K<IJz>OM@-A37/21w��=` ��/lWI69;cGI<?JL3uGI/dGI/�<c�&E2[LE2V&/2-A371uG?<\g=69;=<IW�/l/2143�<?-&E�[PE�V\/�-]3rG?<\g=69;=<IW�/m�

a, b,∈ G⇒ab ∈ G

`�c` ��/wWI6K;UGI<IJz3l/2143xM@1K19;cB�J�MO3KJLC��

a(bc) = (ab)c`

�?`dab[�/ �?JL193K/Q<?-�E�[PE2V&/2-A3d-I/�<?3769/e ∈ G

3K/2[�_]<I/ea = ae = a (∀ a ∈ G)

`�I`x� �NMO_U<?/�E�[PE2V&/2-A3vM <I-�JP-A>=/�6K19/�-I;O37E

a−13K/2[�_]<I/

aa−1 = a−1a = e`

��/Zg=69;=<IW�/�/2143#GIJL3 �� 19Jab = ba (∀ a, b)

`|ab[h/2143#GIJL3 �!��m12eoJP[�WY;=191K8�GI/Z<I--I;=V:iI6K/.�N-?J�GefE�[PE�V\/�-]3K12`�ab[/�193xGIJL3 � � � �� ^19J�[P/m-I;@V5iI6K/mGjefE�[PE2V&/2-A3K1v/�193v�N-IJ�;=<(GIE�-I;=V piI6KM@iI[P/O`Iab[�/2143vGIJL3�� !�� !��19JP-I;@-`{ �c/2V&WI[P/�1 ��=` �ueo;=WYE�67MO3K/2<I6 WNM@6KJz37E

ΠB2;@-I193KJL37<?/lM�>=/2Bl[�efJLGI/2-A3KJL37Ex� [c<?-&g@6K;=<IW�/hGjef;=69GI6K/l�:�

Π−1 = Πq

Π2 =� [�`

�c` �ueo/2-I19/2V5iI[L/wGI/21l376KM@-I19[�MO3KJP;=-I1^CD;@6KV&/.<I-�g=6K;=<?WY/mB�;=-A37JP-]< �T (~a)T (~b) = T (~a+~b)

T (~a)−1 = T (−~a) � �?` �?� ��?` �ueo/2-I19/2V5iI[L/�GI/21Z69;@3�M 37JP;@-I1&CD;@6KV&/ <I- g=69;=<IW�/ B2;=-A3KJP-]<` { - /�yY/�32q^[P/�g=6K;@<IWY/HGI/2169;@3�MO3KJP;=-?1�GNM@-?1

R3/2143wJP1K;@V\;=69W#�I/ MO< g@6K;=<IW�/

SO(3)GI/21mV\MO3769JPB�/21.;=643%�I;@g=;=-NM@[L/21

��

�Πt = Π−1

��GI/�G?E�37/�6KV&JP-NMO-]3(E�gAM@[mF+1

q�/�3�[P/HWI6K;UGI<IJz3ZGI/ GI/2<U��V�M 376KJLB2/�1XGI/SO(3)

/2143l<I-I/.V\MO3K6KJPB�/xG?/SO(3)

` MOJP1R(~a)R(~b) 6= R(~a+~b)

B�MO6h[P/�1^GIJzyjE�6K/2-A3K/21B2;@V\W�;=17MO-]3K/21dGI<�V\;@V\/�-]3dB�JP-IEk37JL_U<?/m-I/wB2;=V&V5<?3K/2-A3dWNM@1d/2-A3K6K/Q/2[P[L/21�`

�� ", ��/ �� "� �-,'��

,.-ZM@WIW�/2[L[P/v6K/2W?6KE219/2-A3�M 37JP;@-ZGI/.GIJLV\/�-I1KJL;=-nGef<?-Zg=6K;=<?WY/v37;=<?3h/2-I19/2V5i?[P/.GI/xV�M 376KJLB2/�1

(n×n)JP19;=V\;@6KW#�I/hM@< g=69;=<IW�/h/�3 37/2[c_]<I/^1KJ

R(a)GIE�1KJLg=-I/^[PMxV\MO3769JPB2/�M@191K;UB2JPE�/lF.[neoE2[LE2V&/2-A3

aGI<�g=69;=<IW�/@qI;=-�M@Jz3m�

R(ab) = R(a)R(b)� �c`��A�@�

{ �?/�V\W?[P/21 ��[P/21xV�M 376KJLB2/�1vG?/SO(3)

CD;=6KV&/2-A3x<I-I/�6K/2W?6KE219/2-A3�M 37JP;@- GI/:G?JPV&/2-I19JP;=-��\GI<g=6K;@<IWY/wGI/�1d6K;@37MO37JL;=-I1^GNMO-I1

R3`

� � ��

�cJ+[neo/219WNM@B2/�GI/:G?JPV&/2-I19JP;=-nGIM@-I1�[L/2_]<I/�[|[L/21vV�MO3K6KJLB2/21dGeo<I-I/�6K/�WI6KE�1K/2-A37MO37JL;=-�M@g=JL1K19/2-A3

WY/�<?3 1�3K6K/#1KE�WNM@6KE&/2- 1K;@<I14p /21KWIM@B2/�1�37/�[P1w_]<I/ 37;=<c37/21Q[P/�1mV�M 376KJLB2/�1m1K;=JL/2-A3QGIJ�M@g@;=-NM@[L/21.WNMO6iI[P;UB@qj[�MX6K/�WI6KE�1K/�-]37MO37JL;=- /2143�G?JL37/ � �� k`��cJuB@eo/2143�JLV\W�;=191KJPi?[P/@q�[�MX69/2WI69E219/2-A3�MO3KJP;=- /2143GIJL3K/ � �� k`

� � � � � � � ��

�c<IWIW�;=19;=-I1._]<I/H19;=JL3�JP-A>OM@6KJPM@-A3�WIM@6�B�/26937M@JP-?/21.;=W�E26KMO37JL;=-I1xGI/519TUV&E�3K6KJP/OqjMO<?3769/2V&/2-A3

GIJL3d_]<eoJP[jB2;=V&V5<?3K/wM}>@/2BQ[P/�1v;@WYE�67MO3K/2<I691U(a)

MO1K1K;UB2JLE21vM@<c�ZE2[PE�V\/�-A371aGjef<I-�g=69;=<IW�/

G`

[H,U(a)] = 0� �?` �=�A�

,.- 19<IWIW�;=1K/#WNMO6mM@JP[L[P/�<I6K1x_U<?/U(ab) = U(a)U(b)

` �c;=Jz3E<I-I/5>OM@[L/2<I6�WI6K;=W?6K/

nCD;=JL1

GIE2g@E2-IE�6KE2/�GI/H�

H|Ψk〉 = E|Ψk〉 , k = 1 . . . n� �?` � �]�

�u6K;=W�;=1KJz37JL;=- �N[P/�1lV�M 376KJLB2/�1D(a)

GIEk�N-IJL/21vWNM@6

Dij(a) = 〈Ψi|U(a)|Ψj〉� �c`�� @�

CD;=6KV&/2-A3d<I-?/Q69/2WI69E219/2-A3�MO3KJP;=-�GI/wGIJPV&/2-?1KJP;@-nGI<�g@6K;=<IW�/

G`

� E2V&;=-I14376KMO37JL;=- ��h;=V&V\/H/�3U(a)

B�;=V\V:<?37/�-]32qI;=-�M(�

HU(a)|Ψk〉 = U(a)H|Ψk〉 = EU(a)|Ψk〉� �c`�� "=�

� � � ��

U(a)|Ψk〉/�193lGI;@-IBx>@/2B�3K/2<I6dWI69;=WI69/�GI/

HGI/�>OM@[P/�<I6hW?6K;=WI69/

E`?ab[jM@WIWNM@6437JL/2-A3^GI;@-IB.M@<

1K;@<I14p /21KWIM@B2/QWI6K;@WI6K/wGI/m>OM@[P/�<I6lWI69;=WI6K/E/k3xW�/2<c3x19/wGIE2B�;=V\W�;=19/26dGNM@-I1dB�/�19;=<I14p /219WNM@B2/&�

U(a)|Ψk〉 =∑

j

|Ψj〉Djk(a)� �?` ��0=�

��W?WI[PJL_U<?;=-I1l<I-I/wGI/�<c�cJP8�V\/w;=W�E26KMO37JL;=-ZGI/w19TUV&E�3769JP/#�

U(b)U(a)|Ψk〉 =∑

j

U(b)|Ψj〉Djk(a)

=∑

j

∑

l

|Ψl〉Dlj(b)Djk(a)

=∑

l

|Ψl〉∑

j

Dlj(b)Djk(a)

M@JL1lB2;=V&V&/U(b)U(a) = U(ba)

qN;=-�M#M@<I1K19Jr�

U(b)U(a)|Ψk〉 =∑

l

|Ψl〉Dlk(ba)� �?` � � �

;=-(/2-�GIE�GI<IJL3v_]<I/#�Dlk(ba) =

∑

j

Dlj(b)Djk(a)� �?` � �A�

;=<(/2-?B2;=69/D(ba) = D(b)D(a)

�� ?`z�� =��xWIWI[PJLB�MO3KJP;=- �c�cJ�<I-*�NM@V&JP[z37;=-?JP/2-#B�;=V\V:<?37/xM�>=/2B�[P/21�E2[LE2V&/2-A371�Geo<I-Xg=69;=<IW�/@qU1K/21hE�3�M 371

WI69;=WI6K/�1hW�/2<?>@/2-A3 1�3K6K/xB2[�MO1K1KE�1h/�-�CD;=-IBk37JP;@-�GI/�1^6K/�WI6KE�1K/2-A37MO37JL;=-I1hJL6K69E2GI<IBk37JLiI[P/�1hM@<U�?_]<I/�[P[L/21JP[L1mM@WIWIM@693KJP/2-?-I/2-A3�` �+M � �� &W�/269V\/k3wGI/#GIE�3K/269V\JL-I/26m<I-?JP_]<I/2V&/2-A3Q1K<?6m[�MiNM@19/XGI/�1 6K/2[PMO37JL;=-I1Q/2-A3769/XE2[LE2V&/2-A371:GI<Sg=6K;@<IWY/\[P/�156K/�WI6KE�1K/2-A37MO37JL;=-I15JL6K6KE�GI<IBk37JPi?[P/21�/�3:[�MC�M�;=-�GI;=-A3#[P/�15>=/2Bk37/�<I6K1\M@1K19;cB�JPE�1#F <I-I/(6K/�WI6KE�1K/2-A37MO37JL;=-�JP6K69E2GI<?B�37JLiI[P/�1K/Z376KM@-I19CD;@6KV&/2-A3WNM@6l[L/21dE2[LE2V&/2-A371dGI<(g=69;=<IW�/@`{ �c/2V&WI[P/ �j[�M�WNMO6KJL3KE5B2;@-I193KJL37<?/ <I- g=6K;=<?WY/#M�>=/�B#[neoJPGI/�-]3KJL3KE��

G = { � [ ,Π} `��h/ g=69;=<IW�/W�;=1K1982GI/�GI/2<U��6K/�WI6KE�1K/2-A37MO37JL;=-I1#JP696KE2G?<IB�3KJPiI[L/215GI/(GIJLV\/�-I1KJL;=- � �t[neo<I-I/ZB�;=6K69/219WY;=-?G�M@<U�E�37MO371dWNMOJP6K1�qI[nefM@<?3K6K/wM@<U�(E�37MO371dJPV&WNM@JL6K1�`

� � ��

��/(g=6K;@<IWY/(GI/�1\69;@3�M 37JP;@-I15W�;=1K1982GI/�GI/21&6K/�WI6KE�1K/�-]37MO37JL;=-I1&JP696KE2G?<IB�3KJPiI[L/21 GI/(GIJLV\/�-I1KJL;=-2l + 1

B�;=-I14376K<?JL37/�1�19<I6�[PM&iNMO1K/:G?/21�Ek3�MO3K1�WI69;=WI69/21 |l, m〉 G?/ ~L /k3 ~L2 `Y�h/�3937/:M 3&6KV\MO37JL;=-6K/�WY;@1K/w1K<I6d[L/m6KE219<I[L37MO3d1K<?JL>OM@-A3Q��u6K;@WY;=19JL3KJP;=- �

〈l′m′|e− i~~α~L|l, m〉 = 0

19Jl 6= l′

� �c`L�� c� �

��

� E2V&;=-I14376KMO37JL;=- �GNM@-?1Q[L/ 1K;=<?14pb/�1KWNMOB2/ |l, m〉 qj;=- WY/�<?3mC�M@JP69/5[P/ B �IM@-Ig=/�V\/�-]3QGI/#iNMO1K/>=/�6K1v[P/21dEk3�MO3K1 |l, m; ~α〉 _U<?J�1K;=-A3xE�3�M 371dWI6K;@WI6K/�1vG?/ ~L · ~α /�3 ~L2 ` M@JP1m�

e−i~~α~L|l, m; ~α〉 = e−

i~αm|l, m; ~α〉 � �?`z�� A�@�

M@<?3K6K/2V&/2-A3vGIJz3d[neo;=WYE�67MO3K/2<I6e−

i~~α~L -I/wB �NM@-Ig=/wWNMO1v[PM:>OM@[P/�<I6lGI/ l q�� `

� <�B2;=<IW�[P/�1lV�M 376KJLB2/�1

D(l)m′m(R(~α)) = 〈lm′|e− i

~~α~L|l, m〉

� �?`z�� =�A�B2;=-?1937Jz37<I/�-A3�<?-I/l6K/�WI6KE�1K/2-A37MO37JL;=-\G?/vG?JPV&/2-I19JP;=-

2l+1GI<&g=69;=<IW�/^GI/21r69;@3�MO3KJP;=-?12`A,.-#W�/2<?3

GIE2V&;=-A3769/26d_]<I/wB2/�1x69/2WI69E21K/�-A3�MO3KJP;=-I1v1K;@-]3vJL6K6KE�GI<IBk37JPi?[P/21h/�3v_]<I/�B�/w1K;=-A3v[P/�1v19/2<I[L/21�`

�� 'd) " 'd' $ +� ^" ' !-+ 'd' $ + ' "S"��(, �Q$ � � ' %('�� * ��! ' $�� )�� $|"� /wV�1�V\/w_]<eoJP[�/�193v<?37JL[P/mGI/wB�[�M@191K/26v[P/�1d;=WYE�67MO3K/2<I691l>=/2Bk37;=69JP/�[P1d/2-�>=/2Bk37/�<I6K1�M �cJ�M@<c�Z;=<

WY;@[�M@JL6K/21 1K<IJz>@MO-]3�B�;=V\V&/2-A3uJL[P1u19/d3K67M@-?19CD;=69V\/�-]3u>UJL14p F p >UJP1uGI/d[PM.WNM@6KJz37EOq=JP[c/2193�3769821r<?3KJP[P/hGI/B2[PM@1K19/26w[P/�1.C�M@V\JL[P[L/21vGef;@WYE�67MO3K/2<I691Q/�- >@/2Bk37/2<?6K1�F��ZB�;=V\W�;=1KM@-A37/21w/k3�W?[P<I1mg=E�-IE26KM@[P/�V\/�-A3/2-�37/2-?1K/2<?6K1xF

2l + 1B2;@V\W�;=17MO-]3K/21d1K<?JL>OM@-A3v[�M5C�M2;@-�G?;=-A3�JL[P1^19/Q3K67M@-I14CD;=6KV&/2-A3d>UJP1 p F pn>cJL1

GI/21v6K;O3�MO3KJP;=-I1�`

�� ,.�,.-0GIJL3�_]<I/(�H;=W�E26KMO37/�<I6K1

Ax, Ay, AzB�;=-I193KJL3K<I/2-A3 [L/21:B�;=V&WY;=1KM@-A37/�15Geo<I-0;=W�E26KMO37/�<I6

>=/�B�37;@6KJP/�[ � ;=<&WI[P<?1�19JPV&WI[P/�V\/�-A3rGef<?-\>=/�B�3K/2<I67� ~A 1KJ�qA1K;=<I1�[nefM@Bk37JP;@- Gef<?-I/�69;@3�MO3KJP;=- G?E��N-IJL/WNM@6

~αqN;=-�M��

R†(~α) ~AR(~α) = R~α~A

� �?`z�� ]�{ �?/�V\W?[P/ �@GjesM@W?6K821r[PM.GIE��N-?JL37JL;=- V�12V&/^GI/ R(~α)

q=;=-&M5� R†(~α)~rR(~α) = R~α~r`(x, y, z)B2;=-?1937Jz37<I/�-A3xG?;=-IBw<I-(>=/2Bk37/�<I62`

�u6K;=W�;=1KJz37JL;=- � ~A /2193t<I-:>=/�B�3K/2<I6u19Jc/k3t1K/�<I[P/�V\/�-A3t1KJcJP[A>=E�6KJz�N/h[L/21 6982g=[L/21 GI/^B�;=V\V:<?3�MO3KJP;=-

[Ai, Lj] = i~εijkAk� �?`z�� =�

� E2V&;=-I14376KMO37JL;=- � � efM@WI69821�[PM&G?E2V&;=-I193K67MO3KJP;=-�C�M@JL3K/wWY;=<?6xEk3�M@iI[LJP6d[�M CD;=69V\/wGI/�[neo;=WYE�67M p37/�<I6dGI/w6K;@37MO37JL;=-q?;@-HMZ�

ei~~α~L~re−

i~~α~L = R~α(~r)

⇔ [xi, L] = i~εijkxk� �?`z�� "A�

��

,.6dB2/�3937/QGIE2V&;=-I14376KMO37JL;=-(-I/QC�MOJL3vM@<IB�<I-I/ �ATcW�;@3 �I8219/21.19<IWIWI[LE2V&/2-A3�M@JL6K/�1v19<I6~r` { [L[P/mW�/2<?3

GI;=-?B 1k3769/QEk37/�-IGI<I/Q17M@-?1xMO<IB2<I-?/�V&;cG?JL�NB2MO37JL;=-ZF537;=<c3d;=WYE�67MO3K/2<I6 ~A = (Ax, Ay, Az)`

{ �c/2V&WI[P/ �~p/2193�<I- >=/�B�3K/2<I6�` { - /�yY/�32qj-I;=<?1mM�>=;=-I1x>c< F&[neo;cB�B�M@19JP;=-�GI/5[PM\6KE�1K;=[L<?37JL;=-

GI/w[neoE2_]<NM 37JP;@-ZGI/��cB �I6 �UGIJL-Ig=/26dGNMO-I1d<I-�W�;@3K/2-A37JL/2[B�/2-A376KM@[+_U<?/[pi, Lj] = i~εijkAk

`

�� "�� ,'�

3v- �� ! � � �!� � � � ;=<S37/�-I1K/�<I6��5GI/Z6KM@-Ign/k3#<I-�/2-I19/2V5iI[L/(GI/

n;=W�E267M 37/2<?6K1

T1, · · · , Tn_]<IJ�1K/w376KM@-I14CD;=6KV&/2-A3d1K<?JL>OM@-A3w�

R(~α)TkR(~α)−1 =∑

l

TlDlk(R~α)� �c`L�� 0=�

;=R([P/�1dV�MO3K6KJLB2/21Dlk(R~α)

B2;@-I193KJL37<?/2-A3v<I-I/w6K/�WI6KE�1K/2-A37MO37JL;=-HGI<�g=69;=<IW�/mGI/21d6K;O3�MO3KJP;=-I1�`6 /2V\M@6K_]<I/#� �=��ef/�193riIJL/2-5<?-I/^g=E�-IE26KM@[PJL17MO3KJP;=-�GI/^[�Mv-I;@3KJP;=-�GI/h>=/�B�3K/2<I6912` { - /�yY/�32q=WY;@<I6

<I-(>=/�B�37/�<I62q�;=-�MZ�R(~α)−1 ~AR(~α) = ¯R~α

~A� �c`L��

;=R ¯R~α

GIE219JPg=-I/�[�M&V�MO3K6KJLB2/��I,(3)

GIE�B26KJz>OM@-A3.[�M&6K;@37MO37JL;=-�GIE��N-?JP/:WIM@6~α/2- B�;c;@6KGI;=-?-IE2/�1

B�MO6937E�1KJL/2-I-I/�12`Y,.-�/�-�G?E2GI<IJz3w�

R(~α)R(~α)−1 ~AR(~α)R(~α)−1 = ¯R~αR(~α) ~AR(~α)−1

~A = ¯R~αR(~α) ~AR(~α)−1� �c`L�� A�

M@JL12q ¯R−1~α = t ¯R~α

qNWI<IJP19_]<I/ ¯R~α

/�193v<I-I/wV\MO3769JPB2/�;=693 �I;=g=;=-IM@[P/O`

R(~α) ~AR(~α)−1 = t ¯R~α~A

R(~α)AiR(~α)−1 =∑

j

(t ¯R~α

)ijAj

R(~α)AiR(~α)−1 =∑

j

Aj

(¯R~α

)ji

� �c`L�=� A�

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U�

|Ψ〉 → U |Ψ〉 � �?`z�=�=� �

��

;=-�GI;=JL3^B �NM@-Ig=/�6d[P/21d;=W�E26KMO37/�<I6K1w�

A→ UAU † � �?`z�=� �=�WY;@<I6l_]<I/�[L/21^>OM@[L/2<I691vV&;}T=/�-I-I/w-I/w1K;=JL/2-A3vWNM@1dV&;cGIJz�NE2/�12` { -�/�yY/�3Q�

〈Ψ|A|Ψ〉 → 〈UΨ|UAU †|UΨ〉= 〈Ψ|U †UAU †U |Ψ〉= 〈Ψ|A|Ψ〉

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{|Ψ〉 → U |Ψ〉A → UAU †

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�� ,'��4��"& ��,.��"�� ,�, �� % $�� ,'/0� �k �

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2k +1;=W�E26KMO37/�<I6K1

T qk_]<IJ�19/�3K67M@-I14CD;=6KV&/2-A3XF [nefM@JLGI/�G?/H[�M 6K/�WI6KE�1K/�-]37MO37JL;=- JP696KE�GI<IB�3KJPiI[L/(GI/

GIJPV&/2-?1KJP;@-2k + 1

�D

(k)mm′(R~α) = 〈km′|e−i~α~L/~|km〉

� �?`z�=� �A�M@<?3K6K/2V&/2-A3vGIJz3d_U<?J�1K/m376KM@-I19CD;@6KV&/2-A3dGefM@WI6K8�1��

R(~α)T qkR(~α)−1 =

k∑

q′=−kT q

′

k D(k)q′q (R~α)

� �?`z�=� =�

�u6K;=W�;=1KJz37JL;=- �

[Jz, Tqk ] = qT qk

[J+, Tqk ] =

√k(k + 1) −m(m + 1)T q+1

k

[J−, Tqk ] =

√k(k + 1) −m(m− 1)T q−1

k

� �?`z�=��"A�

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T−11 =

Ax − iAy√2

qT 0

1 = AzqT 1

1 =Ax + iAy√

2

� �?`z�=�0=�

��

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Lz�

〈j ′m′|T qk |jm〉 = 〈jkmq|jkj ′m′〉〈j ′||Tk||j〉� �?`L�@� � �

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mq�-IJ GI/

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17M@<?C|1KJu�{m′ = m + q|j − j ′| ≤ k ≤ j + j ′

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17MO<?Ct19Ju�

|j − k| ≤ j ′ ≤ j + k ⇔

j − k ≤ j ′

k − j ≤ j ′

j ′ ≤ j + k

⇔

k ≥ j − j ′

k ≤ j + j ′

k ≥ j ′ − j

⇔ |j − j ′| ≤ k ≤ j + j ′� �c`L� � A�

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0q1qI/k3

2�

6KM@-Ig0 → ~A. ~B

� �c`L� �U� �6KM@-Ig

1 → ~A ∧ ~B� �c`L� �@�=�

6KM@-Ig2 → (A1B2 + A2B1, A1B3 + A3B1, A2B3 + A3B2,

A1B1 − A2B2, 3A3B3 − ~A. ~B)� �?`L�}�@�A�

� ��

��

� ��

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(∀ |Ψ〉) 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 ≥ E0 E0

� E2-?/26Kg@JP/wGI<(CD;=-IGNM@V&/2-A37M@[�� ?`L�}�

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� E2V&;=-I14376KMO37JL;=- ��c;=JL/2-A3 |N〉 [L/21lEk3�MO3K1vW?6K;=WI69/21dGI/ H `�,.-�WY/�<?3vE2B�6KJP69/ �

H =∑

N

EN |N〉〈N |

〈Ψ|H|Ψ〉 =∑

N

EN |〈Ψ|N〉|2

≥ E0

∑

N

|〈Ψ|N〉|2 = E0

∑

N

〈Ψ|N〉〈Ψ|N〉

= E0〈Ψ|Ψ〉 �� ` � �� ?`s�=�

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��

|Ψ〉 =∑

i

ci|i〉

⇒ 〈Ψ|H|Ψ〉 =∑

i

|ci|2Ei ≥ E0

∑

i

|ci|2 = E0〈Ψ|Ψ〉� �� ?`f�A�

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H =−~2

2m

d2

dx2+

1

2mω2x2

� �� ?`��]�

/�3t/21K1KM�T=;=-I1rGI/�376K;@<?>=/�6t[P/�V\JL-IJPV5<?V GI/ 〈Ψ|H|Ψ〉〈Ψ|Ψ〉

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Ψa(x) = 1x2+a

M�>@/2Ba > 0

` ;=69V5<I[P/�1l<?37JL[P/�1.�

I1 =

∫ +∞

−∞

dx

x2 + a=

1√a[arctan(

x√a)]+∞∞ =

pi√a

�|;=19;=-I1In =

∫ +∞

−∞

dx

(x2 + a)n

dInda

= −nIn+1

⇒ In+1 = − 1

n

dIndaGef;@R

I2 =dI1da

=π

2a−3/2

I3 =3π

8a−5/2

I4 =5π

16a−7/2

� � � � � ��

∫ +∞

−∞

dx

x2 + a

d2

dx2(

1

x2 + a)dx = −

∫ +∞

−∞

[d

dx(

1

x2 + a)

]2

dx

+

[1

x2 + a

d

dx(

1

x2 + a)

]+∞

−∞d

dx

1

x2 + a=

−2x

(x2 + a)2

∫ +∞

−∞

dx

x2 + a

d2

dx2(

1

x2 + a)dx = −

∫4x2

(x2 + a)4dx = 4[I3 − aI4]

∫x2 dx

(x2 + a)2= I1 − aI2

⇒ 〈Ψ|H|Ψ〉 =−~2

2m(−4)[I3 − aI4] +

1

2mω2[I1 − aI2]

〈Ψ|Ψ〉 = I2

⇒ 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 =

~2

4m

1

a+

1

2mω2a = E(a)

dE

da= 0

W�;=<I6

⇔ amin =~√2mω

⇒ Emin = E(amin) =1√2~ω

� � ?` @�

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H = H0 + V� � ?`!"=�

;=RH0

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1K<IW�E269JP/�<I6vF\� /2-x� `

��

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/�3εp[P/�1vE�-I/269g=JP/�1dWI6K;=W?6K/21w�

H0|ϕp〉 = εp|ϕp〉� �� ?`!0=�

/�3:1K;=Jz3 |ϕn〉 <I-0Ek3�MO3:WI6K;@WI6K/ ! � ! � � � � ��! ��k` �UJ�[PM�W�/26437<I69iNMO37JL;=- /�193 ��WY/k37Jz37/ � q ;@-0WY/�<?3/219WYE�6K/�6�3769;=<?>@/26^<I-\E�37MO3hW?6K;=WI69/�GI/H�kWI6K;UB �I/ �kGI/ |ϕn〉 GI;=-A3�[neoE2-I/�6Kg=JL/x19/267M ��WI6K;UB �I/ ��GI/

εn`�u[P<?1vW?6KE2B�JP19E2V&/2-A3�qNB�;=-I1KJLGIE269;=-I1v[P/mWI69;=iI[L82V&/.WI[P<?1vg@E2-IE�67M@[�GI;@-I-IEwWNM@6

H = H0 + λV� �� ?` � �

�cJλ/�193x1K<.3&1KM@V\V&/2-A3vW�/�37Jz3 � -I;@<I1v>@/2696K;=-I1xF#[�M:�N-�GI<�B2M@[PB�<I[�B2/w_]<eoJP[C�M@<c3x/�-A37/2-?GI6K/

WNM@6�[�FA� q�;=-�W�/2<?3X/21KW�E269/26(E2B�6KJL6K/ [PM 1K;=[L<?37JL;=-�19;=<I1Z[�M CD;=6KV&/ Geo<I-�GIE�>@/2[L;=WIW�/2V&/2-A3(/2-WI<IJL1K1KM@-IB2/�1xG?/

λ�

|Ψn〉 = |ϕn〉 + λ|Ψ(1)n 〉 + λ2|Ψ(2)

n 〉 + . . .

En = εn + λE(1)n + λ2E(2)

n + . . .� �� ?` �A�

�cJr[L/&6KM�T=;=- GI/&B2;=-A>=/�6Kg=/�-IB2/�GI/\B�/�3937/&1KE269JP/#/2143:19<IW�E26KJL/2<I6w;=< E2gAMO[rF �=q�;@- M@<?67M(<I-I/i�;=-I-I/QM@WIW?6K;��cJPV\MO37JL;=-XGI/w[neoE2-?/26Kg@JP/m/2-�1K/w[LJPV&JL3�MO-]3hF#_U<?/2[P_]<I/�1v3K/269V\/�12`{ 1K1KM�T=;=-I1�GI;=-IBwGI/wGIEk37/269V\JL-I/26l[L/21^3K/26KV&/21dGI/wB�/�3K3K/w1KE269JP/O`~dTcW�;@3 �I8219/ ��;=- MZ[�MX[LJPi�/26437E5GI/#B �I;@JP1KJL6m[�MX-I;@6KV\M@[PJL17MO3KJP;=-�GI/ |Ψn〉

`��/#B �I;=Jz��[P/5WI[L<I1B2;@V\V&;cG?/lB2;=-I19JP1437/.FwWI69/2-IGI69/ 〈ϕn|Ψn(λ)〉 = 1

W�;=<I6u3K;=<?3λqUM@<?3K6K/�V\/�-]3�GIJz3 〈ϕn|Ψ(i)

n 〉 =0W�;=<I6l37;=<c3

i`

6 /2W�;=693K;=-I1 [L/21:/ �cWI6K/�1K1KJL;=-I1#GI/ |Ψn〉/k3En

/2-SWI<IJL1K1KM@-IB2/�1#GI/λGIM@-I1 [neoE2_]<NM 37JP;@-0GI/

�cB �I6��cG?JP-Ig=/�6m�

(H0 + λV )|Ψn〉 = En|Ψn〉ab[>UJP/�-]3Q�(H0 + λV )(|ϕn〉 + λ|Ψ(1)

n 〉 + λ2|Ψ(2)n 〉 + . . . )

= (εn + λE(1)n + λ2E

(2)n + . . . )(|ϕn〉 + λ|Ψ(1)

n 〉 + λ2|Ψ(2)n 〉 + . . . )

� �� ?`z�� A��h/k3K37/QE2_]<NM 37JP;@-�G?;=JL3 1k376K/m>U67MOJP/wF B �NM@_]<I/Q;=6KG?6K/w/2-

λ`

�=`x,.6KG?6K/ �

H0|ϕn〉 = εn|ϕn〉� � ?`L�@� �

�c`x,.6KG?6K/#�

H0|Ψ(1)n 〉 + V |ϕn〉 = εn|Ψ(1)

n 〉 + E(1)n |ϕn〉

� � ?`L�}�=�

� � � � � ��

�?`x,.6KGI69/ � � ≥ 2� �

H0|Ψ(k)n 〉 + V |Ψ(k−1)

n 〉 = εn|Ψ(k)n 〉 + E(1)

n |Ψ(k−1)n 〉 + · · ·+ E(k−1)

n |Ψ(1)n 〉 + E(k)

n |ϕn〉� �� ?`z�}�A� M@JP19;=-I1l[L/mWI6K;UGI<IJz3d1KB�MO[�M@JL6K/.GI/Q[neo;=69GI6K/ �wWNM@6 〈ϕn| `��/QW?6K/2V&JP/�6h37/�6KV&/wGI<�V&/2-]iI69/GI/QgAM@<IB�B �I/:/�193�E2gAM@[ F

εn〈ϕn|ψ(1)n 〉 WI<?JP1K_]<I/ H0|ϕn〉 = ε|ϕn〉

q�/�3x1K/�1KJPV&WI[LJL�N/mM�>@/2B[P/mW?6K/2V&JP/�6^37/�6KV&/mGI<�V&/2-]iI69/wGI/wGI6K;=Jz37/O`Iab[>UJP/2-A3dGI;@-IB#�

〈ϕn|H0|Ψ(1)n 〉 + 〈ϕn|V |ϕn〉 = εn〈ϕn|Ψ(1)

n 〉 + E(1)n 〈ϕn|ϕn〉

E(1)n = 〈ϕn|V |ϕn〉

� �� ?`z� �]�� /wV�12V&/@qI19J�-I;=<I1lC�M@JL1K;=-I1l[L/mWI6K;UGI<IJz3d1KB2M@[�M@JL6K/.G?/Q[�ef;@6KGI69/

kM�>@/2B |ϕn〉 qNJP[�>UJP/2-A3Q�

E(k)n = 〈ϕn|V |ϕ(k−1)

n 〉� �� ?`z� =�

�|;@<I6r_U<?/v[�esMO[Pg=;=69JL3 �IV\/r19;=JL3uB�;=V&WI[P/k3�q@JL[cC�M@<c3r17M�>=;=JL6uGIE�3K/269V\JL-I/26 |ϕ(k)n 〉 B�;=-I-NMOJP1K1KM@-A3

E(1)n . . . E

(k)n

/k3Ψ

(1)n . . .Ψ

(k−1)n

` �h;=V&V&/5[P/�1 |Ψ(k)n 〉 1K;=-A3QWNM@6 �ATUW�;@3%�I8�1K/\;=693 �I;=g=;Op-NM@<U� F |ϕn〉

qtJL[h1K<�3&3&GI/(B�;=-I-NM � 3769/Z[�M WI69; � /2Bk37JP;@- G?/ |Ψ(k)n 〉 1K<?6#3K;=<I1 [L/21 |ϕm〉

qm 6= n

` M@JL1K;=-?1lGI;=-IBw[P/mW?6K;UGI<IJL3d19B�M@[PM@JP69/.GI/w[neo;=69GI6K/kM�>=/�B 〈ϕm| `?ab[>cJL/2-A3Q�

〈ϕm|H0|Ψ(k)n 〉 + 〈ϕm|V |Ψ(k−1)

n 〉= εn〈ϕm|Ψ(k)

n 〉 + E(1)n 〈ϕm|Ψ(k−1)

n 〉 + · · · + E(k)n 〈ϕm|ϕm〉

⇒ εm〈ϕm|Ψ(k)n 〉 + 〈ϕm|V |Ψ(k−1)

n 〉= εn〈ϕm|Ψ(k)

n 〉 + E(1)n 〈ϕm|Ψ(k−1)

n 〉 + · · · + E(k−1)n 〈ϕm|Ψ(1)

n 〉

〈ϕm|Ψ(k)n 〉 =

1

εn − εm

(〈ϕm|V |Ψ(k−1)

n 〉 − E(1)n 〈ϕm|Ψ(k−1)

n 〉 − · · · − E(k−1)n 〈ϕm|Ψ(1)

n 〉)

�|;@<I63769;=<?>=/�6 |Ψ(k)n 〉 q2JP[O1K<.3#3 GI;=-?BuGI/uB2;=-?-NM � 3K6K/ |Ψ(1)

n 〉, . . . , |Ψ(k−1)n 〉 /�3 E(1)

n . . . E(k−1)n

`��/21mB�;=-IGIJz37JP;@-I1.1K;@-]3wi?JP/2-�69/2V&WI[PJL/21�`�,.-�-efM(V�12V&/:WIM@1miY/�1K;=JL- GI/

E(k)n`+��eo/2193Q[PM

3%�?E2;=69JP/mGI/�� ! �� `�^M@1dWNM@6437JPB�<I[PJL/26 � M@JP19;=-I1 [L/hB2M@[PB�<I[?F.[�ef;@6KGI69/l�c`@,.-#M.GIE � Fx3K6K;=<c>=E

E(1)n = 〈ϕn|V |ϕn〉

` M@JL1K;=-I1|[P/^W?6K;UGI<IJL31KB2M@[�M@JL6K/mGI/w[�ef;=69GI6K/:��WNMO6 |ϕm〉 `

〈ϕm|Ψ(1)n 〉 =

1

εn − εm〈ϕm|V |ϕn〉

� �� ?`z��"A�B2/w_]<IJ�GIEk�N-IJL3

Ψ(1)n`

��

E(2)n = 〈ϕn|V |Ψ(1)

n 〉=

∑

m

〈ϕm|V |ϕm〉〈ϕn|Ψ(1)n 〉

=∑

m6=n〈ϕn|V |ϕm〉〈ϕm|Ψ(1)

n 〉

WI<?JP1K_]<I/ 〈ϕn|Ψ(1)n 〉 = 0

q?Gef;@R �

E(2)n =

∑

m6=n

∣∣∣〈ϕm|V |ϕn〉∣∣∣2

εn − εm

� � ?`L��0=�

6 /2V\M@6K_]<I/�1 �� w�cJ |ϕn〉 /2193�[P/ CD;=-?GNM@V&/2-A3�M@[�q εn − εm < 0 (∀ m)

` ��/&3K/269V\/&GI/&GI/2<c�cJL82V&/\;=69GI6K/M@iNMOJP1K19/wGI;=-IBm37;@< � ;=<I691l[neoE2-I/�6Kg=JL/mGI<�CD;@-IGNM@V&/2-A3�MO[n`�=�+ab[?M@W?WNM@67M � 3|GNM@-I1 37;=<?3K/21t[L/21 CD;=6KV5<?[P/21 GI/�1tGIE2-?;=V\JL-NMO3K/2<I691 GI<53bTUW�/ 1

εn−εmW�;=<I6

m 6= n`

��eo/2193w<?-I/5JP-?GIJPB2MO37JL;=-�_]<I/ B�/�3937/ M@W?WI6K;UB �I/5-I/:WY/�<?3mWNM@1�12efM@WIWI[LJP_]<I/26�F�<I-�-IJz>=/2M@<�GIE2g@Ekp-IE�6KE@`��h/QB�MO1v19/26KM#[email protected][P/mWNM@6KM@g=6KM@W#�I/w1K<?JL>OM@-A3�`

�|;@<I6^B2;@-]3KJP-]<I/�62q?JL[NC�M@<?3^B2M@[PB�<I[P/�6h[L/xCD;=-IGIM@V\/�-A3�M@[�F:[�ef;=69GI6K/.�c` � esMOWI6K8�1lB2/._]<IJYWI6KE�B282G?/@qIJL[/2143vGI;=-I-IEwWNMO6Q�

〈ϕm|Ψ(2)n 〉 =

1

εn − εm

(〈ϕm|V |Ψ(1)

n 〉 − E(1)n 〈ϕm|Ψ(1)

n 〉)

=1

εn − εm

(∑

p6=n〈ϕm|V |ϕp〉〈ϕp|Ψ(1)

n 〉 − 〈ϕn|V |ϕn〉〈ϕm|Ψ(1)n 〉)

=1

εn − εm

(∑

p6=n

〈ϕm|V |ϕp〉〈ϕp|V |ϕn〉εn − εp

− 〈ϕn|V |ϕn〉〈ϕm|V |ϕn〉εn − εm

)

�� & ��-,'�� "�� ,'��$�$�� (�� 0�"��,

�h/k3K37/wMOWIWI6K;UB �I/./�193l3K6K821h1KJPV&WI[L/x/k3l/k�U376 12V&/2V&/2-A3l<?3KJP[L/ � <I19_]<esF5[�ef;@6KGI69/m�:W�;=<I6h[�efE�-I/26 pg=JL/@` ��<UpbGI/�[�F?q]B@eo/2143^6KM@WIJPG?/2V&/2-A3hJL-I/k�U3K6KJPB2M@iI[L/@` { -XWNM@6437JLB2<I[LJP/26�q=[neo/k�cWI69/2191KJP;@-�W�;=<I6�[P/21r>=/�Bkp37/�<I6K1xWI6K;@WI6K/�1x-eo/2143.WNMO1xWNMO6937JLB2<I[LJP8�6K/2V&/2-A3l3K67M@-I19WNM@69/2-A37/O`,.-HW�/2<c3�;=ic37/2-?JP6v<I-I/Q/k�cWI6K/�14p

� � � � � ��

1KJL;=-�WI[P<I1:1KJPV&WI[L/�GI/([�M�C�M2;=-�19<IJL>OM@-A37/O` 6 /2WNM@6437;=-?1 GI/(H0 + λV )|Ψn〉 = En|Ψn〉

/k3C�M@JP19;=-I1l[L/mWI6K;UGI<IJz3d1KB�MO[�M@JL6K/.WNMO6 〈ϕm| `Iab[>UJP/�-A3

(En − εm)〈ϕm|Ψn〉 = λ〈ϕm|V |Ψn〉� �� ?`z� � �

� >@/2BQ[�M5-I;@6KV\M@[PJL17MO3KJP;=- 〈ϕn|Ψn〉 = 1qN;=-�MZ�

|Ψn〉 = |ϕn〉 +∑

m6=n|ϕm〉〈ϕm|Ψn〉

|Ψn〉 = |ϕn〉 + λ∑

m6=n|ϕm〉

〈ϕm|V |Ψn〉En − εm

� �� ?`z��A�

{ -�Jz37E26KM@-A3dB2/k3K37/wE�_U<IMO37JL;=-qIJL[j>UJP/2-A3Q�

|Ψn〉 = |ϕn〉 + λ∑

m6=n|ϕm〉

〈ϕm|V |Ψn〉En − εm

+λ2∑

m6=n,j 6=n|ϕj〉

1

En − εj〈ϕj|V |ϕm〉

1

En − εm〈ϕm|V |ϕn〉 + . . .

�cJN;=-#C�M@JL3t<I-\G?E�>=/�[P;=W�/2V&/2-A3�GI/En

/�-�WI<?JP1K1KM@-IB�/vG?/λqA;=-#6K/k376K;@<?>=/xiIJP/�-&1��I6�[�M.V�12V&/

3%�IE�;=6KJL/\_]<I/\WI6KE�B2E�GI/2V&V\/�-A3�` M@JP1Q[�M(1KJLV\W?[PB2Jz37E GI< GIE�>@/2[P;@WIWY/�V\/�-A3:CD;=69V\/�[uWY/�6KV&/�3�GI/3769;=<?>=/�6x19JPV&WI[P/�V\/�-A3l<I-I/wE2_]<NM 37JP;@-�W�;=<I6l[�efE�-I/269g=JP/wF <I-�;=69GI6K/QM@69iIJL3K67M@JL6K/2V&/2-A3h/2[P/k>=EO`6 /k>=/�-I;=-I1v/2-�/�yY/�3xF [neoE2_]<NMO3KJP;=-(GI/��cB �?6 �UGIJP-Ig@/26m�

(H0 + λV )|Ψn〉 = En|Ψn〉� �� ?`s� A�

��/mWI6K;UGI<IJz3d1KB�MO[�M@JL6K/.WNMO6 〈ϕn| GI;=-I-I/&�

En = εn + λ〈ϕn|V |Ψn〉� �� ?`s�c� �

�cJh;@- JL- � /2Bk37/X[�ef/ �cWI6K/�1K1KJL;=-�WI6KE�B2E�GI/2-A37/�GI/ |Ψn〉F <I-�;=69GI6K/XG?;=-I-IEOqt;=-�;=i?3KJP/2-A3 <?-I/

E2_]<NMO3KJP;=-Q-I;=-Q[PJL-IE�MOJP6K/|WY;@<I6En

GI;@-]3 [PMd19;=[P<c37JP;@-m/2193|JPG?/2-A37JL_U<?/rFv[PMd19;=[P<c37JP;@-mGI/ 6 M�TU[P/�JPg �cp�cB �I6��cGIJL-Ig=/�6tF�[�ef;=69GI6K/

k+1qf/k3t_]<IJcB2;=-A37JL/2-A3uGI/�1 37/�6KV&/21|Gef;@6KGI69/h1K<?WYE�6KJP/�<I62` � M@6|/k�c/2V&WI[L/@q

[neo;=6KG?6K/nWY;@<I6 |Ψn〉

B�;=-IGI<?JL3vFZ�

En = εn + λ〈ϕn|V |ϕn〉 + λ2∑

m6=n

∣∣∣〈ϕm|V |ϕn〉∣∣∣2

En − εm

� �� ?`s�=�=�

�h/�3937/d3%�IE�;=6KJL/v1�esMOWIWY/�[P[L/v[PMmV\Ek3%�I;UGI/vGI/�� ! � � � � !"��k`@ab[�C�MO<?3�6KE219;=<IGI69/ � /2-\g=E2-IE p67M@[ -]<IV&E26KJL_]<I/2V&/2-A3 �d<I-I/5E�_]<NMO37JL;=- -I;@- [PJL-IE�M@JL6K/OqNV�M@JL1vF�<I-�;=69GI6K/�GI;=-I-IE�[P/�1�6KE�1K<I[z3�MO3K1

� ��

1K;@-]3d/�-(g=E2-IE�67M@[jV\/�JP[P[L/2<I691h_]<I/mB�/2<c�(G?/Q[PM�3%�?E26K;@JP/.GI/ 6 M�TU[P/2JLg �cp �cB �I6 �UGIJL-Ig=/26�B�M@6l[�efE�-I/26 pg=JL/&B�;=-A37JL/2-A3:E�gAM@[L/2V&/2-A3:<?-I/\JL-?�N-IJz37E#GI/#37/�6KV&/21:Gjef;=69GI6K/&1K<?WYE�6KJP/�<I6QGI< GIEk>=/�[P;=WIW�/2V&/2-A3GI/ 6 M}TU[L/2JPg �cp4�cB �?6 �UGIJP-Ig@/262`

�� -,�� /0$��/ ��

�cJr[�ef;=-�>@/2<?3�B�M@[LB2<I[L/26m[PM�>@MO[P/2<?6QV&;}T=/�-I-I/&Gef<?-I/#;=iI1K/�69>OM@iI[L/#GNM@-I1Q[neoE�37MO3wWY/�693K<I6Ki�E@q�JL[C�M@<?3dE�B269JP6K/#�

< O >=〈Ψn|O|Ψn〉〈Ψn|Ψn〉

� � ?` �O�A��h;=V&V&/�[PM#-?;=6KV&/�G?/ |Ψn〉

M#E�3KE��?�cE�/:WNM@6 〈ϕn|Ψn〉 = 1q 〈Ψn|Ψn〉 6= 1

`�ab[�C�M@<?3xGI;=-IBB�MO[PB2<?[P/26lB�/�3937/w_]<NM@-A37Jz37E@` � esM@W?6K821v[P/�GIE�>@/2[P;@WY/�V\/�-]3

|Ψn〉 = |ϕn〉 + λ|Ψ(1)n 〉 + λ2|Ψ(2)

n 〉 + . . .� �� ?`s� �]�

;=-�MZ�

〈Ψn|Ψn〉 = 1 +

+∞∑

k,l=1

λk+l〈Ψ(k)n |Ψ(l)

n 〉� � ?` � =�

,.-(>=;=Jz3d_U<?/�[L/mWI6K/�V\JL/26h3K/26KV&/m/2143�Gjef;=69GI6K/��c` � B�/�3v;=69GI6K/OqI;=-�M(�

〈Ψn|Ψn〉 = 1 + λ2〈Ψ(1)n |Ψ(1)

n 〉 � � ?` � "A�19;=JL3

〈Ψn|Ψn〉 = 1 + λ2∑

m6=n〈Ψ(1)

n |ϕm〉〈ϕm|Ψ(1)n 〉

� � ?` � 0=�

〈Ψn|Ψn〉 = 1 + λ2∑

m6=n

∣∣∣〈ϕm|V |ϕn〉∣∣∣2

(εn − εm)2

� � ?` � � �

�� "�� "� � /0$�� 0��$��

ab[]-eo/2143tWNM@1+W�;=1K19JPiI[L/rGI/�G?E�37/�6KV&JP-I/�6+[P/r67M�T=;@-:GI/�B2;=-A>@/26Kg@/2-IB�/^GI/�[�Ml1KE�6KJP/�W�;=<I6+[neoE2-I/�6Kg=JL/B�MO6x;=-H-I/:B�;=-I-NM � 3�WIM@1x[PM&CD;=69V\/w/ �?W?[PJPB�JL3K/�GI<�GIE�>@/2[L;=WIW�/2V&/2-A3�`23x-I/�B2;=-IG?JL37JL;=- ��67M@JL1K;=-Up-NM@i?[P/ � GI/&>@MO[PJPG?JL37E:/2193:_]<I/�[L/\B�;c/ 3\B�JP/�-]3�GI< 3K/269V\/

λ21K;=Jz3QWY/k37JL3QWNMO6�6KM@WIW�;=693�F�B�/2[L<IJ

/2-λ`�,.62q?19J�;=-(GIE219JPg=-?/mWNM@6

∆[PM5GIJLyYE269/2-IB�/w/2-A3769/

εn/�3v[P/�-IJL>@/�M@<([P/mW?[P<I1lWI69;cB �?/@qN;=-�M

iIJL/2-(1 �I6w�

� � � � � ��

∣∣E(2)n

∣∣2 =

∣∣∣∣∣∣∣

∑

m6=n

∣∣∣〈ϕm|V |ϕn〉∣∣∣2

(εn − εm)

∣∣∣∣∣∣∣

≤∑

m6=n

∣∣∣〈ϕm|V |ϕn〉∣∣∣2

|εn − εm|

≤ 1

∆

∑

m6=n

∣∣∣〈ϕm|V |ϕn〉∣∣∣2

=1

∆

(∑

m

〈ϕn|V |ϕm〉〈ϕm|V |ϕn〉 −∣∣∣〈ϕn|V |ϕn〉

∣∣∣2)

=1

∆

(〈ϕn|V 2|ϕn〉 − 〈ϕn|V |ϕn〉2

)

��JL-I1KJ

E(2)n

E(1)n

� 1

⇔ ∆ �∣∣∣∣∣〈ϕn|V 2|ϕn〉〈ϕn|V |ϕn〉

− 〈ϕn|V |ϕn〉∣∣∣∣∣

� �� ?`s��A�

� /rC�M@B2;=-Qg=E�-IE26KM@[P/Oq�376K;@-I_]<I/26 [�Ml1KE�6KJL/�GI/�W�/2693K<I6KiIMO37JL;=-QMvGI/�1 B �NMO-IB2/�1 Ge�1�3K6K/�<I-I/�iY;=-?-I/M@WIWI69;��?JLV�M 37JP;@-�1KJj[P/21hE2[P/�V\/�-A371^GI/mV\MO3K6KJPB�/�GI/�[�M:W�/26437<I69iNMO37JL;=-Z1K;@-]3^WY/k37JL3K1^WNM@6h67M@WIW�;=643F [neoE2B�MO693d/2-A3K6K/Q-IJL>@/�M@<c��N[P/21lB�;c/ 3&B2JP/�-A371vGI/ |Ψ(1)

n 〉 1K;=-A3vW�/�37Jz371lGI8�1x_]<I/∣∣∣∣∣〈ϕm|V |ϕn〉εn − εm

∣∣∣∣∣� 1� �� c`s� =�

��\�� (, �w$ * ' %(' + � ' $ " ' $�� "+* ��!-+ % , � ,d!(,^$ , ' +

�� & ��-,.� � ��$ � � �� ./0� ,'�

�c<IWIW�;=19;=-I1�_]<eo<I-�-IJz>=/2M@<εnGI/

H0

19;=JL3xGIE�g=E2-IE�6KEOqj/�3�GIE219JPg=-?;=-I1�WIM@6ϕni

qi = 1 . . . k<I-I/liNMO1K/^;=643%�I;@-I;=6KV&E2/hGI/hCD;=-IBk37JL;=-I1 W?6K;=WI69/21uGI/

H0

GeoE2-?/26Kg@JP/εn` �+M�3 �IE2;=69JP/�WI6KE�B2E2G?/2-A37/

/2143./2-�E2B �I/�B5GI8�1�[P/wB2M@[PB�<I[+GI/Ψ

(1)n

B�M@6vJP[+M@W?WNM@67M � 3dGI/�1x37/�6KV&/21 1εni

−εnj

〈ϕni|V |ϕnj

〉 M�>@/2B<I-XGIE�-I;=V&JP-NM 37/2<?6�_]<IJY/2193h-]<I[n`�1xE2M@-IV&;=JP-I1�q]B2/21hGIE2-I;@V\JL-NMO37/�<I6K1�-efM@WIWNMO67M@JL1K1K/�-A3^_]<efM}>@/2B[P/m-]<IV&E26KMO37/�<I6 〈ϕni

|V |ϕnj〉 `��h/�[�M#1K<Ig=g@826K/w_]<I/Q1KJ�[PM#iIM@1K/QGI<�1K;=<I1 pb/�1KWNM@B�/�GjefE�-I/269g=JP/ En/2143�B �?;=JP19JP/lGI/^3K/2[P[L/^1K;@6937/l_]<I/d[L/21uE2[LE2V&/2-A371rGI/lV\MO3K6KJPB�/^-I;=-#GIJPM@g=;=-IM@<c�:GI/

V1K;=JL/2-A3r-]<I[P1�q

� � � � ��

;=-�-I/:GI/k>U67M@Jz3�WI[L<I1xM}>@;=JP6vGI/�WI6K;=i?[P82V&/@` { -HGjesM@<c376K/�1�37/�6KV&/21�q�B�/2[PM\1K<?g=g=8269/:GI/�B �I/269B �I/26[P/�1lE�37MO371dW�/2693K<I6Ki�E21v1K;@<I1d[�M5CD;=6KV&/5�

{|Ψn〉 =

∑i〈ϕni

|Ψ(0)n 〉|ϕni

〉 + λ|Ψ(1)n 〉 + λ2|Ψ(2)

n 〉 + . . .

En = εn + λE(1)n + λ2E

(2)n + . . .

(H0 + λV )(∑

i

〈ϕni|Ψ(0)

n 〉|ϕni〉 + λ|Ψ(1)

n 〉 + λ2|Ψ(2)n 〉 + . . . )

= (εn + λE(1)n + λ2E(2)

n + . . . )(∑

i

〈ϕni|Ψ(0)

n 〉|ϕni〉 + λ|Ψ(1)

n 〉 + λ2|Ψ(2)n 〉 + . . . )

� � ?`s�c� ��=`x,.6KG?6K/

0�

H0

∑

i

〈ϕni|Ψ(0)

n 〉|ϕni〉 = εn

∑

i

〈ϕni|Ψ(0)

n 〉|ϕni〉 , �

�c`x,.6KG?6K/1�

H0|Ψ(1)n 〉 +

∑

i

〈ϕni|Ψ(0)

n 〉V |ϕni〉 = εn|Ψ(1)

n 〉 + E(1)n

∑

i

〈ϕni|Ψ(0)

n 〉|ϕni〉

M@JL1K;=-I1l[L/mWI6K;UGI<IJz3l1KB2M@[�MOJP6K/�WNM@6 〈ϕnj| `Iab[>UJP/�-]3Q�

〈ϕnj|H0|Ψ(1)

n 〉 +∑

i

〈ϕni|Ψ(0)

n 〉〈ϕnj|V |ϕni

〉 � � ?`s�=�=�

= εn〈ϕnj||Ψ(1)

n 〉 + E(1)n (∑

i

〈ϕni|Ψ(0)

n 〉〈ϕnj|ϕni

〉

Geo;=R∑

i

〈ϕnj|V |ϕni

〉〈ϕni|Ψ(0)

n 〉 = E(1)n 〈ϕnj

|Ψ(0)n 〉

� � ?`s�@�A�

�x<?3769/2V&/2-A3.GIJz3�qE

(1)n� � � �� !"� � �� !� �� 〈ϕnj

|V |ϕni〉 q�/�3[L/21dB2;=V&W�;=17M@-A3K/21dGI<�>@/2Bk37/2<?6�WI69;=WI6K/QM@191K;UB2JLEQ19;=-A3v[P/21dB�;=V\W�;=1KM@-A37/21vGI/ |Ψn〉GIM@-I1

[PMXiNM@1K/ |ϕni〉 `��ueo/2-I19/2V5i?[P/#GI/21w>=/�B�3K/2<I691:WI69;=WI6K/�1 |Ψn1〉, . . . , |Ψnk

〉 GI/#[�MZV\MO3769JPB�/〈ϕnj

|V |ϕni〉 B�;=-I193KJL3K<I/ <I-I/ -I;=<?>@/2[P[L/ iNM@1K/ GI<�1K;=<?14pb/�1KWNMOB2/ GjefE�-I/269g=JP/

εnql/�3�JP[L1

1KMO37JL19CD;=-A3w�

〈Ψni|V |Ψnj

〉 = E(1)niδij

� � ?`s� �]��eo/2143�GI;@-IBwiIJP/�-�[PM5iNM@19/�_]<IJ+M@-?-U<?[P/m[P/�1dE2[PE�V\/�-A371d-I;=-(GIJPM@g=;=-NMO<c�ZGI/

V`

6 /2V\M@6K_]<I/ �E(1)ni

6= 〈ϕni|V |ϕni

〉 � � � ?`s� =�

� � � � � ��

{ -Zg=E2-IE�67M@[�q?[L/21^E�-I/269g=JP/�1E

(1)ni

19;=-A3l37;=<?3K/21lGIJzyjE�6K/�-]3K/21�`�,.-ZGIJL3^M@[P;=691h_]<I/m[PM:WY/�693K<I6KiNM p37JL;=-�M � � � � � � � � � ��! �� !"��k`

{ �?/�V\W?[P/ ��h;=-I19JPGIE�6K;=-I1&�H-?JL>=/2M@<c�0GIE2g=E�-IE269E21 |ϕ1〉

/�3 |ϕ2〉` �+M [P/�>@E2/ZGI/ZG?E2g=E�-IE269/21KB�/2-IB�/�/�193#;=iUp

37/�-U<?/:/2-�GIJ�MOg=;=-NM@[LJP1KM@-A3l[�M V\MO3769JPB2/Vij = 〈ϕni

|V |ϕnj〉 qNB�/:_]<IJ+B2;@6K6K/�1KW�;=-IGHF\[neoE2_]<NM 37JP;@-M@<c�Z>OM@[L/2<I691lWI6K;@WI6K/�1��

∣∣∣∣V11 − E V12

V21 V22 − E

∣∣∣∣ = 0

(V11 − E)(V22 − E) − |V12|2 = 0

E2 − E(V11 + V22) + V11V22 − |V12|2 = 0

E± =1

2[V11 + V22 ±

√(V11 + V22)2 − 4V11V22 + 4 |V12|2]

=1

2(V11 + V22) ±

1

2

√(V11 − V22)2 + 4 |V12|2

� �� ?`f� "A�{ -�WNMO6937JLB2<I[LJP/�62qU1KJ�[P/�1dE2[PE�V\/�-A371lGIJPM@g=;=-NMO<c�Z19;=-A3xE�gAM@<c� �

V11 = V22 = V� qNJP[j>UJL/2-A3w�

E(1)± = V ± |V12|

� �� ?`f��0=�⇒ E

(1)+ − E

(1)− = 2|V12|

� �� ?`f� � ��+M.[P/�>@E2/lGI/lG?E2g=E�-IE269/21KB�/2-IB�/�/�193 � � ! �� lGNMO-I1u[P/�1t37/269V\/�1t-I;=-#GIJPM@g=;=-IM@<c�Y`A�h/l69E21K<?[L3�M 3

/2143 FHB�;=-A3767MO1937/�65M�>=/�BX[�M�B2;=-A3K6KJPi?<?37JL;=- G?/215E�[PE2V&/2-A3K1�GI/\V�M 376KJLB2/&-I;=- GIJ�M@g@;=-NM@<c� M�>@/2BGI/21vE�37MO371d-I;@-�G?E2g=E�-IE269E21.MO<�G?E2WNM@643�qN_]<IJ�/�193x/2[P[L/�37;@< � ;=<I691v_]<NMOGI67MO3KJP_]<I/O`�cJV12 = 0

;@-(3K6K;=<c>=/{E

(1)+ = V11

E(1)− = V22

M@JL1l1KJV12 6= 0

qIB2/w-eo/2143�WIM@1d[P/mB�MO1 � >=;=JL6l6K/�V�M@69_]<I/wB2Jzp GI/2191K<I17� ��|;=<?6�M@[L[P/�6dF\[neo;=69GI6K/w1K<?JL>OM@-A3�q�B2;@V\V&/2-IB�;=-I1dWNM@6vGIEk37/269V\JL-I/26 |Ψ(1)

n 〉 ` { - WI6K/�-NM@-A3x[P/wWI6K; pGI<IJz3#1KB�MO[�M@JL6K/XGI/([�ef;=69GI6K/��WNM@6 〈ϕm| � M�>=/�B εm 6= εnWI<IJL1K_]<I/ |ϕm〉

/�193\/2- GI/ �I;@6K1#GI<1K;=<?14pb/�1KWNMOB2/:G?E2g=E�-��KE��kqNJP[Y>cJL/2-A3Q�

〈ϕm|H0|Ψ(1)n 〉 +

∑

i

〈ϕni|Ψ(0)

n 〉〈ϕm|V |ϕni〉

= εn〈ϕm|Ψ(1)n 〉 + E(1)

n

∑

i

〈ϕni|Ψ(0)

n 〉〈ϕm|ϕni〉︸︷︷︸

=0

� �#� � ��

〈ϕm|Ψ(1)n 〉 =

∑

i

〈ϕni|Ψn〉〈ϕm|V |ϕni

〉εn − εm

� � ?`s��A�

�ueoE2_]<NMO3KJP;=-�F [neo;=6KGI69/2GI;=-I-?/�MO[P;=691m�

H0|Ψ(2)n 〉 + |Ψ(1)

n 〉 = εn|Ψ(2)n 〉 + E(1)

n |Ψ(1)n 〉 + E(2)

n

∑

i

〈ϕni|Ψ(0)

n 〉|ϕni〉

� � ?`�� A�

�u6K; � /�37;@-I1x19<I6 〈ϕnj| `��cJ�;=-(C�M@Jz3l[ne��ATcW�;@3 �I8219/Q_]<I/ 〈ϕnj

|Ψ(1)n 〉 = 0

qoJP[�>UJP/2-A3Q�

〈ϕnj|V |Ψ(1)

n 〉 = E(2)n 〈ϕnj

|Ψ(0)n 〉

� � ?`��?� �1K;@JL3

∑

m,i

〈ϕnj|V |ϕm〉〈ϕm|V |ϕni

〉εm − εm

〈ϕni|Ψ(0)

n 〉 = E(2)n 〈ϕnj

|Ψ(0)n 〉

� � ?`��A�=�

�x<?3769/2V&/2-A3�G?JL3�qE

(2)n

GI/�>U6KM@JL3 1k376K/m>OM@[L/2<I6lWI69;=WI69/QG?/Q[PM V�MO3K6KJLB2/

∑

m


〉εm − εn

� � ?`��=�A�

��/vWI6K;=i?[P82V&/@q@B@eo/2193�_]<I/v[P/^>=/�B�3K/2<I6�WI6K;@WI6K/vB2;=696K/�1KW�;=-IGNM@-A3 � � � � 1�3K6K/dE2g=M@[�F 〈ϕnj|Ψ(0)

n 〉 `,.6uB2/^>@/2B�3K/2<I6�WI69;=WI6K/l/�193�GIE � Fm>@/2Bk37/2<?6�W?6K;=WI69/dGI/ 〈ϕni|V |ϕnj

〉 q@/�3�FmV\;@JP-I1|_U<?/vB�/21rGI/�<c�V\MO3769JPB2/�1m-I/#B2;=V&V5<?3K/2-A3�qjB2/#_]<IJu-eo/2193�WNM@1w[L/5B�M@1w/�- g=E�-IE26KM@[nqj[P/21m>@/2B�3K/2<I691:WI69;=WI69/21w-I/1K;@-]3vWNMO1vB�;=V&V5<I-I1�`� eo;=RX>UJP/�-A3l[P/vWI6K;@iI[P8�V\/��.ab[�>UJP/2-A3lG?/�-I;@3K6K/ �]TUW�;@3%�?821K/m_]<I/m[L/21�B2;=V&WY;@17M@-A37/�1hGI/�1h>=/�Bkp

37/�<I6K1wWI69;=WI69/21w1K<I6 |ϕni〉 1K;@-]3QWI<?6K/2V&/2-A3�Geo;=6KGI69/ ?` { - C�MOJL3�qYWY;@<I6m;=i?3K/2-IJL6m<I-I/ 3%�IE�;=6KJL/

B2; �IE269/2-A37/OqYJL[C�M@<?3x1K<IWIW�;=19/26�_]<I/�B2/�1xB�;c/ 3\B�JP/�-]3K1�W�;=1K1982GI/�-A3m/2<c�]p V�1�V\/�1x<I-�GIE�>@/2[L;=WIW�/kpV&/2-A3t/�-

λ` �r[L<?3 O3t_]<I/^GI/�37/2-A3K/26u<I-5G?E�>=/�[P;=W?WY/�V\/�-]3r14TU1937E�V�M 37JP_]<I/hGNM@-I1tB�/�3937/^GIJL6K/�B�37JL;=-

� JL[+/k�cJP1437/5GI/�1.V&E�3%�?;cGI/�1�WI[L<I1�B�;=V&WNM@B�3K/21.W�;=<I6x[P/wC�M@JP69/}� q�B�;=-A37/�-]3K;=-I1 pb-I;=<?1mWY;=<?6�[P/�V\; pV&/2-A3&Gef;@i?37/�-IJP6&<I-I/�M@WIWI69;��?JLV�M 37JP;@-SB2; �IE�6K/�-]3K/ F [neo;=6KG?6K/��c` �|;@<I6\B�/2[PM?qu6K/�WNM@693K;=-I1&GI/[neoE2_]<NM 37JP;@-ZGI/��cB �I6 �UGIJL-Ig=/26w�

(H0 + V )|Ψn〉 = En|Ψn〉� � ?`�� ]�

GIEk>=/�[P;=WIW�;=-I1 |Ψn〉1K<I6d[�M5iIM@1K/wWI69;=WI6K/wGI/

H�

|Ψn〉 =∑

i

〈ϕni|Ψn〉|ϕni

〉 +∑

m

〈ϕm|Ψn〉|ϕm〉� � ?`�� =�

/k3xB �?/26KB �?;=-I1�<?-�Ek3�MO3dWI69;=WI69/ |Ψn〉GeoE2-I/�6Kg=JL/

EnWI69;cB �?/QG?/

εn`��h;=V&V&/

� � � � � ��

H0|Ψn〉 = εn∑

i

〈ϕni|Ψn〉|ϕni

〉 + εm∑

m

〈ϕm|Ψn〉|ϕm〉� �� ?`��"A�

/�3H|ψn〉 = En|ψn〉 = En

∑i〈ϕni

|Ψn〉|ϕni〉 + En

∑m〈ϕm|Ψn〉|ϕm〉

qI-I;@<I1vM}>@;=-I1Q�

(H − H0)|Ψn〉 = V |Ψn〉� �� ?`�� 0=�

⇒∑

i

(En − εn)〈ϕni|Ψn〉|ϕni

〉 +∑

m

(En − εm)〈ϕm|Ψn〉|ϕm〉 =

∑

i

〈ϕni|Ψn〉V |ϕni

〉 +∑

m

〈ϕm|Ψn〉V |ϕm〉

M@JP19;=-I1^[L/mWI6K;UGI<IJz3d1KB�MO[�M@JL6K/mM�>=/�B 〈ϕnj| `Nab[Y>cJL/2-A3Q�

(En − εn)〈ϕnj|Ψn〉 =

∑

i

〈ϕni|Ψn〉〈ϕnj

|V |ϕni〉

+∑

m

〈ϕm|Ψn〉〈ϕnj|V |ϕm〉

��<�WI6K/�V\JL/26^;@6KGI69/Q19JPg=-?JL�NB2MO37JzC q

〈ϕm|Ψn〉 ≈ 〈ϕm|Ψ(1)n 〉

=∑

i

〈ϕnj|Ψn〉〈ϕm|V |ϕni

〉εn − εm

(En − εn)〈ϕnj|Ψn〉 =

∑

i

〈ϕnj|V |ϕni

〉〈ϕni|Ψn〉

+∑

i

∑

m


〉εn − εm

〈ϕni|Ψn〉

�c;=Jz3(En − εn)〈ϕnj

|Ψn〉 =∑

i

(〈ϕnj

|V |ϕni〉 +

∑

m


〉εn − εm

)〈ϕni

|Ψn〉� �� ?`��

JP-IM@[P/�V\/�-]32qc[PM B2;=696K/2Bk37JL;=-�F#[�efE�-I/269g=JP/m/�193d>OM@[P/�<I6lWI69;=WI6K/wGI/w[PM5V�MO3K6KJLB2/

〈ϕnj|V |ϕni

〉 +∑

m


〉εn − εm

� �� ?`�� A�

� ��

/k3x[L/21dB2;U/43&B�JP/2-A3K1 〈ϕni|Ψn〉

1K;=-A3v[P/�1l>=/2Bk37/�<I6K1vWI6K;@WI6K/�1�M@191K;UB2JLE212`

�h;=-?1KE2_]<I/�-IB2/ �+�cJtWNMO6m/k�c/2V&WI[P/ W�;=<I6mG?/21w67M@JL1K;=-?1.GI/#19TUV&E�3769JP/Oq 〈ϕnj|V |ϕni

〉 = 0qY[�M

GIE�g=E2-IE�6K/�1KB2/�-IB2/d-I/h1K/26KM�[P/k>=E�/h_]<efM@< 1K/�B2;=-IG ;@6KGI69/@q /�3t[L/21 >@MO[P/2<?6K1 WI6K;=W?6K/21 B2;=696K/219WY;@-IGI/2-A3F [�M GIJPM@g=;=-NMO[PJP1KMO37JL;=-&GI/w[�M5V\MO3769JPB2/

∑

m


〉εn − εm

� � ?` A�6 /2V\M@6K_]<I/ � �cJ�M@< WI6K/�V\JL/26w;=69GI6K/#<I- -IJL>@/�M@< /2193�-I;=- GIE�g=E2-IE�6KEOq|;=- W�/2<?3�<?37JL[PJL1K/26�[�M

3%�?E2;=69JP/5GI/�1.W�/2693K<I6KiIMO37JL;=-I1.-I;@-�GIE2g=E�-IE269E2/�1QW�;=<I6.M@[P[L/26xFX[neo;=6KGI69/:19<IJL>OM@-A3�`�c;=Jz3 |Ψnj〉 [L/>=/�B�3K/2<I6dWI69;=WI6K/mB�;=6K69/219WY;=-?GNM@-A3�qI/k3

εn +E(1)nj

1K;=-(E�-I/269g=JP/O`��h;=V&V\/ 〈Ψni|V |Ψnj

〉 /�193v-]<I[W�;=<I6^37;@<I1d[P/21 |Ψni〉 JP1K19<I1d1K<�1K;=<?14pb/�1KWNMOB2/QGIE2g=E�-IE269E@q�;=-(3769;=<?>=/#�

E(2)nj

=∑

m


〉εn + E

(1)nj − εm

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H = H0 + V/k3�B2;=-I19JPGIE�6K;=-?1�<I-&-IJz>=/�MO<

ε0GIE2g@E2-IE�6KE�G?/

H0

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E0

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H|Ψ0〉 = Eo|Ψ0〉� � ?` @�=�

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;=W�E26KMO37/�<I6l37/2[j_U<?/

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(PHP PHQQHP QHQ

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)= E0

(P |Ψ0〉Q|Ψ0〉

)

� � � � � ��

ab[/2-�GIE2B�;=<I[L/ �

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{ 3dGI/�[PM5WI6K/�V\JL8269/ �

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� �� ?`! �]�� ;=-IB

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= PH0P + PV P + PHQ1

E0 −QHQQHP

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` JP-IM@[P/�V\/�-]3.�

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� � � � ��

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t0

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t0

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t0

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t0

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t0

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t0

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[P/�1|B �NMOWIJL3K6K/21 WI69E2B2E�GI/2-A3K1 � >@;=JP6|� �NM@W?JL3769/ �?`��]� ` 1x;=<?1|/�-:67MOWIWY/�[P;=-?1 JLB2JA[L/21 WI6KJL-IB2JLWNM@[P/�1�WI6K; pWI6KJLE�3KE212q+M�>OM@-A3wGeoJP-A3769;cG?<IJP69/5<I-I/&M@<?3K6K/ 69/2WI69E21K/�-A3�MO3KJP;=- GIJz37/ 6K/�WI6KE�1K/�-]37MO37JL;=- JP-A37/�67M@Bk37JP;@-_]<IJ�B2;=-?GI<IJL3xM@<(GIEk>=/�[P;=WIW�/2V&/2-A3xB �I/269B �IE@`

��

�0�0�� ", �� ./ ��0� � ��'& ,�� 0��,

��eo/2143l[�MQ6K/2W?6KE219/2-A3�M 37JP;@-XGI/�>@/2-]<I/m376KM@GIJz37JP;@-I-I/2[L[P/O` ��/21^Ek3�MO3K1 |ϕS(t)〉 � [�efJL-IGIJLB2/ S C�M@JL3�JPB�J6KEkCDE269/2-IB�/:F [�M 69/2WI69E21K/�-A3�MO3KJP;=-�GI/��cB �I6��cGIJL-Ig=/�6��hE�>@;=[P<?/2-A3�19<IJL>OM@-A3

i~∂

∂t|ϕS(t)〉 = HS(t)|ϕS(t)〉

� �=�=`z� =��/21m;@iI1K/�69>OM@iI[L/21.19;=-A3wGI/21.;@WYE�67MO3K/2<I691��?�cE21w<I-I/�CD;=JP1xWY;=<?6x37;@<?37/�12` { [P[L/21xWY/�<?>=/�-]3wG?Ekp

W�/2-IGI69/xG?<\37/�V\W?12q]V�MOJP1uJP[N1�esMOg=JL3�M@[L;=6K1uGeo<I-I/xGIE2W�/2-?GNM@-IB�/�JP-A3K6KJP-?1K82_]<I/v[PJLE2/dFQ[�Mm_]<NM@-A37Jz37E_]<eo;=-�>@/2<?3xV&/21K<?6K/26�` { [P[P/�-ef/�193xWNM@1d6KE�g=JP/mWIM@6l[nef~�M@V&JP[z37;=-?JP/2-j`

�0�0�� ", �� ./ ��0� � � �� % ��, �

{ -#6K/�WI6KE�1K/2-A37MO37JL;=-#GI/l~x/�JP19/2-]iY/�6KgIq@[P/21|>=/2Bk37/�<I6K1rGeoE�37MO371u19;=-A3rJP-IGIE�WY/�-IGNM@-A3K1uGI<537/�V\WI1�q/�3v[L/21v;=WYE�67MO3K/2<I691dGIE2W�/2-IG?/2-A3.GI<�37/�V\WI1d19/2[P;@- � [neoJP-IGIJLB2/

HC�M@JL3^6KE�CDE�6K/�-IB2/:F&[PM#69/2WI69E219/2-cp

3�M 37JP;@-ZGI/Q~x/2JL1K/�-Ui�/269g]�.�{|ϕH >= |ϕS(t0) >OH(t) = U †

S(t, t0)OS(t)US(t, t0)

� �@�=`L� "A�

�+>=/2BQB2/�1xG?E��N-IJz37JL;=-I12q?;=-�M&iIJL/2- �

< ϕS(t)|OS(t)|ϕS(t) >=< ϕS(t0)|OH(t)|ϕS(t0) >� �=�=`z�0=�

�Q�� ' �Z$ , + ' ! " � "+*��! *2! " ' $ �Q) "+*��:! �

�c<?WIWY;@1K;=-I1v_]<I/Q[ne��NM@V&JP[L3K;=-IJL/2-&1K;=Jz3dGI/w[�M:CD;=69V\/ �

H(t) = H0 + V (t).

�+M:6K/�WI6KE�1K/�-]37MO37JL;=-XJL-]3K/26KM@B�3KJP;=-\12eo;=i?37JL/2-A3l/�-�376KM@-I19CDE�67M@-A3l[PM�GIE�WY/�-IGNM@-?B2/.37/�V\W�;=69/2[P[L/dGI<I/FH0

19<I6d[P/21d;=i?1K/264>OM@iI[P/�1w�

OI(t) = eiH0(t−t0)/~OS(t)e−iH0(t−t0)/~

�|;@<I6^_]<I/m[P/�1h>OM@[P/�<I6K1�V\;}T@/2-I-I/�1dGI/21^;@iI1K/�69>OM@iI[L/21l6K/�193K/2-A3dJP-IB �IM@-Ig=E�/212q?JP[�C�M@<?3^E�gAM@[L/2V&/2-A3V&;cGIJz�N/26h[P/21lEk3�MO3K1w�

|ϕI(t) >= eiH0(t−t0)/~|ϕS(t) >,.-�MXiIJL/2- 1��I6 |ϕI(t0) >= |ϕS(t0) >= |ϕH >`�,.- G?E��N-IJz3.[�ef;=W�E26KMO37/�<I6�GjefEk>=;=[L<?37JL;=- /2-

6K/�WI6KE�1K/�-]37MO37JL;=-�JP-A37/�67M@Bk37JP;@-�WIM@6m�

|ϕI(t) >= UI(t, t0)|ϕI(t0) >

��

ab[�/2143xG?;=-I-IEwWNM@6UI(t, t0) = eiH0(t−t0)/~US(t, t0)� E�3K/269V\JL-I;=-I1d[neoE2_]<NM 37JP;@-ZGIJLyYE269/2-A37JL/2[P[L/._]<IJ�6KE�g=JL3d19;=-�E�>=;@[P<?3KJP;=-��

∂

∂tUI(t, t0) = eiH0(t−t0)/~ iH0

~US(t, t0) + eiH0(t−t0)/~

(−i~

)H(t)US(t, t0)

= eiH0(t−t0)/~

(−i~

)(H(t) − H0

)US(t, t0)

= eiH0(t−t0)/~

(−i~

)(V (t)

)US(t, t0)

=−i~eiH0(t−t0)/~V (t)e−iH0(t−t0)/~eiH0(t−t0)/~US(t, t0)

=−i~VI(t)UI(t, t0)

⇒ i~∂

∂tUI(t, t0) = VI(t)UI(t, t0)

� �=�=`z� � �

��ef/�193w[PM�V�1�V\/�E2_]<NMO3KJP;=-H_U<?/ B2/2[L[P/�;=i?37/�-]<I/5W�;=<I6.[�ef;=W�E26KMO37/�<I6�GjefEk>=;=[L<?37JL;=-US(t, t0)

/2-6K/�WI6KE�1K/2-A37MO37JL;=-�G?/Q�cB �I6 �UGIJL-Ig=/26�q?V\M@JP1hB@eo/2193d[PM5WY/�693K<I6KiNM 37JP;@-Z/2-(6K/�WI6KE�1K/2-A37MO37JL;=-(JP-A37/�67M@B p37JL;=-

VI(t)_]<IJ � ;=<I/w[P/m6� =[P/mG?/Q[�es~xM@V&JP[L3K;=-IJL/2-`

� <0B�;=<IWq�37;@<I1:[L/21QB�M@[LB2<I[L1QC�M@Jz371wW�;=<I6Q/k�cWI6KJLV\/�6US(t, t0)

F�[nefM@JPG?/&G?/H(t)

W�/2<c>=/2-A31�3K6K/m376KM@-I1KW�;=19E21Q�

UI(t, t0) =� [

++∞∑

n=1

(−i~

)n ∫ t

t0

dt1

∫ t1

t0

dt2 · · ·∫ tn−1

t0

dtnVI(t1)VI(t2) · · · VI(tn−1)

;=<UI(t, t0) =

� [+

+∞∑

n=1

(−i~

)n1

n!

∫ t

t0

dt1

∫ t

t0

dt2 · · ·∫ t

t0

dtnT(VI(t1)VI(t2) · · · VI(tn−1)

)

;=<UI(t, t0) = T

[exp

(−i~

∫ t

t0

dt′VI(t′)

)] � �=�@`L��=�

�uef;=W�E26KMO37/�<I6US(t, t0)

12eo/2-�GIE�GI<IJL3vWNMO6

US(t, t0) = e−iH0(t−t0)/~UI(t, t0)

,.-�M GI;=-IB�;=i?3K/2-]<�<I-�GIE�>@/2[L;=WIW�/2V&/2-A3�G?/US(t, t0)

/�-�WI<IJL1K1KM@-IB2/�1�GI/V (t)

`��cJV (t)/2143Z<I-I/ W�/�3KJL3K/�W�/26437<I69iNMO37JL;=-qr;=-�W�/2<c3X;=i?3K/2-IJL6�GI/�1ZM@WIW?6K;��cJPV\MO37JL;=-I1#67M@JL1K;=-?-NM@iI[L/21&/2-

-I/#B2;=-?1K/264>OM@-A3�_]<I/#B�/26937M@JP-?1.37/269V\/�1mGI< GIE�>@/2[L;=WIW�/2V&/2-A3�` � MO-I1w[P/ WNM@6KM@g=6KM@W#�I/519<IJL>OM@-A32q-I;=<I1xM@[L[P;=-I1l-?;=<I1vB2;=-A37/�-A37/26xGeo<?37JL[PJP19/26lB�/�3K3K/QCD;@6KV5<I[L/QM@< � �� ;@6KGI69/@`Y�c;=-�<?37JL[PJP1KMO37JL;=-X14Tc1 p37E�V�MO3KJP_]<I/�M@<c� ;=69GI6K/�1�19<IJL>OM@-A371\/2143\3K67M@Jz37E2/�GNM@-?1�[P/(B�;=<I6K1\GI/�3 �IE2;@6KJP/(_]<NM@-A3KJP_]<I/ GI/�1B �NM@V&WI12`

��#� � � � ��

�Q�� $ � � � �(* ��* "�, %(' "�$ � !-+�*k"+*��! ' " $�� ' % � �w$ %('�� ' $ � *�c<?WIWY;@1K;=-I1v_]<eo<I- �NMOV\JL[L37;@-IJP/�-�1K;@JL3dGI/w[�M:CD;@6KV&/

H = H0 + V (t)� �=�=`s� A�

M�>=/�BV (t) =

{0

19Jt ≤ t0

V (t)19J

t > t0

� �@�=` �U� �

�xWIWY/�[P;=-?1 |i〉 [P/�1dE�3�M 371dWI6K;@WI6K/�1vG?/ H0

�H0|i〉 = εi|i〉

� �@�=` �@�=�/�351K<IWIW�;=19;=-I1#_]<I/Z[P/\19TU193K82V&/Z1K;=Jz3 GNM@-I1:[neoE�3�M 3 |i〉 F t = t0

` � <?- JL-I1937M@-A3t > t0

q [L/19TU143782V&/Q1K/26KM&GIM@-I1d<I-�E�3�M 3

|ϕs(t)〉 =∑

n

cn(t)|n〉� �=�=`s�@�A�

�+M WI6K;=iIM@iIJP[LJL3KE\GI/Z3K67M@-I19JL3KJP;=-SGI/ |i〉 → |n〉 /�193&GI;=-?-IE2/�WNM@6 |cn(t)|2 `tab[l1�esM@g@JL3#iIJL/2-Geo<I-I/wWI69;=iNM@iIJL[PJz37E.WI<?JP1K_]<I/ ∑n |cn(t)|

2 = 1` � efM@WI6K8�1xB�/w_U<?J�WI6KE�B282G?/�q

|ϕs(t)〉 = US(t, t0)|i〉⇒ cn(t) = 〈n|ϕs(t)〉 = 〈n|US(t, t0)|i〉

= 〈n|e− i~H0(t−t0)UI(t, t0)|i〉

= e−i~εn(t−t0)〈n|UI(t, t0)|i〉

⇒ Pi→n = |cn(t)|2 =∣∣∣〈n|UI(t, t0)|i〉

∣∣∣2 � �@�=` � �]�

� efM@WI69821vB2/w_]<IJ+WI6KE�B28�GI/@q � � � �� !V (t)

qN;=-HMZ�

UI(t, t0) =� [ − i

~

∫ t

t0

dt1VI(t1)

〈n|UI(t, t0)|i〉 = − i

~

∫ t

t0

dt1〈n|VI(t1)|i〉 M@JL1

VI(t1) = ei~H0(t1−t0)V (t1)e

− i~H0(t1−t0)

⇒ 〈n|UI(t, t0)|i〉 = − i

~

∫ t

t0

dt1ei(En−Ei)(t1−t0)/~〈n|V (t1)|i〉

Pi→n =

∣∣∣∣−i

~

∫ t

t0

dt1ei(En−Ei)(t1−t0)/~〈n|V (t1)|i〉

∣∣∣∣2 � �@�=` � =�

�� *��

�0� � � � � � � � $��/ ��

�c<IWIW�;=19;=-I1d_]<I/V (t)

19;=JL3dGI;@-I-IEwWNM@6

V (t) =

{0

1KJt ≤ 0

VJL-IGIE�WY/�-IGNM@-A3vGI<(37/2V&WI1 19J

t > 0

� �=�=`s� "A�

Pi→n =

∣∣∣∣−i

~〈n|V |i〉

∫ t

t0

dt1ei(En−Ei)t1/~

∣∣∣∣2 � �=�=`s� 0=�

M@JL1

∫ t

t0

dt1ei(En−Ei)t1/~ =

[~ei(En−Ei)t1/~

i(En − Ei)

]t

0

=~

i

ei(En−Ei)t1/~ − 1

En − Ei� ;=-IB

Pi→n =∣∣∣〈n|V |i〉

∣∣∣2∣∣∣∣1 − ei(En−Ei)t1/~

En − Ei

∣∣∣∣2

,.6∣∣1 − eix

∣∣2 = (1 − eix)(1 − eix)

= (e−ix/2 − eix/2)(eix/2 − e−ix/2)

= −2i sin(x/2)2i sin(x/2) = 4 sin2(x/2)

��JL-I1KJ

Pi→n =4∣∣∣〈n|V |i〉

∣∣∣2

(En − Ei)2sin2 (En − Ei)t

2~

� �=�=`s� � �

{ 37<IGIJL;=-I1 [PM GIE2W�/2-IGIM@-IB2/(37/�V\W�;=69/2[P[L/�GI/(B�/�3K3K/ZCD;=-IB�3KJP;=-j` �|;@<I6#B2/ZC�M@JP69/@q|JP[h/�193&<?3KJP[P/GeoE�37<?GIJP/�6v[PM:CD;=-IBk37JP;@-

f(ω) = 4ω2 sin2(ωt

2)WI<IJL1K_]<I/

Pi→n = ~2∣∣∣〈n|V |i〉

∣∣∣2

f(En − Ei

~)

,.-�>@E26KJz�N/QM@JP19E2V&/2-A3d[P/�1dWI6K;=W?6KJPEk37E�1l1K<IJz>@MO-]3K/21w�

f(ω) = 019J ωt

2= kπ ⇒ ω =

2πk

t(k 6= 0)

limω→0

f(ω) = t2

��

f(ω) =4

ω2

19J ωt

2=π

2+ kπ

� ;=<I6^<I-?/�>OM@[P/�<I6�G?/tGI;=-I-?E2/@qc[P/�1�376KM@-I19JL37JL;=-I1�[P/�1hWI[L<I1�C�M�>=;@6KJP19E2/�1^1K;=-A3�37/2[L[P/�1h_]<I/

ω ≤2πt

`

−4 −3 −2 −1 0 1 2 3 4x

0

0.2

0.4

0.6

0.8

1

~ 0.047

sin2(xπ)/(xπ)

2

�cJw[�ef;=-�/�1K17MOJP/ GI/ GIE�3K/26KV&JP-?/26([�MSGIJzyjE�6K/2-?B2/ GeoE2-I/�6Kg=JL/ /�-]3K6K/ <I- E�3�M 3�JL-IJL3KJ�M@[x/�3�<I-E�37MO3l�N-NMO[j/2-�M@W?WI[PJL_U<IM@-A3l<I-I/mW�/26437<I69iNMO37JL;=-ZW�/2-IGNMO-]3d<?-(3K/2-IW?1

∆tqI;@-�W�;=<I6967M:C�M@JL6K/xGI/21

376KM@-I19JL37JL;=-I1�>=/2691lGI/21lEk3�MO3K1lGeoE2-I/�6Kg=JL/En−Ei . 2π~

∆t

` �x<?3769/2V&/2-A3dGIJz3�qU[�M:WI69E2B�JP1KJL;=-XGI/�[�MV&/21K<?6K/wGI/

∆E-?/QW�;=<I6967M 1�3K6K/wV&/2JP[L[P/�<I6K/x_U<?/ 2π~

∆t

`

∆E &2π~

∆t

∆E∆t & 2π~� �=�=`s��A�

�h/k3K37/Q6K/�[�MO3KJP;=-Z6KM@WIW�/2[L[P/.[PM56K/2[PMO37JL;=-ZGeoJP-IB�/26437JL3K<IGI/

∆x∆p &~

2

� �@�=`s� A��h/5-jef/�193mWNMO1�<I-I/:6K/2[PMO37JL;=-�GefJL-IB2/�693KJL37<?GI/:MO< 1K/�-I1.193K6KJLB�3.B2M@6v[neoE2-I/�6Kg=JL/�/k3m[P/w37/�V\WI1x-I/

1K;@-]3vWNMO1vG?/21v;=iI1K/�69>OM@iI[L/21dV\M@JP1^BOef/�193v<I-I/Q6K8�g=[P/.3K6K8�1v<c37JP[L/@`� ;=<I6v<I-I/m>OM@[P/�<I6lGI/

ωGI;=-I-IE�/@q�;=-�M WNM@6dM@JL[P[P/�<I6K1��

f(ω) ≤ 4

ω2

⇒ f(ω)

f(ω = 0)≤ 4

ω2t2→ 0

_U<IM@-IGt→ +∞

�cJ�[P/d37/�V\W?1�37/�-IG�>@/26K1+∞ qc[L/21r3767MO-I1KJz37JP;@-I1hF

ω = 019;=-A3^1KE�[P/�B�37JL;=-I-IE�/21�`#�|;@<I6hGI;@-I-I/26

<I-I/mCD;=69V5<I[PMO37JL;=-XWI69E2B2JL1K/QF B2/k3K37/�M 3&6KV\MO3KJP;=-qUGIE�V\;=-A3K6K;=-I1l3K;=<?3dGefM@i�;=6KG�_]<I/

�� *��

limt→∞

1

tω2sin2(ωt) = πδ(ω).

� �@�=`s�c� �ab[jC�M@<?3vEk3�M@iI[LJP6^_]<I/

limt→+∞

1

t

∫ +∞

−∞

1

ω2sin2(ωt)ϕ(ω)dω = πϕ(0)

� >@/2BQ[P/mB �NMO-Ig=/2V&/2-A3vGI/m>OM@6KJPM@iI[L/x = ωt

JL[Y>UJP/�-]32q

1

t

∫ +∞

−∞

t2

x2sin2(x)ϕ

(xt

) dxt

=

∫ +∞

−∞

1

x2sin2(x)ϕ(

x

t)dx

limt→∞

∫ +∞

−∞

sin2(x)

x2ϕ(x

t)dx = f(0)

∫ +∞

−∞

sin2(x)

x2dx = πϕ(0)

��/m6KE219<I[L37MO3xM@-I-I;@-IB2Ew12eo/2-�GIE2G?<IJL3d/�-�69/2V&WI[�M@B2M@-A3En − Ei

WNMO6ω/�3tWNM@6 t

2~

`��;=6K19_U<?/

t→ +∞ q?;=-HM GI;@-IB#�

1

(En − Ei)2sin2

((En − Ei)t

2~

)≈ πt

2tδ(En − Ei)

�h/�3937/Q/k�cWI6K/�1K19JP;=- /�193 � � ! �� w/2-t` �xJP-I19JnqI[PM WI6K;=iIM@iIJP[LJL3KE.GI/w3767MO-I1KJz37JP;@-�WIM@6d<I-IJL3KEwGI/

37/�V\WI1d/�193vGI;=-I-?E2/QWNM@6m�

wi→n =2π

~

∣∣∣〈n|V |i〉∣∣∣2

δ(En − Ei)� �=�=`f�A�=�

��ef/�193x[�M � � �� `

�c<IWIW�;=19;=-I1dGIE219;=6KV\M@JL1l_U<?/V (t)

1K;@JL3dGI;=-?-IEwWNM@6

V (t) =

{0

19Jt ≤ 0

V eiωt + V †e−iωt19J

t > 0

� �=�=`f�=�A�

Pi→n =

∣∣∣∣−i

~

∫ t

t0

dt1ei(En−Ei)t1/~(〈n|V |i〉eiωt1 + 〈n|V †|i〉e−iωt1)

∣∣∣∣2

=

∣∣∣∣∣1 − e−i(

En−Ei~

+ω)t

En − Ei + ~ω〈n|V |i〉 +

1 − e−i(En−Ei

~−ω)t

En − Ei − ~ω〈n|V |i〉

∣∣∣∣∣

2 � �=�=`f� �]�

��<c�(3K/2V&WI1v[P;=-Ig@12qN[P/�1l3767MO-I1KJz37JP;@-I1l>=/2691�[L/21dE�37MO371vGeoE2-I/�6Kg=JL/En = Ei ± ~ω

1K;=-A3�C�M�>@;Op6KJL1KE2/�12`��?MO-I1x69/�C�M@JL6K/.[L/21dB�MO[PB2<?[P12q?;=-(;=i?37JL/2-A3w�

��

wi→n =2π

~

∣∣∣〈n|V |i〉∣∣∣2

δ(En − Ei + ~ω) +2π

~

∣∣∣〈n|V †|i〉∣∣∣2

δ(En − Ei − ~ω)� �=�=`f�� =�

3v-I/ZWY/�693K<I6KiNM 37JP;@- �NM@6KV&;=-IJL_]<I/�GI/XCD69E2_]<I/�-IB2/ωJP-IG?<IJL3:GI/2153K67M@-I19JL3KJP;=-I1:/2-A3769/ZE�3�M 371

GeoE2-I/�6Kg=JL/21l37/�[P[L/21l_]<I/∆E = ±~ω

`

� ��

� � �Z��

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6K/�-]3K/�G?/�[PM GI/219B269JPW?3KJP;=-�Geo<I-I/(WNM@6437JLB2<I[L/ZB2;=V&V\/�[neoE2[P/�B�3K6K;=-` � M�WNMO6937JL/Z1KWNM 37J�MO[P/XGI/([PMCD;=-IBk37JP;@-�Gjef;=-?GI/w/2193v69E2g=JL/mWNM@6d[neoE2_]<NMO3KJP;=-ZG?/:�UB �I6 �UGIJP-?g=/26w�

−~2

2m4Ψ(~r) + V (~r)Ψ(~r) = EΨ(~r)

� � �U`L� �{ 1K17M�T@;=-I1tGI/�g=E2-?E267MO[PJP19/26 B2/k3K3|E2_]<NMO3KJP;=-:M@<:B2M@1 GI/�GI/2<U�:WIM@693KJPB2<?[P/21 JPG?/2-A37JL_U<?/21 /�-:JL-]3K/26 p

M@B�3KJP;=-m>UJ�Ml<I-�WY;O37/2-A3KJP/2[]FlGI/�<c�QB2;@6KWI1U(~r1, ~r2)

� WNM@6+/ �?/�V\W?[P/r[�M^69E2WI<I[L1KJL;=-wB2;=<I[L;=V5iIJL/2-I-?//2-A3769/�G?/2<c�(E�[P/�B�3769;=-I1 e2

|~r1−~r2|�k`

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Ψ1(r1)Ψ2(r2)1KMO37JL19C�M@191K/�[neoE2_]<NMO3KJP;=-wGI/h�cB �I6��cGIJL-Ig=/�6 M@W?WI6K;=W?6KJPE�/d�

(− ~2

2m41 −

~2

2m42 + V (~r1) + V (~r2) + U(~r1, ~r2)

)Ψ1(~r1)Ψ2(~r2) = EΨ1(~r1)Ψ2(~r2)� �}�c` �@�

M@JL1l<I-I/m37/�[P[P/.M@WIWI69;cB �I/Q1K<?WIWY;@1K/w_]<I/�[�efE�_]<NMO37JL;=-ZGIJzyjE�6K/2-A3KJP/2[L[P/

(− ~2

2m41 −

~2

2m42 + V (~r1) + V (~r2) + U(~r1, ~r2)

)Ψ(~r1, ~r2) = EΨ(~r1, ~r2)

� �}�c`s�=�

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{ -X/�yY/�32q |Ψ(~r1, ~r2)|2/�193�[�MwWI6K;=iIM@iIJP[LJL3KE^GI/x3K6K;=<c>=/26�<I-I/xWNM@6437JPB�<I[P/dMO<�W�;=JP-A3

~r1/k3h[�esMO<?3769/

� � � � � � ��

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|Ψ(~r1, ~r2)|2 = |Ψ(~r2, ~r1)|2

⇒ Ψ(~r2, ~r1) = eiθΨ(~r1, ~r2)

Ψ(~r1, ~r2) = eiθΨ(~r2, ~r1) = ei2θΨ(~r1, ~r2)

⇒ ei2θ = 1

eiθ = ±1� � �c`��]�

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GIE��N-?J WIM@6m�

(P12Ψ)(~r1, ~r2) = Ψ(~r2, ~r1)� � �c`! =�

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Ψ�

/26KV&JP;@-I1 �x-A37JP14TUV\Ek376KJL_]<I/ 12, 3

2. . .

�h;=19;=-I1 �UTUV&E�3769JP_]<I/0, 1, 2, . . .

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� ��

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[PJLiY/�6937E�Gef<?-�CD/�6KV&JP;=-` �|;=<?6v<?-�E�[P/2Bk3769;=-qxi = (~ri, σi)

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37JLB2<I[L/X�ϕn(xi)

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Ψ(x1 . . . xN) =∑

n1,n2...nN

cn1,n2...nNϕn1(x1) . . . ϕnN

(xN )� �}�c`!"=�

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WY;=<?6�[L/21��mB2;U;=6KG?;=-I-IE�/212q][L/6KE�1K<I[z3�MO3r/�193r19TUV&E�3K6KJP_]<I/m�

ϕn(x1)ϕn(x2) = ϕn(x2)ϕn(x1)À�UJI[neo;=- <?3KJP[PJL1K/�GI/2<c�5CD;=-IB�3KJP;=-?1

GIJLyYE269/2-A37/�1ϕn1(x1)

/�3ϕn2(x2)

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{ϕn1(x1)ϕn2(x2) − ϕn1(x2)ϕn2(x1)

��-A3KJP19TUV&E�3K6KJP_]<I/ϕn1(x1)ϕn2(x2) + ϕn1(x2)ϕn2(x1)

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∣∣∣∣ϕn1(x1) ϕn1(x2)ϕn2(x1) ϕn2(x2)

∣∣∣∣�+Mwg@E2-IE�67M@[LJP17M 37JP;@-#M@<#B�M@1�F

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1√N !

∣∣∣∣∣∣∣

ϕn1(x1) . . . ϕn1(xN)``` ` ` ` ```ϕnN

(x1) . . . ϕnN(xN )

∣∣∣∣∣∣∣;=R 1√

N !

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� � � �� !�� `Nab[�-I/w1�esM@W?WI[PJL_U<?/miIJP/�-�1��I6d_]<esMO<c�(CD/269V\JL;=-I12`

� � � � � � ��

� �� ,��0�� "� � �� "�N

� ��,�� 0� � ��/0�"�!��,'/0��.��0�

�c<?WIWY;@1K;=-I1�_U<?/N

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H =N∑

i=1

Hi

� � �c`!0=�;=R

Hi

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� E�1KJLg=-I;=-I1vWNM@6ϕn

[P/21^CD;@-IB�3KJP;=-I1lGjef;=-?GI/21v/�3En

[P/�<I6lE2-I/�6Kg=JL/#�

Hiϕn(xi) = Enϕn(xi)� � �c` � �

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Hϕn1(x1)ϕn2(x2) . . . ϕnN(xN ) = (En1 +En2 +· · ·+EnN

) (ϕn1(x1)ϕn2(x2) . . . ϕnN(xN))

MOJP1�1KJ�;=-\C�M@Jz3�-efJLV\W�;=6437/l_]<I/2[L[P/vWY/�6KV5<c3�MO3KJP;=-#GI/�1^B2;U;=6KG?;=-I-IE�/212qU[�Mw1K;@V\V&/dGI/21�E2-I/�64pg=JL/21l6K/�193K/�JL-IB �NM@-Ig@E2/#�

Hϕn1(xp(1))ϕn2(xp(2)) . . . ϕnN(xp(N)) =

(En1 + En2 + . . . EnN)(ϕn1(xp(1))ϕn2(xp(2)) . . . ϕnN

(xp(N)))

,.-�/2-�GIE2GI<?JL3d_]<I/&�

H

∣∣∣∣∣∣∣

ϕn1(x1) . . . ϕn1(xN)``` ` ` ` ```ϕnN

(x1) . . . ϕnN(xN )

∣∣∣∣∣∣∣= (En1 + En2 + . . . EnN

)

∣∣∣∣∣∣∣

ϕn1(x1) . . . ϕn1(xN)``` ` ` ` ```ϕnN

(x1) . . . ϕnN(xN )

∣∣∣∣∣∣∣

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��

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� ��

� � � �� 12�h;=-I19JPGIE�6K;=-I1��0E2[L/2Bk376K;@-I12`��/21(WI69;=WI69JPE�3KE21(GefM@-A37JL19TUV\Ek3769JP/ B�;=-IB2/�6K-I/�-A3�FS[�M CD;=JL1Z[P/�1

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Ψ(~r1, σ1, ~r2, σ2) = ϕ(~r1, ~r2)χ(σ1, σ2)� �}�c`��=�

�cJh[PMHCD;=-IBk37JP;@-0Geo;=-IGI/(GI/Z19WIJP-S/�193#19TUV&E�3K6KJP_]<I/ZGIM@-I1 [neoE2B �IM@-Ig=/σ1 ⇔ σ2

qt[PMHWNM@6437JL/;=6Ki?JL3�MO[P/ GI;=Jz3 1�3K6K/\M@-A37JP14TUV\Ek376KJL_]<I/@q+/�3Q69E2B2JLWI6K;U_]<I/2V&/2-A32q 19Jr[�MZCD;=-IB�3KJP;=- Geo;=-IG?/�GI/&1KWIJL-/2143�MO-]3KJP14TcV&E�3K6KJL_U<?/@qN[�M5WIM@693KJP/m;=69iIJL37M@[P/.G?;=JL3 1k3769/Q14TUV\Ek376KJL_]<I/@`,.62q^W�;=<I6XG?/2<c� 19WIJP-I1 1

2

qh[neo/21KWIM@B2/ GI/ ~xJL[Pi�/2643X-efMS_U<?/ ��Ek3�MO3K12q^/k3�JL[�/�193(C�M@B2JL[P/�GI/B2;=-?193769<IJP69/w<I-I/wiNM@19/Q-?/QB�;=V&WI6K/�-NM@-A3v_U<?/�GI/�1dE�3�M 371d19TUV&E�3769JP_]<I/�1x/k3vM@-A37JP14TUV\Ek376KJL_]<I/21�`

��-A37JL19TUV&E�3769JP_]<I/ �

1√2(η1/2(σ1)η−1/2(σ2) − η−1/2(σ1)η1/2(σ2)) ⇔

| ↑↓〉 − | ↑↓〉√2

� � �c`z�� A�� ;=-IBw<I-�1KJL-Ig=<I[L/�3lG?;=JL3vM�>=;@JP6l<I-I/wWNMO6937JL/m;=6Ki?JL3�MO[P/.14TcV&E�3K6KJL_U<?/@`�UTUV\Ek3769JP_]<I/ �

η1/2(σ1)η1/2(σ2) ⇔ | ↑↑〉η−1/2(σ1)η−1/2(σ2) ⇔ | ↓↓〉

1√2(η1/2(σ1)η−1/2(σ2) + η−1/2(σ1)η1/2(σ2)) ⇔

| ↑↓〉 + | ↑↓〉√2

��/.376KJLWI[P/k3lGI;=Jz3dGI;=-IBQM�>=;@JP6l<I-I/wWNMO6937JL/m;=6Ki?JL3�MO[P/mM@-A3KJP19TUV&E�3K6KJP_]<I/O`�h/21\6K/2V\M@69_U<?/21&1K;=-A3�F [neo;=69JPg=JL-I/XGeo<I-�W �IE2-I;@V\8�-I/�3K6K821&JPV&W�;=6937M@-A3&M@WIW�/2[PE � � ��

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d� qc[L/21l1KW?JP-I1^G?/mB2/21lE�[P/�B�3769;=-I1dM@<?6K;=-A3^3K/2-IGNMO-IB2/QF�CD;=69V\/�6

<I- E�3�M 3�G?/\19WIJP- V\M �cJPV5<IV�`��h;@-I1KJLGIE269;=-I1w/2- /�yY/�3QGI/2<U� E2[P/�B�3K6K;=-I1�`|�YefJL[P1.19;=-A3:GIM@-I1w<I-E�37MO3w376KJLWI[P/k3�q[PMZWNM@693KJP/#;=69iIJL37M@[P/5/�193:M@-A37JL19TUV\Ek3769JP_]<I/@q+MO[P;=691Q_]<I/&12eoJP[L1m1K;=-A3�GNM@-I1Q<I- E�3�M 31KJL-Ig=<I[L/�3�q�[PM&WNMO6937JL/�19WNMO3KJ�M@[L/Q/�193�19TUV\Ek3769JP_]<I/@` � WIM@693KJP6dGI/:G?/2<c��CD;=-IBk37JL;=-I1vGef;@-IGI/

ϕ1(~r)/�3ϕ2(~r)

qI;=-�W�/2<?3vB2;@-I193K6K<IJL6K/m<I-I/wCD;=-IBk37JL;=-(M@-A37JL19TUV\Ek3769JP_]<I/#�

ΨA(~r1, ~r2) =1√2

(ϕ1(~r1)ϕ2(~r2) − ϕ1(~r2)ϕ2(~r1))� � �U`L�=�}�

/�3v<I-I/mCD;=-?B�37JL;=-�M@-A37JL19TUV\Ek3769JP_]<I/ �

� � � � � � ��

ΨS(~r1, ~r2) =1√2

(ϕ1(~r1)ϕ2(~r2) + ϕ1(~r2)ϕ2(~r1))� � �c`z� �=�

MOJP1mB2;@V\V&/ΨA(~r1 = ~r, ~r2 = ~r) = 0

qj[PM�>OM@[P/�<I6�V&;}T=/2-?-I/#GI/ [�M�69E2WI<?[P1KJL;=- B�;=<I[P;@V#piIJL/2-I-I/ e2

|~r1−~r2|2/�193 -I/k3K3K/2V&/2-A3 WI[L<I1:W�/�3KJL37/\_]<I/ZGNM@-I1:[neoE�3�M 3

ΨS

q|W�;=<I6�[P/�_U<?/2[ΨS(~r1 =

~r, ~r2 = ~r) 6= 0`��uefEk3�MO3

ΨA

M�>@/2BQ[�M:CD;=-IBk37JL;=-(;=6KiIJz3�M@[L/mM@-A37JL19TUV&E�3769JP_]<I/m/�193vGI;=-IBmC�M�>=;@6KJP19E@`

� � � ��

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/k3ϕ2

�

Ψ(x1, x2) =1√2

(ϕ1(x1)ϕ2(x2) − ϕ1(x2)ϕ2(x1))� � �c`z�}�A�

�+M >@MO[P/2<?6lV\;}T=/�-I-I/wGeo<I-I/Q;=iI19/269>OM@i?[P/m_]<I/2[LB2;=-?_U<?/O(x1, x2)

1�efE�B269JL3Q�

1

2

∫dx1

∫dx2

[ϕ∗

1(x1)ϕ∗2(x2)O(x1, x2)ϕ1(x1)ϕ2(x2)

+ ϕ∗1(x2)ϕ

∗2(x1)O(x1, x2)ϕ1(x2)ϕ2(x1)

− ϕ∗1(x1)ϕ

∗2(x2)O(x1, x2)ϕ1(x2)ϕ2(x1)

− ϕ∗1(x2)ϕ

∗2(x1)O(x1, x2)ϕ1(x1)ϕ2(x2)

] � �}�c`L� �]��/21uG?/2<c�5GI/269-IJP/�6K1 3K/26KV&/21 B2;=696K/219WY;@-IGI/2-A3�F�G?/21uJP-A3KE2g=6KM@[P/�1 ;=R ;=-5M.JP-A37/�69>=/�693KJ?[P/�1 CD;=-IB p

37JL;=-I1u69/2[�M 37JL>@/21�Fx1

/k3x2

FwgAM@<IB �I/d/k3hFwGI69;=JL3K/lGI/d[neo;=WYE�67MO3K/2<I6O(x1, x2)

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/k3tGeo<I- ;=W�E26KMO37/�<I66K/�[�MO3KJLC+F

x2

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< H > =

∫dx1ϕ1(x1)H1(x1)ϕ1(x1)

+

∫dx2ϕ2(x2)H2(x2)ϕ2(x2)

−∫dx1ϕ1(x1)H1(x1)ϕ2(x1)

∫dx2ϕ2(x2)ϕ1(x2)

−∫dx2ϕ2(x2)H2(x2)ϕ1(x2)

∫dx1ϕ1(x1)ϕ2(x1)

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dxϕ1(x)ϕ2(x) 6= 0`+�cJt;@- B�;=-I19JPGI8�6K/#GI/2<U� MO37;@V\/�1mGe��ATUGI69;=g=82-?/

� ��

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ϕ1s(~r)` M@JP1�[P/21vCD;=-IBk37JP;@-I1

ϕ1s(~r)B2/�-A376KE�/21QM@<?3K;=<I6.G?/21

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� $#�� + ��!-+

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ϕn(x1)ϕn(x2)/2193v14TcV&E�3K6KJL_U<?/

ϕn1(x1)ϕn2(x2) + ϕn2(x1)ϕn1(x2)/2193v14TcV&E�3K6KJL_U<?/ � � �c`z��"A�

{ 1K17M�T@;=-I1 GI/rB2;=-I143769<IJP69/�<?-I/uCD;=-IB�3KJP;=-wFd�lWNM@6437JLB2<I[L/21�M�>=/�B^�hCD;=-?B�37JL;=-I1�Geo;=-IG?/ϕ1

/k3ϕ2

�

ϕ1(x1)ϕ1(x2)ϕ2(x3) + ϕ2(x1)ϕ1(x2)ϕ1(x3) + ϕ1(x1)ϕ2(x2)ϕ1(x3)� � �c`z�0=�

{ [P[P/�B2;=-A37JL/2-A3x�537/269V\/�12`� /�C�M@B�;=-Zg=E�-IE26KM@[P/Oqc19J;=-XB �I;@JP1KJz3

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FNWNMO6937JLB2<I[L/21 � ;=R

niWY/�<?39p 1�3K6K/5E�gAM@[|F

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M�>@/2B [�M\B2;=-A>=/�-A37JP;@-0! = 1

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� �� 0�"�!��/0�"� ��,'/0� ��0�

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CD;=JP1ϕ1

qn2

CD;=JP1ϕ2

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H =∑

iHi

qdGeoE2-?/26Kg@JP/E =∑

i niEni

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HGjefE�-I/269g=JP/

E =∑

i niEni

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i niEni

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0/�3

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ϕ0(xi)[P/XCD;=-?GNM@V&/2-A3�M@[�GI/([ne��NM@V&JP[z37;=-IJL/2- F�<I-I/(WNM@6437JLB2<I[L/XGeoE2-I/�6Kg=JL/E0

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� �'� � � � ��

ϕ0(x1) . . . ϕ0(xN )� �}�c`L� � �

GeoE2-I/�6Kg=JL/NE0

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� $&� � � � ' +�� Q) ' %(' � � )��+MZGI;=-I-?E2/ GI/�1wCD;@-IB�3KJP;=-I1.FX<I-I/ WNM@6437JLB2<I[L/:<?3KJP[PJL1KE�/21.W�;=<I6.B�;=-I14376K<?JP6K/ <?- Ek3�MO3�37;@37M@[P/ p

V&/2-A3#M@-A37JL19TUV&E�3769JP_]<I/XW�;=<I6:[P/21�CD/26KV&JP;@-I1:;=<03K;@3�MO[P/2V&/2-A3�19TUV&E�3769JP_]<I/ZW�;=<I65[L/215i�;=19;=-I11KW�E2B�JL�N/dB�;=V\W?[P8�3K/2V&/2-A3r[neoE�37MO3�F

NWNM@693KJPB�<I[P/�12`��|M@6937M@-A3hGeo<I-I/diNMO1K/vGI/lCD;=-IB�3KJP;=-?1�W?6K;=WI69/21

F��hWNM@693KJPB�<I[P/ϕ1, ϕ2 . . . ..

qO;=-�WY/�<?3tG?;=-IBhB�;=-I14376K<?JP6K/�B2/�_U<jef;=-5M@WIW�/2[P[L/ � �� FNWIM@693KJPB2<?[P/21|;=R [P/�1 �O/�371r19;=-A3u1KJLV\WI[L/2V&/2-A3|6K/2W�E269E21rWNM@6|[P/�-I;=V5iI69/^GI/^CD;=JL1|_]<eo<I-I/lCD;=-?Bkp

37JL;=-ZJP-A3K/269>UJL/2-A3�` /269V\JL;=-I1 �

|n1, n2, . . .〉, ni = 0;=<

1� � �c`z��A�

�h;@1K;=-I1 �

|n1, n2, . . .〉, ni = 0, 1, 2, . . .� � �c`s� A�

3v- ;=WYE�67MO3K/2<I6�_U<?/2[PB�;=-I_]<I/�W�/2<?3�12eo/k�cWI69JPV&/26X19JPV&WI[P/�V\/�-]3\GNM@-?1X[neo/219WNM@B2/HGI/ ;cB ��F[nefM@JLGI/^Geo;=W�E267M 37/2<?6K1 ��B269E�MO3KJP;=- �

c†i/�3 �2M@-I-?J �IJL[�MO3KJP;=- �

ci_U<?J�W�/269V\/k3K3K/2-A3rGI/dB2JL6KB2<?[P/26uGIM@-I1

[neo/219WNM@B�/.GI/ ;cB ��`?,.-ZWY/�<?3lWNM@6h/k�c/2V&WI[P/�GIE2V&;=-A3769/26h_]<I/m19Jj<I-Z;@WYE�67MO3K/2<I6^/�193d[PM�1K;@V\V&/Geo;=W�E267M 37/2<?6K15F�<I-0B2;@6KWI1\�

O =∑

i Oi(xi)q ;=R 37;=<I1�[P/�1

Oi

1K;=-A35JLGI/2-A3KJP_]<I/21�q|;@-0WY/�<?3E2B�6KJL6K/ �

O =∑

ij

Oijc†icj

� �}�c` �U� �

M�>=/�BOij =

∫dxϕ∗

i (x)O(x)ϕj(x)FXB�;=-IGIJz37JL;=-H_]<I/ [L/21.;=W�E26KMO37/�<I6K1

c†iqci1KMO37JL19C�M@191K/2-A3

[P/�1l6K8�g=[P/�1v19<IJL>OM@-A3K/21w�

/269V\JL;=-I1 �{c†i |n1, . . . , ni . . .〉 = (−1)

P

(1,i−1)(1 − ni)|n1, . . . , ni + 1, . . .〉ci|n1, . . . , ni . . .〉 = (−1)

P

(1,i−1)ni|n1, . . . , ni − 1, . . .〉;=R ∑

(1, i − 1)|n1, . . . , ni . . .〉 = (n1 + · · · + ni−1)`u�c<I6 [�M�iNM@1K/(GI/ZB�/�3937/ZGIEk�N-IJL3KJP;=-jq

[�M#1K/�<I[P/Q_]<IJ gAM@67MO-]3KJL3v[neo/k�cWI69/2191KJP;@-H1KJLV\W?[P/wB2J pbGI/�1K1K<?1�W�;=<I6v[P/21d;=W�E26KMO37/�<I6K1�qY;@-HGIE�V\;=-A3K6K/_]<I/&�

� �� #� � ��

{ci, cj} = {c†i , c†j} = 0

{ci, c†j} = δij� � �c`s�=�=�

;=R {A,B} = AB +BA/2143v[nefM@-A37JPB�;=V&V5<?3�M 37/2<?62`

�h;=19;=-I1{c†i |n1, . . . , ni . . .〉 =

√ni + 1 |n1, . . . , ni + 1, . . .〉

ci|n1, . . . , ni . . .〉 =√ni |n1, . . . , ni − 1, . . .〉

�h/21dGIEk�N-IJL3KJP;=-?1lJPV&WI[PJL_U<?/2-A3l_]<I/

[ci, cj] = [c†i , c†j] = 0

[ci, c†j] = δij

� � �c`s�@�A�;=R

[A,B] = AB − BA/�193&[P/�B2;=V&V5<?37MO37/�<I6 �NMOiIJL3K<I/2[�`u�h/21 6982g=[L/21 19;=-A3#JPGI/�-]3KJP_]<I/�1#F

B2/�[P[P/�1l_]<ef;@-H;=i?3KJP/�-IGI67MOJL3lW�;=<I6l<I-?/QB�;=[P[L/2Bk37JP;@-XGeo;=1KB�JP[P[PMO37/�<I6K1 �IM@6KV&;=-IJL_U<?/212`�h/�3937/.C�M@B�;=-ZGI/m6K/�WI6KE�1K/�-]3K/26v[P/�1^19TU193K82V&/21xF5WI[P<?1KJP/�<I6K1hWNM@693KJPB�<I[P/�1^12efM@WIW�/2[L[P/.[PM � �� ! ��

� � � !�� !N` { [P[P/:/2193Q/�- WI6KMO37JL_]<I/ JP-IG?JP1KW�/2-?17M@iI[L/5W�;=<I6.376KM@JL3K/26.G?/21wWNM@6437JPB�<I[P/�1m/2- JP-cp37/�67M@Bk37JP;@-` { [P[L/.1K/�67M#/2-I19/2JPg@-IE2/QGNM@-I1dGIJzyjE�6K/�-]3K1dB2;=<I691vM}>OM@-?B2E21 � W �]TU1�`�143�MO32`IaKa�qNW �]TU19JP_]<I/GI<�1K;=[LJPGI/ma9a�q?3%�IE�;=6KJL/mGI/21vB �NM@V&WI1d_]<NM@-A37JL_]<I/21v6K/�[�MO3KJL>UJP1437/217�k`

� � � � � � ��

� � � � � �

� � � � �S� � � � � �� X�

& �� ,d!(,^$ � �k*k"�, +

� �� 3v-Q/�1KWNMOB2/rGI/r~xJL[Pi�/2643/2143 <I-w/21KWIM@B2/w>=/�B�37;@6KJP/�[ 1K<?6�[P/tB�;=6KWI1+GI/�1 B�;=V\W?[P/k�c/�1V5<I-IJ�Gjef<I- WI6K;UGI<IJz3d1KB�MO[�M@JL6K/.GIEk�N-IJ+W�;=1KJz37JLC `� �� 3v-�/�1KWNMOB2/wGI/Q~xJL[Pi�/2693h/2143xG?/QG?JPV&/2-I19JP;=-X�N-?JP/m1KJj;=-(WY/�<?3d376K;@<?>=/�6v<?-I/

iNM@19/�M�T=M@-A3d<I-(-I;=V5iI69/.�N-IJG?/.>=/2Bk37/�<I6K1�`j�UJP-I;=-jq?JP[�/2143vGI/mGIJPV&/2-?1KJP;@-XJP-?�I-IJP/O`N�?M@<cC|V&/2-Up37JL;=-XB2;@-]3K67M@JL6K/Oqc[L/21^WI69;=WI69JPE�3KE21^B�Jzp GI/2191K;=<I1h1K;=-A3l>OM@[�MOiI[P/�1�W�;=<I6�GI/21l/�1KWNM@B�/21dGI/m~vJP[LiY/�693�GI/GIJPV&/2-?1KJP;@-Z_U<?/2[PB�;=-I_]<I/Oq��I-IJP/w;=<(JP-c�N-IJP/O`� � � � �� { -�V&E2B2M@-IJP_]<I/Z_]<NMO-]3KJP_]<I/OquJP[h/�193��NM@i?JL37<?/2[^GI/Z-?;@37/�6&[L/21 E�[PE2V&/2-A3K1 GI/

[neo/219WNM@B2/vGI/v~xJL[Pi�/2643uWNM@6rGI/�1�14TUV5iY;@[P/21 |ϕ〉 q |ψ〉 q |χ〉 · · · ��/dWI69;cGI<?JL3r1KB2M@[�MOJP6K/lG?/ |ϕ〉 M�>=/2B|χ〉 /2143�-I;O37E 〈χ|ϕ〉 `� � �� reo/k�c/�V\WI[L/XGI/ZWI[L<I15-NM 37<I69/2[^Geo/219WNM@B�/�G?/(~xJP[LiY/�693�GI/ZGIJLV\/�-I1KJL;=-

N/�193

GI;=-I-?EQWIM@6dGI/21l>=/�B�3K/2<I691�B2;@[P;=-I-?/21vFNB2;=V&WY;@17M@-A37/�1dB2;=V&WI[P/ �c/21w�

|ϕ〉 =

a1```aN

|χ〉 =

b1```bN

� ��`L�}�

��/mWI6K;UGI<IJz3d1KB�MO[�M@JL6K/ 〈χ|ϕ〉 /2143xG?E��N-IJ+WNMO6m�

〈χ|ϕ〉 = b∗1a1 + b∗2a2 + · · ·+ b∗NaN� ��` �@�

= (b∗1 · · · b∗N)

a1```aN

� ��`s�=�

� ��

� � � ��

�=` 〈χ|ϕ〉 /2143x[LJP-IE2M@JP69/ WNMO6v6KM@WIW�;=693lF |ϕ〉 �

〈χ|ϕ1 + λϕ2〉 = 〈χ|ϕ1〉 + λ〈χ|ϕ2〉� �:`��]�

�c` 〈χ|ϕ〉 = 〈ϕ|χ〉∗�?` 〈χ|ϕ〉 /2143�MO-]3KJP[LJP-IE2M@JP69/ WNMO6v6KM@WIW�;=693vF |χ〉 �

〈χ1 + λχ2|ϕ〉 = 〈χ1|ϕ〉 + λ∗〈χ2|ϕ〉� �:`! =�

� ��

〈χ1 + λχ2|ϕ〉 = 〈ϕ|χ1 + λχ2〉∗

= 〈ϕ|χ1〉∗ + λ∗〈ϕ|χ2〉∗= 〈χ1|ϕ〉 + λ∗〈χ2|ϕ〉

�I` 〈ϕ|ϕ〉 /�193\6KE�/2[lB�MO6 〈ϕ|ϕ〉 = 〈ϕ|ϕ〉∗ `�,.-�-I;@37/�M@<?1K1KJ 〈ϕ|ϕ〉 = ||ϕ||2� -I;=6KV&/(M@<

B2M@6K69E}�k` c` �|M@6 �]TUW�;@3%�?821K/Oq 〈ϕ|ϕ〉 ≥ 0

qI/k3 〈ϕ|ϕ〉 = 01KJ�/�3d19/2<I[L/2V&/2-A3v1KJ |ϕ〉 = 0

`� � � � �� 3K;=<?3K/21�B2/21�WI6K;@WI6KJLE�37E�1r1K;=-A3�376KJz>UJ�M@[L/2V&/2-A3t>=E�6KJz�NE2/�1hWNM@6r[L/vW?6K;UGI<IJL3�19B�M@[PM@JP69/

GIEk�N-IJ�1K<?6x[�ef/�1KWNM@B�/wGI/21l>=/�B�3K/2<I691�B2;@[P;=-I-?/21vF B2;U/43&B2JL/2-A371vB2;=V&WI[L/k�c/21�`

� � � � � ��

3v-�;=W�E26KMO37/�<I6 [PJL-IE�MOJP6K/\/2143\<?- ;@WYE�67MO3K/2<I6#M@191K;UB2JPM@-A3&F�3K;=<?3#E�[PE2V&/2-A3#GI/Z[�ef/�1KWNMOB2/(GI/~xJL[Pi�/2643r<I-XM@<c376K/vE2[LE2V&/2-A3�GI/x[�ef/�1KWNMOB2/�G?/�~vJP[Pi�/2643�q=/k3h1KMO37JL19C�M@JL17M@-A3�[�MwB2;@-IGIJL3KJP;=-#GI/v[PJL-IE�M p6KJz37E �

|A(ϕ+ λχ)〉 = |Aϕ〉 + λ|Aχ〉� �:`�"A�

� � � � �� [L/21wWI69;cGI<?JL371mWIM@6m<I-I/&V�M 376KJLB2/(N × N)

FZB2;U/43&B2JL/2-A371QB2;=V&WI[L/k�c/21w1K;@-]3[P/�1l;=W�E267M 37/2<?6K1d[PJL-IE�M@JL6K/�1^GNM@-I1d[�ef/�1KWNM@B�/�G?/21l>=/�B�37/�<I6K1xB2;=[L;=-I-I/�1lGI/Q[P;=-Ig@<I/2<I6

N`

� � ��

{ -5V&E2B2M@-IJP_]<I/�_]<NM@-A37JL_]<I/@qO[�Md_U<IM@-A37JL3KE 〈ϕ|A|ϕ〉 /�193uMOWIWY/�[PE�/ � �� !�!"��GI/�[neo;=W�Ekp67M 37/2<?6AGNM@-I1Q[neoE�37MO3 |ϕ〉 ` �uef;=69JPg=JL-I/ GI/\B2/�3937/&37/269V\JL-I;=[L;=g=JP/�/2143:/ �?W?[PJPB�JL3KE2/&M@< B �IM@WIJL3K6K/�I`

� � ��(��

� ��

��/mB2;=- � <?g=<IE �I/269V\Jz37JP_]<I/�Gef<?-�;=W�E26KMO37/�<I6A/2143�-I;O37E

A† Ìab[�/�193xGIE��I-IJ�WNM@6m�

〈χ|A†ϕ〉 = 〈Aχ|ϕ〉� �:`!0=�

;=<�/2-IB�;=6K/OqNB2/w_]<IJ+/2193vE�_U<?JL>OM@[P/�-A3w�

〈χ|A†ϕ〉 = 〈ϕ|Aχ〉∗ � ��` � �� [P/uB2;@- � <Ig=<IE �I/�6KV&JL3KJP_]<I/uGeo<I-I/rV\MO3769JPB2/u/�193 [�M^B�;=- � <Ig@<IE2/�GI/r[�Mh3K67M@-I19WY;@1KE2/O`

��<?3K6K/�V\/�-]3vGIJz3�q

A =

a11 a12 · · · a1N

a21``àN1 · · · aNN

⇒ A† =

a∗11 a∗21 · · · a∗N1

a∗12``à∗1N · · · a∗NN

� ��`��=�

� �� (AB)† = B†A†

� � � ��

〈χ|(AB)†ϕ〉 = 〈ABχ|ϕ〉 = 〈Bχ|A†ϕ〉 = 〈χ|B†A†ϕ〉� ��`L� A�

� � � � � ��

3x-\;=WYE�67MO3K/2<I6r/�193 �?/26KV&JL3KJP_]<I/l1�efJL[�/�193�E2gAMO[YFw1K;@-�B2;@- � <Ig=<IE �I/�6KV&JL37JL_]<I/@qABOef/�1939pbF pbG?JP6K/d19J

A† = A� ��`L�@� �

� �� JL[YT(M GI/�1x19<Ii?3KJP[PJz37E�1^/2-�GIJPV&/2-?1KJP;@-ZJP-?�N-?JP/@`'1x;=<I1lTZ69/�>UJP/�-IGI69;=-I12`� � � � �� ;=-(-I;@3K/QJL-IGIJzyjE�6K/2V&V&/2-A3

A|ϕ〉 = |Aϕ〉 /�3 〈χ|Aϕ〉 = 〈χ|A|ϕ〉 `

� � � �� ,.-5GIJL3 _]<I/ |ϕ〉 /2143t>=/�B�3K/2<I6uWI69;=WI69/hGI/ A q@/k3u_]<I/ a (∈ C)

/2193t[�Mv>OM@[L/2<I6|WI6K;@WI6K/hM@1K19;cB�JPE2/1KJ

A|ϕ〉 = a|ϕ〉� �:`z� �=�

� �� /21l>OM@[L/2<I691lWI6K;=W?6K/21vGeo<I-�;=WYE�67MO3K/2<I6 �?/26KV&JL3KJP_]<I/m19;=-A3x69E2/�[P[P/�12`

� � � � � ��

� �� c<IW?WY;=19;=-I1v_]<I/A|ϕ〉 = a|ϕ〉 /k3v_U<?/ A† = A

�〈ϕ|A|ϕ〉 = a〈ϕ|ϕ〉

� ��`L�}�=� M@JL1

〈ϕ|A|ϕ〉 = 〈ϕ|A†|ϕ〉 = 〈Aϕ|ϕ〉 = 〈aϕ|ϕ〉 = a∗〈ϕ|ϕ〉⇒ a = a∗ ⇒ a

6KE2/�[ � �:`z� �]�� /21#>=/2Bk37/�<I6K1ZWI69;=WI69/21&Gef<?-�;=W�E267M 37/2<?6��I/�6KV&JL3KJP_]<I/�M@1K19;cB�JPE�1XF GI/21

>OM@[P/�<I6K1lW?6K;=WI69/21vGIJLyYE269/2-A37/�1d1K;=-A3v;=693 �I;=g=;@-NM@<c�Y`� �� c<IW?WY;=19;=-I1v_]<I/

A|ϕ〉 = a|ϕ〉 q A|ψ〉 = b|ψ〉 q A† = A/�3a 6= b

q〈ψ|A|ϕ〉 = a〈ψ|ϕ〉

� ��`L� @� M@JL1

〈ψ|A|ϕ〉 = 〈ψ|A†|ϕ〉 = 〈Aψ|ϕ〉 = (〈ϕ|A|ψ〉)∗ = b∗(〈ϕ|ψ〉)∗ = b∗〈ψ|ϕ〉 � �:`z��"A� M@JL1lB2;=V&V&/

A/2193 �I/�6KV&JL37JL_]<I/@q

b∗ = b` �xJP-I19Ja〈ψ|ϕ〉 = b〈ψ|ϕ〉

⇒ 〈ψ|ϕ〉 = 019J

a 6= b� ��`L�0@�

� �� ;=<?3d;@WYE�67MO3K/2<I6 �I/�6KV&JL37JL_]<I/./2143vGIJ�M@g@;=-NM@[LJP17MOiI[P/O`� �� >=;@JP6lB2;@<I6K1dGI/wV\MO3%�I1�`� �� cJ�GI/�<c�X;@WYE�67MO3K/2<I691^GIJ�MOg=;=-NM@[LJP1KM@iI[P/�1uB2;=V&V5<?3K/2-A3�q];=-�WY/�<?3^[L/21hG?J�M@g=; p

-NM@[LJP19/26^GNMO-I1d<I-I/QiNM@1K/wB�;=V\V:<I-I/@`� �� 1K<?WIWY;@1K;=-I1vGI;=-?BQ_]<I/

AB = BA

� �� 1KJ |ϕ〉 /�193u>=/�B�3K/2<I6�WI6K;@WI6K/lGI/ A GI/l>OM@[L/2<I6tW?6K;=WI69/aqB|ϕ〉 /2143hMO<I1K19J?>=/�B�37/�<I6WI69;=WI6K/wGI/

AGI/m>OM@[P/�<I6^WI69;=WI69/

a`

� �� AB|ϕ〉 = BA|ϕ〉 = Ba|ϕ〉 = aB|ϕ〉

� ��

�

�r69/2V&JP/�6 B�M@1Z�A|ϕ〉 = a|ϕ〉 qt/�3 a /2193&-I;@-cpbGIE�g=E2-?E26KE�/@quBOef/�1939pbF pbG?JP6K/Z_]<eoJP[h-ef/ �cJP193K/M@<IB2<?-�M@<?3769/\>@/2B�3K/2<I6#-I;@-0-]<I[hGI/ H

� FH<I-SB�;c/ 3&B2JP/�-A3 B2;=V&WI[L/k�c/�WI698217��GI/XV�12V&/>@MO[P/2<?6ZWI6K;=W?6K/@` �x[P;=6912q�WI<IJP19_]<I/

B|ϕ〉 /�193HM@<I191KJ.>=/�B�3K/2<I6�WI6K;@WI6K/ GI/ A GI/�>OM@[P/�<I6WI6K;=W?6K/

aqcJP[�/2143l1K;@JL3�WI6K;=W�;=6437JP;@-I-I/2[�F |ϕ〉 q]1K;@JL3^-]<I[�qcG?;=-IB�G?/.[�MwCD;=6KV&/ B|ϕ〉 = b|ϕ〉 qM}>@/2B

bE�>@/2-A37<?/2[P[L/2V&/2-A3vE2gAMO[+F

0`

� � ��(��

�

�U/2B2;@-IG�B2M@1m�?ab[j/k�cJP1437/m<I-(1K;@<I14p /21KWIM@B2/QGI/ H -I;@3KE Ha

3K/2[_]<I/ |ϕ〉 ∈ Ha ⇒ A|ϕ〉 =a|ϕ〉 `

� M@-I1 B2/lB�MO12q@W�;=<I6|37;=<?3 |ϕ〉 ∈ Ha

qB|ϕ〉 ∈ Ha

` �reo;=W�E26KMO37/�<I6B/2143uGI;=-?B^<I- ;=W�E26KMO37/�<I6

GI/ Ha

`��r<?JP1K_]<eoJP[j/2193xWNM@6 �ATUW�;@3%�I8�1K/�GIJ�M@g@;=-NM@[LJP17MOiI[P/OqU;@-�W�/2<c3x[L/mGIJ�MOg=;=-NM@[LJP19/26hGIM@-I1 Ha

`��/21.>=/�B�3K/2<I691QW?6K;=WI69/21.B�;=6K69/219WY;=-?GNM@-A3w1K;=-A3mi?JP/2-�1��I6.E�gAM@[P/�V\/�-A3�>=/�B�3K/2<I691QW?6K;=WI69/21.G?/

AGI/m>OM@[P/�<I6lWI6K;@WI6K/a`

� �� ;=-(M@W?WY/�[P[P/xB2;=V&V5<?37MO37/�<I6^GI/mG?/2<c�X;=WYE�67MO3K/2<I691l[neo;=W�E267M 37/2<?6^GIE��I-IJWNM@6��[A,B] = AB −BA

�cJ�GI/�<c��;=W�E267M 37/2<?6K1dB2;=V&V5<?3K/2-A3�q?[P/2<?6lB2;=V&V5<?37MO37/�<I6l/2143x-]<I[�`

� ��

��/21dB2;=V&V5<?37MO37/�<I6K1l1KMO37JL19CD;=-A3d[L/21dV�1�V\/�1l6K8�g=[P/�1l_U<?/�[L/21lB269;cB �?/�371vGI/ �|;@JP1K19;=- �

• [A,B] = − [B,A]• [A +B,C] = [A,C] + [B,C]• [AB,C] = A [B,C] + [A,C]B

� E2V&;=-I193K67MO3KJP;=- �[AB,C] = ABC − CAB

A [B,C] + [A,C]B = ABC − ACB + ACB − CAB = ABC − CAB

� ��`L� � �

• abG?/2-A37Jz37EwGI/��=M@B2;@iIJu�[A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0

� E2V&;=-I193K67MO3KJP;=- �(ABC − ACB − BCA+ CBA) + (BCA−BAC − CAB + ACB)

+ (CAB − CBA− ABC +BAC) = 0

� ��`L��=�

� � � � � � � � � � � � � � ��

{ -�GIJLV\/�-I1KJL;=-X�N-IJL/ � ;=-(GIE��I-IJL3v[�M:3K67M@B�/wGef<?-�;=W�E26KMO37/�<I6dWNM@6m�� 6A =

∑

n

〈n|A|n〉

;=R�[P/21lEk3�MO3K1 |n〉 CD;=69V\/�-A3d<I-I/wiNM@19/�;=643%�I;@-I;=6KV&E2/mG?/Q[�ef/�1KWNMOB2/wGI/Q~xJL[Pi�/26932`�|;=<?6v[L/21lV\MO3769JPB�/212q?;=-�M&GI;@-IB#�

� 6A =

∑

i

Aii

� �#� � � ��

� �� [�M#376KM@B2/�Gef<?-�WI6K;UGI<IJz3.GI/5V\MO3K6KJPB�/21v/2143mJP-A>OM@6KJPM@-A37/�WNM@6�W�/269V5<?37MO37JL;=-B2JL6KB�<I[�M@JL6K/.G?/21dV�M 376KJLB2/�12`� ��

� 6(A(1) · · ·A(n)) =

∑

i

(A(1) · · ·A(n))ii

M@JL1l[neoE2[LE2V&/2-A3dGI/wV�M 376KJLB2/(A(1) · · ·A(n))ij

1�efE�B26KJz3w�∑

i1,··· ,in−1

A(1)i,i1A

(2)i1,i2

· · ·A(n−1)in−2,in−1

A(n)in−1,j

⇒ � 6(A(1), · · · , A(n)) =

∑

i

∑

i1,··· ,in−1

A(1)i,i1A

(2)i1,i2

· · ·A(n−1)in−2,in−1

A(n)in−1,i

=∑

i1,··· ,inA

(1)in,i1

A(2)i1,i2

· · ·A(n−1)in−2,in−1

A(n)in−1,in

=∑

i1,··· ,inA

(n)in−1,in

A(1)in,i1

· · ·A(n−1)in−2,in−1

=� 6

(A(n)A(1) · · ·A(n−1))� ��` � A�

� � � ��

• � 6(AB) =

� 6(BA)

• � 6(A +B) =

� 6(A) +

� 6(B)

• � 6([A,B]) = 0

• � 6(ABC) =

� 6(CAB) =

� 6(BCA)

& �%$ � +�� ) ' + %('�� * � � ' $|" %(' % * � ' !-+�* ��! *�!��!Z* ' �

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� �� ;=<I1d[P/�1d/21KWIM@B2/�1x19E2WNM@6KM@iI[L/21d1K;=-A3vJP19;=V&;=6KW#�?/212`� �� ;=JL6^B2;=<?6K1vGI/wV�MO3 �`� 6K;=JL1m/k�c/2V&WI[L/21wGeo/21KWIM@B2/�1�GI/&~vJP[Pi�/2643m1KE�WNM@6KM@iI[P/�1m1K;=-A3QB2;@<I67M@V&V&/2-A3w<?37JL[PJL1KE21x/2- V\E p

B�MO-IJP_]<I/w_]<NM@-A37JL_]<I/@`

� � � � � ��

� � �� `(2)

� 3x-#E�[PE2V&/2-A3 |ϕ〉 ∈ `2/2143rGIE��I-IJNWNM@6t<?-I/l1K<IJz37/^G?/l-I;=V5iI69/21tB�;=V&WI[P/ �?/�1

c1, . . . , cn, . . .37/�[P[P/�_U<?/

‖ϕ‖2 =

+∞∑

n=1

|cn|2 < +∞� ��` �U� �

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M�>=/2B |ψ〉 = (d1, . . . , dn, . . .)/2193.GIE��N-?J

WNM@6w�〈ψ|ϕ〉 =

+∞∑

n=1

d∗ncn� ��` �=�@�

� ��

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[�M@_]<I/�[P[L/.1K/wGIE�V\;=-A3K6K/wB2;=V&V&/m1K<IJz3w�

〈ϕ− λψ|ϕ− λψ〉 = ‖ϕ‖2 − λ∗〈ψ|ϕ〉 − λ〈ϕ|ψ〉 + |λ|2 ‖ψ‖2 ≥ 0 ∀λ ∈ C

{ -�B �I;@JP1KJL1K1KM@-A3λ = ‖ϕ‖2

〈ϕ|ψ〉 ⇒ λ∗ = ‖ϕ‖2

〈ψ|ϕ〉qI;=-�;=i?37JL/2-A3w�

‖ϕ‖2 − ‖ϕ‖2 − ‖ϕ‖2 +‖ϕ‖4 ‖ψ‖2

|〈ψ|ϕ〉|2≥ 0

⇒ ‖ϕ‖4 ‖ψ‖2 ≥ ‖ϕ‖2 |〈ψ|ϕ〉|2

�cJ ‖ϕ‖ = 0qI[neoJP-IE�gAM@[LJL37Ev/2143x>U6KM@JP/O`��cJ ‖ϕ‖ 6= 0

qN;=-(GIJz>cJL1K/mWNMO6 ‖ϕ‖2

⇒ |〈ψ|ϕ〉|2 ≤ ‖ϕ‖2 ‖ψ‖2

� ��

|ϕ〉, |χ〉 ∈ `(2) ⇒ |ϕ+ λχ〉 ∈ `(2)� ��` � @�

� � � ��

〈ϕ+ λψ|ϕ+ λψ〉 = ‖ϕ‖2 +∣∣λ2∣∣ ‖ψ‖2 + λ∗〈ψ|ϕ〉 + λ〈ϕ|ψ〉

M@JL12q〈ϕ− λψ|ϕ− λψ〉 ≥ 0 ⇒ ‖ϕ‖2 +

∣∣λ2∣∣ ‖ψ‖2 ≥ λ∗〈ψ|ϕ〉 + λ〈ϕ|ψ〉

/�32q‖ϕ+ λψ‖2 ≤ 2

(‖ϕ‖2 +

∣∣λ2∣∣ ‖ψ‖2) < +∞

� ��

� � �� (2)([a, b])�

�h;=-?1KJPG?E26K;@-I1�[�ef/�1KWNMOB2/dGI/21rCD;=-IBk37JL;=-I1uB2;@V\WI[L/k�c/21ϕ(x)

1K<I6r[�efJL-]3K/264>@MO[P[P/[a, b]

3K/2[L[P/21u_]<I/w�∫ b

a

dx |ϕ(x)|2 < +∞� �:`s� "A�

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ϕ(x), ψ(x) ∈ L2 ([a, b]) ⇒ ϕ(x) + λψ(x) ∈ L2 ([a, b]))� �:`s� 0=�

� �� { [P[L/mGIE2B�;=<I[P/mG?JP6K/�B�3K/2V&/2-A3vGI/w[neoJP-IE�gAM@[PJz37E

‖ϕ‖2 +∣∣λ2∣∣ ‖ψ‖2 ≥ λ∗〈ψ|ϕ〉 + λ〈ϕ|ψ〉

� �� cJϕ(x)

qχ(x) ∈ L(2)([a, b])

qN[P/mWI69;cGI<?JL3l19B�M@[PM@JP69/5�

〈ψ|ϕ〉 ≡∫ b

a

dx ψ∗(x)ϕ(x)� �:`s� � �

/2143viIJP/�-�GIEk�N-IJ� �� { -�/kyj/k3�qN[�efJL-IE2gAMO[PJL3KE:�

∫ b

a

(ϕ∗ − λ∗χ∗)(ϕ− λχ) ≥ 0 (∀λ ∈ C)

B2;@-IGI<IJz3vF‖ϕ‖2 + |λ| ‖χ‖2 − λ∗〈χ|λ〉 − λ〈ϕ|χ〉 ≥ 0

_]<IJnqIM@WIWI[LJP_]<IEwFλ = ‖ϕ‖2

〈ϕ|χ〉qIB�;=-IGI<?JL3vB2;=V&V&/.WI6KE�B2E�GI/2V&V\/�-A3.M@<(69E21K<?[L3�M 3d6K/2B �?/26KB �?E@`

� �� ueo/21KWIM@B2/L2 ([a, b])

/2143x19E2WNMO67M@iI[L/@`� �� eo/2193&<I-?/�B�;=-I19E2_]<I/2-?B2/HGjef<I-S3 �IE2;=6982V&/Z1937M@-IGNM@69G�G?/Z[nefM@-NM@[zTc19/

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ϕ(x) =

+∞∑

n=−∞cn

1√b− a

e2iπnxb−a

M�>@/2Bcn =

1√b− a

∫ b

a

dx ϕ(x)e−2iπnxb−a

� ��` � �A�

� � � � � � � ��

��/21rCD;=-IB�3KJP;=-?1ϕn(x) = 1√

b−ae2iπnxb−a

B2;=-I1437Jz37<I/�-]3�<I-?/xiNMO1K/d;=693 �I;=-I;@6KV&E2/lGIE�-I;=V5iI6KM@iI[L/lGI/L2 ([a, b])

` { -�/�yY/�3�q

‖ϕn(x)‖2 =1

b− a

∫ b

a

dx e−2iπnxb−a e

2iπnxb−a = 1

� ��`s� =�/�3

〈ϕn|ϕm〉 =1

b− a

∫ b

a

dx e2iπx(m−n)

b−a

=1

b− a

(b− a)

2iπ(m− n)

[e

2iπb(m−n)b−a − e

2iπa(m−n)b−a

]� ��`s�?�}�

M@JL1e

2iπb(m−n)b−a − e

2iπa(m−n)b−a = e

2iπa(m−n)b−a

[e

2iπ(b−a)(m−n)b−a − 1

]= 0

� ��`s�=�=�

� � �� L2 (R)

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/2193 69/2V&WI[�M@B�EWNM@6

R` � ;=<I1Z[L/21\6KE219<I[L37MO371�WI6KE�B2E2G?/2-A371�12efM@WIWI[LJP_]<I/�-]32qh/k3Z;=- W�/2<?3ZG?E2V&;=-A376K/�6X_]<I/�B2/k3

/219WNM@B2/v/2143h19E2WNMO67M@iI[L/@qABOef/�1939pbF pbG?JP6K/l_]<eoJP[N/ �cJP193K/dGI/21�iNM@19/21�GIE2-I;@V5iI67MOiI[P/�12` { -�WNMO6937JLB2<I[LJP/�62q[P/�1dCD;@-IB�3KJP;=-I1lW?6K;=WI69/21vGI/w[neo;=1KB�JP[L[�MO3K/2<I6 �NM@6KV&;=-IJL_]<I/mB2;=-?1937Jz37<I/�-A3x<?-I/�iIM@1K/wGIE�-I;=V5iI6KM@iI[L/GI/wB2/k3x/�1KWNMOB2/QGI/Q~xJP[LiY/�6932`

& � � $�� 'd) " 'd' $ + ' " � ' �($, + ' !#" � "+*��:! � � 'd) "�$ � � ' �� * � ' !-+�* ��! � *2!Z* '

�c;=Jz3 |ϕ〉 <I-(E2[LE2V&/2-A3lGeo<I-(/219WNM@B�/mGI/w~xJL[Pi�/2693hGI/m-I;=69V\/:�=` � ;=<?3dE�[PE2V&/2-A3 |ψ〉 W�/2<c3d1K/GIE2B�;=V&WY;=19/26d1K<?JL>OM@-A3Q�

|ψ〉 = a|ϕ〉 + |ϕ⊥〉M�>=/�B 〈ϕ|ϕ⊥〉 = 0

� ��`s�=�=�� ab[�1K<.3#3vGI/QB �I;=JL1KJP6

{〈ϕ|ψ〉 = a

|ϕ⊥〉 = |ψ〉 − a|ϕ〉⇒ 〈ϕ|ϕ⊥〉 = 〈ϕ|ψ〉 − a = 0

� ��`s� �A�

�� /QW?6K; � /�B�37/�<I6 1K<I6v[neoE�37MO3 |ϕ〉 -I;@3KE P|ϕ〉/2143vGIE��N-?J�WNM@6

P|ϕ〉|ψ〉 = a|ϕ〉 = 〈ϕ|ψ〉|ϕ〉 � ��`s�� @�

� � � � � ��

� � ��

��/ WI6K;UGI<IJz3Z1KB2M@[�MOJP6K/�GI/ |ϕ〉 =

a1

. . .an

/�3 |ψ〉 =

b1. . .bn

/2143(GIE��I-IJmWNM@6 〈ψ|ϕ〉 =

b∗1a1 + . . .+ b∗nan`c�h/2[PM:1K<?g=g=8269/�GeoJP-A3K6K;UGI<IJP69/d[neo/21KWIM@B2/ � � � � GI/21h>@/2Bk37/2<?6K1l[PJLg=-I/�1v� 〈ψ| ≡(

b∗1 . . . b∗n) `r�h;=V&V\/\[P/XW?6K;UGI<IJL351KB�MO[�M@JL6K/�/�193&M@-A3KJP[PJL-IE�MOJP6K/&W�;=<I65[PM0�

� � >OM@69J�M@i?[P/@q ;=- MiIJL/2- 1��I6 〈λψ| = λ∗〈ψ| `j�c/5iIM@17M@-A3m19<I6m[P/Q37/�6KV&/5M@-Ig@[�M@JL1��i?67M@B �O/�3 � � B26K;UB �I/k3 �kq � JP6KM@B MJP-A3K6K;UGI<IJL3^[�M:37/�6KV&JP-I;@[P;=g=JL/��

|ϕ〉 =�O/�3 〈ψ| =

iI67M � ��`s� "A�

� � � � �� #� �

� efM@WI69821vB2/w_]<IJ+WI6KE�B28�GI/@qP|ϕ〉|ψ〉 = 〈ϕ|ψ〉|ϕ〉 = |ϕ〉〈ϕ|ψ〉

⇒ P|ϕ〉 = |ϕ〉〈ϕ|

� ��`s� 0=�

� � � � ��

�c;@JL3 {|ϕi〉, i = 1, . . . , N} <?-I/�iIM@1K/Q;=693 �I;=-I;@6KV&E2/mGI/Q[neo/219WNM@B2/QGI/�~xJL[Pi�/26932`'�+M#1K;=V&V\/GI/�1xWI69; � /2Bk37JL;=-I1lGI/m37;@<I1d[P/21dE�[PE2V&/2-A3K1lGI/w[�M iNM@19/Q/�193vE2g=M@[P/wF [neoJPG?/2-A37Jz37E@`� �� h;=-I19JPGIE�6K;=-I1u<?-#>@/2Bk37/2<?6 |ψ〉 _]<I/2[LB2;=-I_]<I/dGI/ H

`=,m- W�/2<c3uE2B�6KJP69/.�

|ψ〉 =N∑

i=1

ci|ϕi〉� ��`s� � �

�h;=V&V\/�[�M5iNM@19/m/2143x;@693%�?;=-I;=69V\E�/@q 〈ϕi|ϕj〉 = δij` ��JL-I1KJ�q ∑N

j 6=i cj|ϕj〉/�193d;=643%�I;@g=;=-NM@[L/mF

|ϕi〉` � ef;=R

P|ϕi〉|ψ〉 = ci|ϕi〉

⇒N∑

i=1

P|ϕi〉|ψ〉 =

N∑

i=1

ci|ϕi〉 = |ψ〉� ��`s��A�

�h/�3937/wWI69;=WI6KJLE�3KEQ1�efE�B269JL3�q?/2-H-?;@3�MO3KJP;=-ZG?/ � JP67MOB �� [

=∑

n

|ϕn〉〈ϕn| � ��`�� A�

{ [P[L/./2143�B�;=-I-]<I/w1K;=<?1x[L/m-I;=V GI/w69/2[�M 37JP;@-XGI/mCD/269V\/k37<I69/ `

� � � � � � � ��

� � ��

3x- ;=W�E26KMO37/�<I6A/2193rG?E��N-IJcWNM@6t[PM�GI;=-?-IE2/^G?/l1K/21uE�[PE�V\/�-]3K1tGI/^V\MO3K6KJPB�/�GIM@-I1u<I-I/hiNM@1K/��

Amn = 〈ϕm|A|ϕn〉`

� ��

A =∑

m,n

Amn|ϕm〉〈ϕn| � ��`��I�}�

� � � �� /21tE�[PE�V\/�-]3K1tGI/�V�MO3K6KJLB2/�GI< V\/�V5iI69/�GI/hGI6K;=Jz37/�1K;=-A3uG?;=-I-IE�1tWNM@6��

〈ϕi|(∑

m,n

Amn|ϕm〉〈ϕn|)|ϕj〉 =

∑

m,n

Amn 〈ϕi|ϕm〉︸︷︷︸δim

〈ϕn|ϕj〉︸︷︷︸δjn

= Aij� ��`��]�@�

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1K;=-A3vGI;=-?BQE�gAM@<c�Y`

� � � � � ��

�h;=-I19JPGIE�6K;=-I1d<I-�;=W�E26KMO37/�<I6 �I/269V\Jz37JL_U<?/A`

�=`x�c<IW?WY;=19;=-I1x_U<?/:1K/�1x>OM@[L/2<I691x19;=JP/�-]3v37;@<?37/�1�-I;=-�GIE2g@E2-IE�6KE2/�12`��+/�1x>@/2B�3K/2<I691.WI69;=WI6K/�11K;@-]3uM@[P;=691|;@693%�?;=g=;=-NMO<c��GI/2<c�5F�GI/�<c�Yq@/k3uM@WI69821t-?;=6KV\M@[PJL17MO3KJP;=-jq2JP[L1|B�;=-I1437JL3K<I/2-A3u<?-I/iNM@19/�;=643%�I;@-I;=6KV&E2/O` � E219JPg=-I;@-I1hB�/�3K3K/�iNMO1K/�WIM@6 {|ϕi〉, i = 1, . . . , N} `��/�1hE2[LE2V&/2-A371GI/wV\MO3769JPB2/�GI/

AGIM@-I1dB2/k3K37/wiNMO1K/Q1K;=-A3d37;=<?1vG?J�M@g=;@-NM@<c�Y` { -�/�yY/�3�q

〈ϕj|A|ϕi〉 = ai〈ϕj|ϕi〉 = aiδji = 019Jj 6= i

� ��`��A�=�,.-�/�-�G?E2GI<IJz3d_]<I/

A =∑

i

ai|ϕi〉〈ϕi| � ��`�� A�

�c`x�c<IW?WY;=19;=-I1m_]<I/ [L/21�>OM@[L/2<I691�WI69;=WI6K/�1mWI<IJL1K1K/�-A3 1�3769/ GIE2g@E2-IE�6KE2/�12q�/k3wM@WIW�/2[L;=-I1G(i)[�M�GIE2g=E�-IE269/219B2/2-?B2/�G?/0[�M�>OM@[L/2<I6�WI6K;@WI6K/

aiqx;=R

iWI69/2-IG V&;=JP-I1�GI/

N>OM@[L/2<I691

1KJv[neo<I-I/�M@<�V\;=JL-I1#GI/21&>OM@[P/�<I6K1&WI6K;@WI6K/�1�/2143XGIE�g=E2-?E26KE�/@` � E219JPg=-I;@-I1�GI/�WI[P<?1\WNMO6ϕ

(r)i , r = 1, . . . , G(i)

<I-I/(iNMO1K/�;=693 �I;=-I;=69V\E�/XGI< 1K;=<?14pb/�1KWNMOB2/H/�-Ig=/�-IGI6KE(WNMO6&[L/21>=/�B�3K/2<I691�WI69;=WI6K/�1vM@1K19;cB�JPE21vF [�M5>OM@[L/2<I6lWI69;=WI69/

ai`�,.-HM#M@[L;=6K1

• 〈ϕ(r)i |A|ϕ(r′)

j 〉 = 019Ji 6= j

B�MO6�[L/21m>=/�B�37/�<I6K1:WI6K;=W?6K/21�M@191K;UB2JPE�1:F(GI/�<c� >@MO[P/2<?6K1W?6K;=WI69/21dGIJzyjE�6K/2-A3K/21d1K;=-A3x;=693 �I;=g=;@-NM@<c�Y`

� ��

• 〈ϕ(r)i |A|ϕ(r′)

i 〉 = aiδrr′B�MO6 〈ϕ(r)

i |ϕ(r′)i 〉 = δrr′

`,.-(/2-�GIE2G?<IJL3v[�M569/2WI69E219/2-A3�MO3KJP;=-�1KW�/2Bk376KM@[P/ GI/

A�

A =∑

i

G(i)∑

r=1

ai|ϕ(r)i 〉〈ϕ(r)

i |� ��`�� =�

� /lC�M�;=- B2;@V\WNMOB�37/Oq=;=-#-I;@3K/21u19;=<?>=/�-A3A =

∑i ai|ϕi〉〈ϕi|

M�>=/2Bv[�MxB2;=-A>=/�-A37JP;@-\_]<I/〈ϕi|ϕj〉 = δij

V�12V&/m1KJai = aj

`

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<I-I/ZiIM@1K/ZB�;=V\V:<I-I/@`t�cJ^<I-S;@WYE�67MO3K/2<I6AM G?/215>OM@[P/�<I6K1:WI6K;=W?6K/21 G?E2g=E�-IE269E2/21�qr;=-SWY/�<?3

<?3KJP[PJL1K/�6�B�/�3937/.WI69;=WI69JPE�3KE�W�;=<I6�[P/k>=/26h[neoJP-IGIEk37/�6KV&JP-NMO3KJP;=-�� !�� !�uGI/�[PM�iNM@19/.GNM@-?1^[P/v1K;=<?14p/219WNM@B�/�WI69;=WI69/uGef<?-I/u>@MO[P/2<?6�WI6K;=W?6K/uGIE�g=E2-IE�6KE�/@`O�h;=-I19JPGIE�6K;=-I1+/�-�/kyj/k3|<?-Q;@WYE�67MO3K/2<I6

B_]<IJ

B2;@V\V5<c37/.M�>@/2BAqc/�3hGIE219JPg=-I;@-I1^WNM@6 {|ϕn〉, n = 1, . . . , N} <I-I/�iNM@1K/�WI6K;=W?6K/.B�;=V\V:<I-I/:�

{A|ϕn〉 = an|ϕn〉B|ϕn〉 = bn|ϕn〉

� ��`�� "A�

�cJ(an, bn) 6= (am, bm)

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1K/�V5iI[L/ Gjef;=W�E26KMO37/�<I6K1A1, . . . , Ap

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Ai|ϕn〉 = a(i)n |ϕn〉

� ��`��0=�B2/�[�M 1K<?WIWY;@1K/Q_U<?/ (

a(1)n , . . . , a

(p)n

)6=(a

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(p)m

) GI8�1v_]<I/m 6= n.

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U †U = U U † =� [

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� ��`��

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� � � � ��

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qI;=-�M(�

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/�3 〈U(ϕ+ λψ)|U(ϕ+ λψ)〉 = 〈Uϕ|Uϕ〉 + |λ|2 〈Uψ|Uψ〉 + 2< / (λ〈Uϕ|Uψ〉)� ��` =��uefE�gAM@[LJL37E � �:`�� A�hJLV\W?[PJP_]<I/

< / (λ〈ϕ|ψ〉) = < / (λ〈Uϕ|Uψ〉)λ = 1 ⇒ < / (〈ϕ|ψ〉) = < / (〈Uϕ|Uψ〉)λ = i⇒ = V (〈ϕ|ψ〉) = = V (〈Uϕ|Uψ〉)

� ��` c�}�

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� [� ��` =�@�

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U−1/�3hF�gAM@<?B �I/

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U †U =� [xJPV&WI[PJL_]<I/�_]<I/�[L/

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� �� *��

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� � � ��

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n〉 = 〈Uϕm|Uϕn〉 = 〈ϕm|U †U |ϕ〉 = 〈ϕm|ϕn〉 = δmn� ��` @�=�

� � � � � ��

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q�GIE��I-IJP191K;=-I1<I-�;=W�E267M 37/2<?6

UWNM@6

U |ϕn〉 = |ϕ′n〉`

� �� U/�193v<I-IJz3�M@JL6K/O`

� �� c;=Jz3 |ψ〉 <I-�E2[PE�V\/�-A3dGI/ H`�,.-�WY/�<?3dE2B�6KJL6K/#�

|ψ〉 =∑

n

cn|ϕn〉 ⇒ |Uψ〉 =∑

n

cn|ϕ′n〉

⇒ 〈Uψ|Uψ〉 =∑

m,n

c∗mcn 〈ϕ′m|ϕ′

n〉︸︷︷︸δmn

=∑

n

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n〉 = U |ϕn〉q�/�3

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c′n =∑

m

U †nmcm

� ��` =��

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∑

m

〈ϕ′n|ϕm〉〈ϕm|ψ〉

=∑

m

〈Uϕn|ϕm〉〈ϕm|ψ〉

=∑

m

〈ϕn|U †|ϕm〉〈ϕm|ψ〉

=∑

m

U †nmcm

� ��` "A�

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� ��` 0=�

� � � � � � � ��

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WNM@191K/26mG?/ {|ϕi〉} F {|ψi〉}qjBOef/�1939pbF pbG?JP6K/Q37/�[ _]<I/ |ψi〉 = U |ϕi〉

qY/k3mGIE�1KJPg@-I;=-I1.WIM@6Uij

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U/�3A1KMO37JL19CD;=-A3

(U †AU

)ij

= diδij� ��` � �

� � � ��

diδij = 〈ψi|A|ψj〉=∑

m,n

〈ψi|ϕm〉〈ϕm|A|ϕn〉〈ϕn|ψj〉

=∑

m,n

U †imAmnUnj =

(U †AU

)ij

� ��` �=�

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A|ϕn〉 = an|ϕn〉q |ϕn〉 ∈ H

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n

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A =∑

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7GI<�[LJL>U6K/�GI/ �+/ �h/�[P[PM@B@`

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x/k3p`

� � � � � �� x

1v;=<I1xM}>@;=-I1l>U<�_]<I/Q[�M � CD;=-IB�3KJP;=- �δ(x− x0)

GIEk�N-IJP/wWNMO6δ(x− x0) = lim

a→0ϕ(a)x0

(x)

M�>=/2Bϕ(a)x0

(x) =

{0

19Jx /∈

[x0 − a

2, x0 + a

2

]

119J

x ∈[x0 − a

2, x0 + a

2

]� ��`!"=�=�

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WNM@6

〈x0|ϕ〉 =

∫ +∞

−∞dx δ(x− x0)ϕ(x) = ϕ(x0)

� ��`!"@�A�;=RZ[PM�CD;=-IBk37JP;@-

δ(x− x0)W�/2<?3 1k3769/.WI6KJL1K/.B�;=V&V\/x[�M�[LJPV&JL3K/vGjesM@<c376K/�1^CD;=-IBk37JP;@-I1h_]<I/mB�/2[P[L/

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∫ +∞

−∞dx |x〉〈x| =

� [ � ��`!" �]��

〈ϕ|∫ +∞

−∞dx |x〉〈x||ψ〉 =

∫ +∞

−∞dx 〈ϕ|x〉︸︷︷︸

ϕ∗(x)

〈x|ψ〉︸︷︷︸ψ(x)

= 〈ϕ|ψ〉 � ��`!" =�

� ��/� � �� ueo/2-I19/2V5i?[P/.GI/�1lE�37MO371 {|x〉, x ∈ R} B2;=-I1437Jz37<I/.<?-I/miNM@19/mGI/m[neo/219WNM@B2/GI/w~vJP[Pi�/2643�` { -(/kyj/k3�qNB2/k3K3K/Q69/2[PMO37JL;=-XGI/.CD/�6KV&/�37<?6K/mW�/26KV&/�3dG?/mGIE2B�;=V\W�;=19/26^37;@<?3 �@/k3 |ϕ〉1K<?JL>OM@-A3w�|ϕ〉 =

∫ +∞

−∞dx |x〉〈x|ϕ〉 =

∫ +∞

−∞dx ϕ(x)|x〉

� �:`�" "A�� /��

x� � esM@W?6K821�B2/\_U<?J�WI69E2B�82GI/Oq ;=- >=E269JL�N/�M@JP19E2V&/2-A3

_]<I/Q[neo;=WYE�67MO3K/2<I6xW�;=1K1982GI/Q[�M 69/2WI69E21K/�-A3�M@JL;=-�19WY/�B�3K67M@[L/m1K<IJz>@MO-]3K/#�

x =

∫ +∞

−∞dx x|x〉〈x|

� ��`!" 0=�

� � � � � � � ��

� � � � � ��p

�uef;=W�E26KMO37/�<I6p/2143xG?E��N-IJ+WNMO6

pϕ(x) = −i~∂ϕ∂x

� ��`!" � �

��/21lCD;=-IBk37JP;@-�W?6K;=WI69/21dGI/p17M 37JP14CD;=-A3

−i~∂ϕ∂x

= pϕ(x)

⇒ ϕ(x) = Ceipx/~

� ��`!" �=�

WI<IJL1K_]<I/ −i~ ∂∂x

(eipx/~

)= −i~ ip

~eipx/~ = peipx/~

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` { -�/�yY/�32q∫ +∞

−∞dx |ϕ(x)|2 = C2

∫ +∞

−∞dx

GIJL>@/269g=/ � ��` 0 =�

� � � ��

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ϕ(x)<I-?/.CD;=-IB�3KJP;=-ZG?/

L2(R)`N�?M � � � �� /2143dGIE��N-?JP/

WNM@6

ϕ(k) =1√2π

∫ +∞

−∞dx e−ikxϕ(x)

� ��` 0c�}�

�h/�3937/w6K/�[�MO3KJP;=-Z1�efJL-]>@/2691K/Q/2-

ϕ(x) =1√2π

∫ +∞

−∞dk eikxϕ(k)

� ��` 0=�@�

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ϕp(x) =1√2π~

eipx/~� ��` 0@�=�

[�M 6K/�[�MO3KJP;=-�GI/mCD/26KV&/�3K<I6K/w1�efE�B26KJz3

� [=

∫ +∞

−∞dp |p〉〈p|

� ��` 0 �A�

� ��

� ��

〈x|∫ +∞

−∞dp |p〉〈p||ϕ〉 =

∫ +∞

−∞dp 〈x|p〉〈p|ϕ〉

=

∫ +∞

−∞dp

1√2π~

eipx/~∫ +∞

−∞dx′

1√2π~

e−ipx′/~ϕ(x′)

=1

2π~

∫ +∞

−∞dp

∫ +∞

−∞dx′ eipx/~e−ipx

′/~ϕ(x′)

=1

2π~

∫ +∞

−∞dp eipx/~

∫ +∞

−∞dx′ e−ipx

′/~ϕ(x′)

=1

2π~

∫ +∞

−∞dp eipx/~

√2πϕ

(p~

)

=1√2π

1

~

∫ +∞

−∞dk ~eikxϕ(k)

= ϕ(x)

� ��` 0 =�

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−∞dp |p〉〈p||ϕ〉 = 〈x|ϕ〉

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� � � � � � � � � �� x� �

p

,.-�M#G?E2V&;=-A376KEw_]<I/QWY;@<I6^37;=<c37/.CD;=-IBk37JL;=-ϕ(x)

;=-�MZ�∫ +∞

−∞dx′ ϕ(x′)

(1

2π~

∫ +∞

−∞dp eipx/~e−ipx

′/~

)= ϕ(x)

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δ(x− x′)qN;=-(/2-�GIE2GI<?JL3v_]<I/

1

2π~

∫ +∞

−∞dp ei

p(x−x′)~ = δ(x− x′)

� ��` 0 � ��^M@[LB2<I[L;=-I1lGIE�1K;=69V�M@JL1l[P/�1lWI6K;UGI<IJz371l1KB2M@[�MOJP6K/�1 〈x|x′〉 /k3 〈p|p′〉

〈x|x′〉 =

∫ +∞

−∞dp 〈x|p〉〈p|x′〉 =

1

2π~

∫ +∞

−∞dp eipx/~e−ipx

′/~ = δ(x− x′)

⇒ 〈x|x′〉 = δ(x− x′)

� ��` 0 �A�

� � � � � � � ��

〈p|p′〉 =

∫ +∞

−∞dx 〈p|x〉〈x|p′〉 =

1

2π~

∫ +∞

−∞dx eip

′x/~e−ipx/~ = δ(p′ − p)

⇒ 〈p|p′〉 = δ(p− p′)

� ��` � =�

B�M@6δ(p′ − p) = δ(p− p′)

`

� � ��

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� [=∑

n

|n〉〈n| +∫dν |ν〉〈ν| � ��` � �}�

/�3

A =∑

n

an|n〉〈n| +∫dν a(ν)|ν〉〈ν| � ��` � �@�

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〈m|n〉 = δmn�UTUV5iY;@[P/mGI/ �m6K;=-?/2B �@/�6

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δ

� ��` � �=�

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/�3 H2

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GI;=-A3[P/�1vE�[PE�V\/�-]3K1l1K;=-A3xGI/w[�M:CD;=69V\/

|Ψ〉 =∑

m,n

bm,n|m⊗ n〉� ��` � �]�

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〈m′ ⊗ n′|m⊗ n〉 = δmm′δnn′

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/�3 H2

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� � � � � ��

|Φ〉 =∑

m

cm|m〉; |χ〉 =∑

n

dn|n〉� ��` � "A�

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|Φ〉 ⊗ |χ〉 =∑

m,n

cm dn|m⊗ n〉� �:` � 0=�

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/�3 H2

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|i〉 =∑

mRim|m〉|j〉 =

∑n Sjn|n〉

� ��` �@� �

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˜|Φ〉 ⊗ |χ〉 =∑

i,j

ci dj|i〉 ⊗ |j〉� �:` � �A�

�u6K;@WY;=19JL3KJP;=- �

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� E�V\;@-I193K67MO3KJP;=- �

� � � � � � � ��

|Φ〉 =∑

m

cm|m〉 =∑

m

cm∑

i

R−1mi |i〉

⇒ ci =∑

m

cmR−1mi

GI/wV�12V&/dj =

∑

n

dnS−1nj

⇒ ˜|Φ ⊗ χ〉 =∑

i

ci |i〉 ⊗∑

j

dj |j〉

=∑

i

∑

m

cmR−1mi |i〉 ⊗

∑

j

∑

n

dnS−1nj |j〉

=∑

i

∑

m

cmR−1mi

∑

m′

Rim′ |m′〉 ⊗∑

j

∑

n

dnS−1nj

∑

n′

Sin′ |n′〉

=∑

m

cm∑

m′

(R−1R)mm′ |m′〉 ⊗∑

n

dn∑

n′

(S−1S)nn′|n′〉

=∑

m

cm|m〉 ⊗∑

n

cn|n〉 = |Φ ⊗ χ〉� �:` �?� �

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|Ψ〉 =∑

m,n

bm,n|m⊗ n〉� ��`��=�=�

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M+N-I;=V5iI69/21

cm�m = 1...M

� /�3dn

�n = 1...N

��37/2[L1 _]<I/bmn = cm dn

`�h;=V&V\/�JP[AT5M

M NE2_]<NM 37JP;@-I1 W�;=<I6

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/�3 H2

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� ;=<?3m;@WYE�67MO3K/2<I6AM@g=JP1917M@-A3�GNM@-I1 H1

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�

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q Hy

/�3 Hz

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xqy/k3zqU[neo/219WNM@B2/�GI/.~vJP[Pi�/2643

W�;=<I6h[L/�V&;=<?>@/2V&/2-A3lg=[L;=iNM@[�/2143l[P/xWI6K;UGI<IJz3�3K/2-I19;=6KJL/2[|� H~r = Hx⊗Hy ⊗Hz

q?/k3l[P/�1 �@/k371|~r〉 1K;@-]3v69/2[PJLE21dM@<U� �@/k371 |x〉 q |y〉 /�3 |z〉 WIM@6

|~r〉 = |x〉 ⊗ |y〉 ⊗ |z〉� ��`�� =�

�cJ�[�ef;@-�6KM � ;=<?3K/w[P/.19WIJP-qN;@-�M#GI/wV�12V&/

|~r, σ〉 = |x〉 ⊗ |y〉 ⊗ |z〉 ⊗ |σ〉� �:` � "A�

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_]<IJj/�193d[neo/219WNM@B�/.GI/.GIJLV\/�-I1KJL;=-M +N

/2-Ig=/�-IGI69E.WNM@6[PM56KE2<?-IJP;=-(GI/�1xiIM@1K/�1xGI/ H1

/�3 H2

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� � � � � �

� �� Z�Z��

� �2� Y! "�, �w$ � � ' � �Q' +�+�* ' !(! ' � , ' � � '

� ��

∫ +∞

−∞dx e−

12Ax2

=

√2π

A

� ��`L�}�

� E2V&;=-I14376KMO37JL;=- � �|;=19;=-I1z =

∫ +∞−∞dx e

− 12Ax2

z2 =

∫ +∞

−∞dx e−

12Ax2

∫ +∞

−∞dy e−

12Ay2

=

∫

R2

dxdy e−12A(x2+y2)

=

∫ +∞

0

dr r

∫ 2π

0

dθ e−12Ar2

= 2π

∫ +∞

0

du e−Au

= 2π

[e−Au

−A

]+∞

0

=2π

A

⇒ z =

√2π

A

� ��` �@�

��

� �%$ �!#"�, �w$ � � ' � �Q' +�+�* ' !�! ' � , ' � � ' � � 'd) ' ! � � ' $ � ' %-'

� � ' $ ) ' �

� � � � ��

∫ +∞

−∞dx e−

12Ax2+Bx =

√2π

Ae

B2

2A

� ��`f�A�

� E�V\;=-?19376KMO37JL;=- �

− 1

2Ax2 +Bx = −1

2A

(x− B

A

)2

+B2

2A

⇒∫ +∞

−∞dx e−

12Ax2+Bx = e

B2

2A

∫ +∞

−∞dx e−

12A(x−B

A)2

= eB2

2A

∫ +∞

−∞du e−

12Au2

︸︷︷︸√2πA

� ��`��]�

� �� !#"�, �w$ � � ' � �Q' +�++* ' !(! ' � � � �� ' )*'

� � � � ��

∫ +∞

−∞dx e−α

2x2

=

√π

α

� ��`! =�

� � ��x ∈ R

�� <e(α2) ≥ 0�� !�� ! � �� −π

4≤ argα ≤ π

4 �� E�V\;=-?19376KMO37JL;=- � �|;@1K;=-I1

α = aeiθM�>=/2B −π

4≤ θ ≤ π

4

/k3a > 0

`∫ +∞

−∞dx e−α

2x2

=1

a

∫ +∞

−∞du e−e

2iθu2

=2

a

∫ +∞

0

du e−e2iθu2 � ��`�"A�

1x;@<I1:MO[P[P;@-I1.B�MO[PB2<?[P/26wB�/�3K3K/\JL-A37E2g@67M@[L/ WNM@6w[�MZV&E�3%�?;cGI/#GI/�1:69E21KJLGI<I1�` �h;@-I1KJLGIE269;=-I1�F�B�/�3/�yY/�3^[�MmCD;=-IB�3KJP;=-

e−z2 M�>=/2B

z ∈ CqU/�3^B2M@[PB�<I[P;@-I1h19;=-XJP-A3KE2g=6KM@[P/v1K<IJz>OM@-A3^[P/vB2;@-]3K;=<I6^G?;=-I-IE

1K<?6v[PM:�Ng=<I69/B.1

`��h;=V&V\/e−z

2 /2193xM@-IM@[LT]37JL_]<I/@qI;@-�M iIJL/2-�1 �I6I1 + I2 + I3 = 0

M�>@/2B

I1 =

∫ R

0

dx e−x2

⇒ limR→+∞

I1 =

∫ +∞

0

dx e−x2

=

√π

2

� ��`!0=�

I2�N;=-(W�;=1K/

z = Reiϕ ⇒ dz = iReiϕdϕqIGjef;=R

z2 = R2e2iϕ = R2 (cos 2ϕ+ i sin 2ϕ)

⇒ I2 = iR

∫ θ

0

dϕ e(−R2(cos 2ϕ+i sin 2ϕ)+iϕ) � ��` � �

� � ��

0

=m(z)

<e(z)I1

I2

I3

θ

R

�� `L��

�h;=-A37;=<?6vW�;=<I6l[�efEk>OM@[P<NM 37JP;@-XGI/e−z

2 GIM@-I1d[P/mWI[PM@-(B2;=V&WI[L/k�c/@`

I3�N;=-(WY;@1K/

z = reiθ ⇒ dz = eiθdr

⇒ I3 = eiθ∫ 0

R

dr e−e2iθr2

⇒ limR→+∞

I3 = −eiθ∫ +∞

0

dr e−e2iθr2

� ��`��=�

�^M@[PB�<I[P;@-I1l[�M5[LJPV&JL37/xGI/I2[P;=691K_]<I/

R→ +∞ �

|I2| ≤ R

∫ θ

0

dϕ e−R2 cos 2ϕ =

R

2

∫ 2θ

0

dϕ e−R2 cosϕ

� �x`L�� =�

;=R�[neo;=-(M&<c37JP[LJP19E�[neoJP-?E2gAM@[LJL3KE ∣∣∣∣∫ b

a

dx f(x)

∣∣∣∣ <∫ b

a

dx |f(x)| `Iab[C�M@<c3dGIJP1437JP-?g=<I/26lG?/2<c�(B2M@1Q��=`

2θ < π2

` � M@-I1dB�/QB2M@12qcosϕ ≥ cos 2θ ⇒ e−R

2 cosϕ ≤ e−R2 cos 2θ

qI/�3

I2 ≤R

22θe−R

2 cos 2θ −−−−→R→+∞

0� �x`L�=�}�

�c`2θ = π

2

` � M@-?1.B2/�B�M@1�qY[L/�V\M � ;=6KM@-A3�GI<HWI6K/�V\JL/26vB�M@1x-I/:19<.3#3.WNM@1xB�M@6cos 2θ = 0

`,.-�>OM 1KE�WNM@69/26d[neoJP-A37E�g=67MO[P/./�-�GI/�<c��WNM@6437JL/21m�

|I2| ≤R

2

∫ ε

0

dϕ e−R2 cos ε

︸︷︷︸≥e−R2 cos ϕ

+R

2

∫ π2

ε

dϕsinϕ

sin ε︸︷︷︸≥1

e−R2 cosϕ

⇒ |I2| ≤Rε

2e−R

2 cos ε +R

2

∫ 0

− cos ε

dueR

2u

sin ε

⇒ |I2| ≤Rε

2e−R

2 cos ε +R

2 sin ε

1 − e−R2 cos ε

R2−−−−→R→+∞

0

� �x`L� �@�

��

;=R�[neo;=-5MdC�M@JL3 [P/�B �NM@-?g=/2V&/2-A3uGI/�>OM@6KJPM@iI[P/u = − cosϕ

Fv[�M�� [LJPg=-?/@`��JL-I1KJ�q limR→+∞ I2 =0q?/�3vJP[69/21437/

limR→+∞

I1 = − limR→+∞

I3 = 0

⇒√π

2= eiθ

∫ +∞

0

du e−e2iθu2

⇒∫ +∞

0

du e−e2iθu2

=

√π

2eiθ

� ��`L��A�

,.-�/�-�G?E2GI<IJz3dWNM@6 � ��`!"=�kq

∫ +∞

−∞dx e−α

2x2

=2

a

√π

2eiθ=

√π

α

� ��`L� �]�

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%(' � � ' $ ) ' �

� � �� <e(α2) ≥ 0� �� ∀β ∈ C

� �� !�� ∫ +∞

−∞dx e−α

2(x+β)2 �� ! � � ��! � � !��

β �

� E�V\;=-?19376KMO37JL;=- � ,.- WY/�<?3 1K/\6K/�193769/2JL-IGI6K/XM@<0B�MO1:;=Rβ/�193 W?<I6K/�V\/�-]3:JPV\M@g=JL-NM@JP69/@` { -

/�yY/�32qO19Jβ = w+iy

Mv<I-I/hWIM@693KJP/r6KE�/2[P[L/r;=-:W�/2<?3 [neoJP-IB�;=6KW�;=69/26�GNM@-?1xq C�M@JP69/�[L/rB �NM@-Ig=/�V\/�-A3

GI/(>OM@6KJPM@iI[P/�1x′ = x + w

/�3X[�efJL-A37E2g@67M@[L/X6K/�1937/�67M JL-IB �NM@-?g=E2/O`^�h;=-?1KJPG?E26K;@-I1\[PM CD;@-IB�3KJP;=-e−α

2z2qN/k3xB2M@[PB�<I[P;@-I1l1K;=-(JL-]3KE2g=6KM@[P/m19<I6d[P/mB�;=-A37;=<I6dGI;@-I-IEw1K<I6d[PM:�Ng=<I69/ �x` �c`

0

=m(z)

<e(z)

β

I1

I2

I3

I4

R−R

�� `s��

�h;=-A3K;=<I6dWY;@<I6d[neoE�>OM@[P<IMO37JL;=-XGI/e−α

2z2GNM@-I1v[P/mW?[�M@-ZB�;=V\W?[P/k�c/O`

� � ��

I1 −−−−→R→+∞

∫ +∞

−∞dx e−α

2x2

=

√π

α

I2 =

∫ β

0

dy ie−α2(R+iy)2 −−−−→

R→+∞0

⇒ I4 −−−−→R→+∞

0

I3 −−−−→R→+∞

−∫ +∞

−∞dx e−α

2(x+β)2

� �x`L� @�

�h;=V&V\/e−α

2z2/2143�MO-NM@[LT]3KJP_]<I/@q?[�M51K;@V\V&/ ∑

i Ii = 0qNGeo;=R([P/m6KE�1K<I[z3�MO3

∫ +∞

−∞dx e−α

2(x+β)2 =

∫ +∞

−∞dx e−α

2x2 � �x`L��"=�� <e(α2) ≥ 0

�� ∀B, β ∈ C �∫ +∞

−∞dx e−α

2(x+β)2+Bx =

√π

αe

B2

4α2� �x`L�0@�

� E2V&;=-I14376KMO37JL;=- �

−α2x2 +Bx = −α2

(x− B

2α2

)2

+B2

4α2

� �x`L� � �� efM@WI6K8�1x[L/m[P/�V\V&/@qc[neoJP-A37E�g=67M@[L/�GI;@-I-I/ √

πα

qNGjef;=R([L/m6KE219<I[L37MO3�`

� � � '-' �/. '-' + � � + � � $ "+* )^' ��* ' $ +� ��

∫ +∞

−∞dx e

i2Ax2

=

√2π

|A|ei �� (A) π

4 ∀A ∈ R� �x`L��=�

� E2V&;=-I14376KMO37JL;=- �

α2 = − iA2

⇒ α =

√|A|2e−i �� (A) π

4

� �x` � =��

∫ +∞

−∞dx e

i2Ax2−iBx =

√2π

|A|ei �� (A) π

4 e−iB2

2A ∀A ∈ R� �x` �c�}�

� E2V&;=-I14376KMO37JL;=- �i

2Ax2 − iBx =

i

2A

(x− B

A

)2

− iB2

2A

� �x` �=�@�

��

� �� !#"�, �w$ � � ' + ) �:! " ' ! � ! " ' ! ' � �Q' +�++* ' !(! '

� � � � ��

∫ +∞

−∞dx x2e−

12Ax2

=√

2πA− 32 ∀A ∈ R

� ��` �O�A�

� E�V\;=-?19376KMO37JL;=- � �|;@1K;=-I1I(A) =

∫ +∞−∞dx e

− 12Ax2 `�,.-�1KM@JL3d_]<I/

I(A) =

√2π

A⇒ dI

dA= −1

2

√2πA− 3

2

� ��` � �]�

M@JL1lGefM@WI6K8�1x19;=-�/k�cWI6K/�1K19JP;=-�1K;=<?1vCD;=69V\/�JP-A37E�g=67M@[L/@q

dI

dA= −1

2

∫ +∞

−∞dx x2e−

12Ax2

⇒∫ +∞

−∞dx x2e−

12Ax2

=√

2πA− 32

� ��` � =�

,.-�W�/2<c3lC�M@B2JL[P/2V&/2-A3lG?E2V&;=-A376K/�6d<I-I/mCD;=6KV:<I[P/.W?[P<I1dg=E�-IE26KM@[P/ �

� � � � �� Jn ≡

∫ +∞−∞dx x

ne−12Ax2 � ��

Jn =

{(n− 1)!!

√2πA−n+1

2��n��

0��n�� ` � "A�

��n!! = 1 · 3 · 5 · . . . · n �� n � � � � � �� ! � � � �� !�� !��

�

� E�V\;=-?19376KMO37JL;=- � �cJn/�193vJPV&WNM@JL6hMO[P;=691l[neoJP-A37E�g=67M@-A3h/2193vJLV\WNMOJP6K/x/�3d[�efJL-]3KE2g=6KM@[P/x/2143x-]<I[L[P/O`

�cJn/2143:WIM@JP6wM@[L;=6K1m;@- WI69;cB�82GI/&WNM@6m69E2B�<I6K69/2-IB�/@` ,m- MZ>c< _]<I/#B@eo/2193�>U67M@JuW�;=<I6

n = 2q

� � ��

1K<IW?WY;=19;=-I1v_]<I/�BOef/�193d>U67M@J�W�;=<I6n/k3xV&;=-A3769;=-I1l_]<I/QB@eo/2143v>U6KM@J�WY;@<I6

n + 2�

dn

dAn[I(A)] =

(−1

2

)n(1 · 3 · . . . · (n− 1))

√2πA−n+1

2

=

(−1

2

)n ∫ +∞

−∞dx xne−

12Ax2

⇒ Jn = (n− 1)!!√

2πA−n+12

dn+1

dAn+1[I(A)] =

(−1

2

)n(1 · 3 · . . . · (n− 1))

√2π

(−n + 1

2

)A−n+1

2−1

=

(−1

2

)n(1 · 3 · . . . · (n− 1))

√2π(n + 1)

(−1

2

)A−n+3

2

=

(−1

2

)n+1

(1 · 3 · . . . · (n− 1) · (n+ 1))√

2πA−n+32

=

(−1

2

)n+1 ∫ +∞

−∞dx xn+2e−

12Ax2

⇒ Jn+2 = (n+ 1)!!√

2πA−n+32

� �x` � 0@�

��

� � � � � �

� ��

,.-�B�;=-I19JPGI8�6K/w<I-�19TU193K82V&/�[LJP-IE2M@JP69/�GI/NE2_]<NMO3KJP;=-?1dF

NJP-?B2;=-I-]<I/�1w�

a11x1 + · · · + a1Nxn = y1

a21x1 + · · · + a2Nxn = y2``` ` ` ` ``` ``àN1x1 + · · · + aNNxn = yN

� ��`z� �

,.- WY/�<?3m69/2WI69E21K/�-A37/26mB�/:14Tc14378�V\/ 19;=<I1�CD;=69V\/QV\MO3769JPB2JL/2[L[P/A~x = ~y

;=RA/2143m[�M&V�MO3K6KJLB2/

N ×NGI/wB2;U/43&B2JL/2-A371

(aij)`��U;=-�G?E�37/�6KV&JP-NMO-]3d1�efE�B269JL3Q�

det(A) =

∣∣∣∣∣∣∣

a11 a12 · · · a1N``` ``` ` ` ` ``àN1 aN2 · · · aNN

∣∣∣∣∣∣∣

� ��`s�=�

,.-�>OM <?37JL[PJL1K/26hGI/2<c�ZWI6K;=W?6KJPEk37E�1vG?<�GIEk37/�6KV&JP-NM@-A3w��=`x�cJ�;=-�V:<I[L3KJPWI[LJP/^<I-?/�B2;@[P;=-I-?/ � [PJPg@-I/}� GI/

AWIM@6�<?-I/�B�;=-I1937M@-A37/

cMO[P;=691�[PMQ>OM@[L/2<I6rGI<

GIEk37/269V\JL-NM@-A3d/2143vV5<I[L3KJPWI[LJPE�/�WNM@6lB�/�3937/wB2;=-?193�MO-]3K/@`�c`x,.-0W�/2<?35M � ;=<?3K/26 F�<I-I/XB�;=[P;=-?-I/ � [PJPg@-I/}�.GI/

AGI/21:V5<I[L3KJPWI[L/21wGefM@<?3K6K/21�B2;@[P;=-I-?/21

� [LJPg=-?/21��GI/A1KM@-I1dB �NM@-?g=/26

det(A)`

,.-�W�/2<?3v<?3KJP[LJP1K/�6h[PM#W?6K/2V&JP8�6K/mWI69;=WI69JPE�3KEmWY;@<I6lE2B�6KJP69/#�

xr det(A) =

∣∣∣∣∣∣∣

a11 a12 · · · xra1r · · · a1N``` ``` ``` ``àN1 aN2 · · · xraNr · · · a1N

∣∣∣∣∣∣∣

� ��`f�A�

;=Rxr

/�193�[�M(6 � JP-?B2;=-I-]<I/&GI< 19TU193K82V&/@` ,m- <?3KJP[PJL1K/ /�-I1K<IJz37/&[�MZ1K/�B2;=-?GI/�WI69;=WI69JPE�3KE#WY;=<?6M � ;=<?3K/26

x1

CD;=JL1^[�M�� B�;=[P;=-?-I/@qx2

CD;@JP1^[PM � � q . . . q xk CD;=JP1^[PM � � B2;=[L;=-I-I/.F5[�M5B2;@[P;=-I-?/ r q

� � � � � � � � ��

W�;=<I6k 6= r

�

xr det(A) =

∣∣∣∣∣∣∣∣∣

a11 a12 · · · (a11x1 + · · ·+ a1rxr + · · ·a1NxN) · · · a1N``` ``` ``` ```aN1 aN2 · · · (aN1x1 + · · ·+ aNrxr + · · ·aNNxN )︸︷︷︸

r ��

· · · aNN

∣∣∣∣∣∣∣∣∣� ��`��]��JL-I1KJ�qI1KJ

x1, . . . , xN/2143x<?-I/�19;=[P<c37JP;@-ZGI<(19TU193K82V&/@q�;=-�M

xr det(A) =

∣∣∣∣∣∣∣

a11 a12 · · · y1 · · · a1N``` ``` ``` ```aN1 aN2 · · · yN · · · aNN

∣∣∣∣∣∣∣= det(Ar)

� ��`! =�

;=RAr

/2143u[�MvV�M 376KJLB2/�;=i?3K/2-]<I/l/2- 69/2V&WI[�M �M@-A3|[�Mx6 � B�;=[P;=-?-I/hGI/ A WNM@6|[P/�>@/2Bk37/2<?6y1```yN

`

{ -�WIM@693KJPB2<?[PJP/�6h19Jdet(A) 6= 0

qN;=-�M

xr =det(Ar)

det(A)r = 1, 2, . . . , n

� ��`!"A�

��eo/2193#B�/X_]<I/Z[neo;=- M@WIW�/2[P[L/�[PM �� ` { -SWI67MO3KJP_]<I/X;@-0[neo<?37JL[PJL1K/#67M@69/2V&/2-A3B�MO6mWY;=<?6.GI/21wW�/�3KJL371�19TU193K82V&/21w;=- WI6K;UB2E�GI/26KMZWI[P<?3� @3mWNMO6.1K<Ii?1937Jz37<?3KJP;=-�/k3w6KE�1K;=[L<?37JL;=- �2F[�M V�M@JL- �Oqu3�MO-IGIJP1\_]<I/�WY;@<I6�GI/�1Zg=67MO-IGI1X14Tc14378�V\/�1�[PM V&E�3 �I;cG?/ GI/21ZW?JL>=;O371XGI/ 5QM@<?1K1-IE�B2/2191KJz37/QV\;=JL-I1^Geo;=W�E26KMO37JL;=-I12`

notes_de_cours quantique

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