106 CHAPTER 6. “®œ&�

6.3 ^¸HHHîîî

ÂNaüPI1:ã. (¡:-I1ø—�ÿ�⇢⌃¬⇣‡aø�Ÿ1/ZeemanHî. vv9ê/˜5P∑

ó�*∞Ñø˝⇢

H0 = �B · (µL + µS) =

eB

2mec(Lz + 2Sz) (6.58)

µL/hS–®ß�Ñ¡È�µS /ÍÀß�Ñ¡È�Mb®∫"üP¡Èˆ�eÜhS–®Ñ¡È�sÉ�

“®œÑs˚. ÍÀ“®œ�7ß�¡È�F/$⇧¡À‘���Ÿ ⇣‡P2Ñ˙∞. ⌦✏-�⌘ÏÚœ‰

¡:π⌘:z.

(° ¡:ˆ�NaüP˜5PÑ»∆�œÔÂô⇣

H0 =P

2

2me

+ V (r) + ⇠(r)~S · ~L, (6.59)

���y/⌦Ç®∫«Ñ¸Ù˝ßæ∆”ÑÑÍÀhS&�˝�Hf ).

• Çú¡:à:�shH 0i � h⇠(r)~S · ~Li�Ô9nyÅpœ0° ⌘Ï˝eHf , dˆÑZeemanHî::

:Zeeman Hî

• Çú¡:à1�Hf¯˘��1/ÕÅÑ�1:ZeemanHî

6.3.1 ::::::ZeemanHHHîîî

H0 =P

2r

2me

+L2

2mer2+ V (r) (6.60)

H = H0 +H0 = H0 +

eB

2mec(Lz + 2Sz) (6.61)

H,H0, L2, Lz, Sz|d˘◆�n, l,m,ms/“}”œPp�s,Å�ÔÂô:

RnlYlm�ms.

¯Õ, J2�H0��˘◆�j�/“}”œPp�,Å��˝ô:Rnl�ljmj

.

⌘ÏÔÂô˙,Åπ↵⇢

HRnlYlm�ms= ERnlYlm�ms

(6.62)

⌃v�:Ñ⌘π↵⇢

[P

2r

2me

+l(l + 1)~22mer

2+ V (r) +

e~B2mec

(m+ 2ms)]Rnl = ERnl (6.63)

⌘ÏÂS

[P

2r

2me

+l(l + 1)~22mer

2+ V (r)]R(0)

nl= E

(0)nl

R(0)nl

(6.64)

Ÿ/° ¡:ˆÑÑ⌘π↵. E(0)nl�´∆: ÚÂ. à�>R

(0)nlÿ/(6.63)Ñ„��«

E = E(0)nl

+eB~2mec

(m+ 2ms)

= E(0)nl

+ µBB(m+ 2ms)

⌘ÏÔÂäŸ*˝ß∞:Enlmms�1€*“}”œPp≥ö.

⇤Q√¡ö⇡⇢

�l = ±1, �m = 0, ± 1, �ms = 0 (6.65)

6.3. ^¸Hî 107

Figure 6.4: ::ZeemanHî.

√¡M�ÍÀ∂��˝9ÿ�ó1ø⌃¬⇣ a.

~! = E(0)3P � E

(0)3S +�Em (6.66)

v-�Em = µBBm,m = 1, 0,�1. ¯ª1øëáÓ⇢

�! =µBB

~ =eB

2mec

˝ßÑ9ÿ≈µå√¡≈µ�DzFig. 6.4@:.

6.3.2 111:::ZeemanHHHîîî

ÜÚ⌦»Õ8ZeemanHî.

Hf = ⇠(r)~S · ~L�H0 = eB

2mec(Jz + Sz)◊◆¯S.

⌘ÏäH0+HfS\∞ÑH0,\:;¸˝œ.ÂH0, J2, Jz, L

2, S

2:åh∆�Ÿ7H0Ñ,Å�ô:Rnl�ljmj.

n, l, j,mj/“}”œPp. H0Ñ,Åπ↵:

H0Rnl�ljmj= E

(0)nlj

Rnl�ljmj(6.67)

˝ß1/æ∆”Ñ˝ß. œ*˝ßÄv¶2j + 1.

⌘Ï ↵3P 12�˝œ(¡:\(↵Ñ˚®.

∞

R31�1 12 ,

12!

(0)n1

R31�1 12 ,�

12!

(0)n2

(6.67)1/

H0 (0)n⌫

= E(0)n (0)n⌫

, ⌫ = 1, 2. E(0)n

= E(0)30 1

2(6.68)

ÄvÆp⇢

(0)n

= c1 (0)n1 + c2

(0)n2 , (6.69)

BEn = E(0)n + E

(1)n .

108 CHAPTER 6. “®œ&�

Ÿ�Å°óH0ÑÈ5C:

H0⌫,⌫0 = ( (0)

n⌫, H

0 (0)n⌫0) (6.70)

1é[Jz, Sz] = 0, ‡dH0˘“(Sz˘“). ⌘ÏÍ�°ó

H011 = (R31�1 1

212, (Jz + Sz)R31�1 1

212)eB

2mec

= (�1 12

12, (~2+ Sz)�1 1

212)eB

2mec

(j = l � 12 ˆ, hSzi = �

12

1+ 12

~2 = �~

6 .

‡dH011 = 1

3µBB. É_1/,Å<�‡dEn = E(0)31 1

2+H

011 ,�1/R31�1 1

212.

�7

H022 = (R31�1 1

2�12, (Jz + Sz)R31�1 1

2�12)eB

2mec= �1

3µBB

‡dEn = E(0)3P 1

2

+H022, ˘î,Å�: R31�1 1

2�12.

π◆®�0�,≈µ⇢

Enljmj= E

(0)nlj

+ E(1)nljmj

E(1)nljmj

=eB

2mec(mj~+ hSzi)

1é

hSzi =~2(�ljmj

,�z�ljmj)

=~2{

mj

j, Sj = l + 1

2

� mj

j+1 , Sj = l � 12

@Â

E(1)nljmj

= BµBmj(1 +hSzimj~

) Ϙ-:⌫∑‡Pg

g = {1 + 1

2j , j = l + 12

1� 12j+2 , j = l � 1

2

˝ßÑ9ÿ≈µÇ˛Fig. 6.3.2@:.

⇤Q√¡ö⇡⇢

�j = 0,±1, �mj = 0,±1

⌘ÏÂS�q⇢˙∞10a1ø��æ∆”Ñ˘î�⌃+D1, D2.

6.4. Öæ∆”Ñ, ÍÀU�� Õ� 109

Figure 6.5: 1:ZeemanHî.

6.4 ÖÖÖæææ∆∆∆”””ÑÑÑ, ÍÍÍÀÀÀUUU������ ÕÕÕ���

⌘Ïe⇤Q"üP�v5På(P˝/ÍÀ 12íP. (PÑ¡È

~µp =gpe

2mpc

~Sp

v-gp = 5.59�'é5PÑg = 2.

(P�5PdÜì—¯í\(��ÿ c‘é ~Sp · ~Se Ñ¡¯í\(, :ÍÀ-ÍÀ&�.

⌘Ï∞(⇤QÍÀ-ÍÀ&�˘"üP˝ßÑÓc. >6;“®œÑsπåvz⌃œîÂ/àRÑ}õf

œ.

⌘ÏäÓò®�0˚✏$*ÍÀ1/2íP. ñHöIv;“®œåÉÑ *⌃œ

~S ⌘ ~S1 + ~S2, (6.71)

Sz = S(1)z

+ S(2)z

, Sx = S(1)x

+ S(2)x

, Sy = S(1)y

+ S(2)y

. (6.72)

◆¡

[Sx, Sy] = i~Sz, [S↵, S2] = 0. ↵ = x, y, z (6.73)

Ÿ7

~S1 · ~S2 =S2 � S

21 � S

22

2=

S2

2� 3

4~2

sS2, Sz˝�~S1 · ~S2˘◆.

Åœ"üPÃb5På(PÑÍÀ∂�, ÄU0↵ €ÕÔ˝"e"p, "e#p, #e"p, #e#p� ÉÏÑ⌅Õ‡†�. Ë✏ŸÃ⌘Ïñ!b4$*íPÑœP�ÑÓò�⌅⌃weàÄU�1/⌅Íœ�çv(�w.

_ÔÂô:|+ie|+ip, |+ie|�ip, |�ie|+ip, |�ie|�ip.ŸÕπ✏vû1/«(˙‚�ms

(1)�m0s(2), ms = ±1/2,m0

s= ±1/2, ÉÏ/S

2(1), Sz(1), S2(2), Sz(2)Ñq

�,Å�. _ÔÂô⇣

↵(1)↵(2), ↵(1)�(2), �(1)↵(2), �(1)�(2)

↵(1) = � 12(1), �(2) = �� 1

2(2); �⇧

|+i1|+i2, |+i1|�i2, |�i1|+i2, |�i1|�i2.

(ÉÏ\˙1œ˚flÑÍÀ∂�:^&�ha.

110 CHAPTER 6. “®œ&�

⌘Ï↵0ÉÏ_/SzÑ,Å˝p⇢

Sz�ms(1)�m0

s(2) = (Sz(1) + Sz(2))�ms

(1)�m0s(2)

= ms~�ms(1)�m0

s(2) +m

0s~�ms

(1)�m0s(2)

= (ms +m0s)~�ms

(1)�m0s(2)

ŸÃË✏⇢$*ó&⌅Í\(0ÍÒzÙÑ�. ⌘Ïó0SzÑ,Å< ~, 0,�~ ÕÔ˝.

⌘ÏÛÂSS2, SzÑq�,Å��,Å<. ÉÏ/⇢

↵1↵2,�1�2,1p2(↵1�2 + �1↵2),

1p2(↵1�2 � �1↵2)

¡�⇢

Sz↵1↵2 = ~↵1↵2, (6.74)

Sz�1�2 = �~�1�2. (6.75)

Sz

1p2(↵1�2 + �1↵2) =

1p2(~2↵1�2 �

~2�1↵2) +

1p2(�~

2↵1�2 +

~2�1↵2) = 0 (6.76)

Sz

1p2(↵1�2 � �1↵2) = 0 (6.77)

ç↵

S2↵1↵2 = (

3~22

+ 2~S1 · ~S2)↵1↵2

= [3~22

+2~24

(�(1)x�(2)x

+ �(1)y�(2)y

+ �(1)z�(2)z

)]↵1↵2

=3~22↵1↵2 +

~22[�1�2 � �1�2 + ↵1↵2] = 2~2↵1↵2

ŸÃ(0�x↵ = �,�y↵ = i�. ↵bÿÅ(0�x� = ↵, �y� = �i↵.

�⌃

S2�1�2 = 2~2�1�2

ç↵, *�

S2 1p

2(↵1�2 + �1↵2)

=3~22

1p2(↵1�2 + �1↵2) +

~22

1p2[(�1↵2 + ↵1�2)⇥ 2 + (�↵1�2 � �1↵2)]

= 2~2 1p2(↵1�2 + �1↵2)

å,€*�

S2 1p

2(↵1�2 � �1↵2)

=3~22

1p2(↵1�2 � �1↵2) +

~22

1p2[(�1↵2 � ↵1�2) + (�1↵2 � ↵1�2) + (�↵1�2 + �1↵2)]

= 0

⌘Ï�e∞˜�SMS, v-S = 0, �⇧1,:;ÍÀœPp�MS 2S + 1Ô˝, :;ÍÀz⌃œœPp. SS = 1,

�SMS=

8>>><

>>>:

↵1↵2 = �11, S = 1,MS = 1

�1�2 = �1�1, S = 1,MS = �1

1p2(↵1�2 + �1↵2) = �10, S = 1,MS = 0

(6.78)

6.5. $*“®œÑ&��C-G˚p 111

:ÍÀ Õ�(triplet). SS = 0ˆ�

�00 =1p2(↵1�2 � �1↵2) (6.79)

�⇧ô: 1p2(|+�i � |�+i), :ÍÀU�(singlet). ^8ÕÅ�

fi«4e↵"üPÑ˝ßÓc. S5P�(Pb⇣ Õ�ˆ˝œ⌦G��b⇣U�ˆ˝œ↵M. ⇣'

¶5.88⇥ 10�6eVÑ˝ßà¬, Ÿ:Öæ∆”Ñ.

6.4.1 œœœPPPKKKœœœÑÑÑBell˙

Ê��Õ˙‚ È/⇢

1p2(↵1�2 + �1↵2) !

+,1p2(↵1�2 � �1↵2) !

�

1p2(↵1↵2 + �1�2) ! �

+,1p2(↵1↵2 � �1�2) ! �

�

Â⌦4*˙1Ñ⇣Bell˙�÷Ï˝/“† �”(entangled), ÉÏ/Sz(1)Sz(2)�Sx(1)Sx(2)Ñq�,Å�.

† ��˝ô⇣$*ÍÀ�Ñ“Ù•XÔ”. ↵bŸ*�/Ù•XÔ�⇢

(c1↵(1) + c2�(1))(c01↵(2) + c

02�(2))

= c1c01↵(1)↵(2) + c2c

01�(1)↵(2) + c1c

02↵(1)�(2) + c2c

02�(1)�(2)

�∫�H Èc1, c2, c01, c

02˝�˝h:

+,

�,�

+,�

�Ÿ7Ñ�.

SchrodingerÑ“+”⇢| i = 1p2(|+ie|;icat � |�ie|{icat), +Ñ{;÷≥é˘5PÍÀÑ“¬fl”�“

Kœ”, 1/xãц �.

6.5 $$$***“““®®®œœœÑÑÑ&&&������C-G˚ppp

KM⌘ÏvÜhS“®œ�ÍÀ“®œÑ&��$*ÍÀ“®œÑ&��⌘ÏÔÂÓ˚✏$*“®

œJ1,J2Ñ&�ƒã/�7Ñ�

æJ1 + J2 = J:;“®œ. π◆¡�

[Jx, Jy] = i~Jz (6.80)

s;“®œ·≥“®œ˘◆s˚.

^&�ha˙‚:|j1m1i|j2m2i. æj1, j2:$*“®œÑœPp�⇡HilbertzÙÑÙ¶:(2j1 + 1)(2j2 +

1).

&�haÑ˙‚îÂ/J21 , J

22 , J

2, Jz Ñq�,Å��∞:|j1j2jmji. ·≥

J21 |j1j2jmji = j1(j1 + 1)~2|j1j2jmji (6.81)

J22 |j1j2jmji = j2(j2 + 1)~2|j1j2jmji (6.82)

J2|j1j2jmji = j(j + 1)~2|j1j2jmji (6.83)

Jz|j1j2jmji = mj~|j1j2jmji (6.84)

∞(ÑÓòˆ⌃$*haÑ˙‚Ñs˚~0. ⌘ÏGæ

|j1j2jmji =X

m1,m2

Cj,mj

m1,m2|j1m1i|j2m2i (6.85)

1éJz = Jz(1) + Jz(2),

Jz|j1j2jmji =X

m1,m2

(m1 +m2)~Cj,mj

m1,m2|j1m1i|j2m2i (6.86)

112 CHAPTER 6. “®œ&�

ÇúBåÍ⇧+m1 +m2 = mjÑy:

|j1j2jmji =X

m1+m2=mj

Cj,mj

m1,m2|j1m1i|j2m2i (6.87)

1Ô·≥mj = m1 +m2, v/JzÑ,Å�.

d�Ÿöj1, j2ÔÂó0Ñjv�/�. ‘ÇhSåÍÀ“®œ&�ˆj = l + 1/2�_ÔÂ/j = l � 1/2.

1ém1�⇢:j1, m2�⇢:j2, @Âmj�⇢ÔÂ/j1 + j2, ŸÙ�j�⇢:

j1 + j2 = jmax. (6.88)

⌘ÏÔÂ⇢«‘É&�ha�^&�haÑHilbertzÙÙ¶e~0jÑ�✏÷<.

jmaxX

j=jmin

(2j + 1) = (2j1 + 1)(2j2 + 1) (6.89)

v-jmaxX

j=jmin

(2j + 1) = (jmax + jmin + 1)(jmax � jmin + 1). (6.90)

⌘Ïó0

j2min

= (j1 � j2)2 (6.91)

@Âjmin = |j1 � j2|. Ÿ_ÔÂ( “b πs˚ba0h:.

(|j1 � j2| j j1 + j2ŸöÑ≈µ↵�mjÔÂ÷�j,�j + 1, . . . , j. Ÿöj,mj , Cj,mj

m1,m2/nöÑ�

:C-G˚p. ÔÂ⌃„:$*ha˙‚ÑÖÔ

Cj,mj

m1,m2= hj2,m2|hj1,m1|j1j2jmji (6.92)

h6.1,6.2Ÿ˙Ü$Õ≈µ.

Table 6.1: ã1: ÍÀåhS“®œ&�. j1 = l, j2 = 1/2;mj = m1 +m2.

m2 = 1/2 m2 = �1/2

j = j1 + 1/2,mj

qj1+mj+1/2

2j1+1

qj1�mj+1/2

2j1+1

j = j1 � 1/2,mj �q

j1�mj+1/22j1+1

qj1+mj+1/2

2j1+1

Table 6.2: ã2: “®œ&�:j2 = 1.m2 = 1, 0,�1. m1 = mj �m2

m2 = 1 m2 = 0 m2 = �1

j = j1 + 1,mj

q(j1+mj)(j1+mj+1)

(2j1+1)(2j1+2)

q(j1�mj+1)(j1+mj+1)

(j1+1)(2j1+1)

q(j1�mj)(j1�mj+1)

(2j1+1)(2j1+2)

j = j1,mj -q

(j1+mj)(j1�mj+1)2j1(j1+1)

qmj

j1(j1+1)

q(j1�mj)(j1+mj+1)

2j1(j1+1)

j = j1 � 1,mj

q(j1�mj)(j1�mj+1)

2j1(2j1+1) �q

(j1�mj)(j1+mj)j1(2j1+1)

q(j1+mj+1)(j1+mj)

2j1(2j1+1)

6.6. EPRo,�BELL�I✏ 113

6.6 EPRooo,,,���Bell���III✏✏✏

Bohm⌃Ûûå.

⇤Q�*-'⇡0ÀPÑpÿ

⇡0 ! e

� + e+. (6.93)

1éÀPÍÀ:ˆ�@Â5Påc5Pb⇣U��fi⌘$π�Gæ5Pfi⌘Êπ, ∞:íP1; c5P⌘Û,

∞:íP2 :1p2(|+i1|�i2 � |�i1|+i2). (6.94)

⌘Ï)($*SG≈nKœ$*5PÑÍÀz⌃œ.

UÏ↵$πÑKœ”ú

hSz(1)i =1

2

~2+

1

2(�~

2) = 0 (6.95)

hSz(2)i =1

2

~2+

1

2(�~

2) = 0 (6.96)

ÙÙ•0⇤Q⇢|+i1|�i2/Sz(1)Ñ,Å��,Å<:~/2; |�i1|+i2_/Sz(1)Ñ,Å��,Å</�~/2; ‡á⌅:1/2. @ÂsGÕ:0.

F/Çúä$πÑ”úXwe�1/Ù˚÷Sz(1)Sz(2), v°óvfl°sG�_1/œPõf��<

hSz(1)Sz(2)i =1p2(2h�|1h+|�2 h+|1h�|)Sz(1)Sz(2)

1p2(|+i1|�i2 � |�i1|+i2) = �~2

4(6.97)

ÙÙ•0⇤Q⇢|+i1|�i2/Sz(1)Sz(2)Ñ,Å��,Å<:�~2/4; |�i1|+i2_/Sz(1)Sz(2)Ñ,Å��,Å

<ÿ/�~2/4; ‡á⌅:1/2. @ÂsGÕ:�~2/4.}6Sz(1), Sz(2)÷±~

2чá˝/12 , F/Sz(1)÷

~2�Sz(2)�ö÷�~

2�ÕK¶Õ.

$*5PKÙ sT�Ÿ:œœœPPPsssTTT���

9nœPõf�π◆¡�

h(~�1 · ~n1)(~�2 · ~n2)i = �~n1 · ~n2 (6.98)

∞:P (~n1,~n2), v-~n1�~n2:$SG1, SG2ÑKœKœπ⌘.

€�e�©⌘ÏæÛ⇢(Êπå⇣ܢ5PÑSzKœ�:⌘⌦�£HdK�˘Ûπc5PKœSz”ú1

8‹:⌘↵. Ÿ/1é‚˝pîÂL)Sz(1)Ñ,Å<:~/2Ñ,Å�; 1é$*íP⌅é(6.94), @ÂûEL

)0

|+i1|�i2. (6.99)

⌘ÏÔÂænSG1, SG2(^8‹Ñ0π�›¡Kœ—�(�ˆ�ÔÓ�t · c < �s, c:I���s:›ª).

¯˘∫Ì��$π/“{z”Ñ�î° ‡ús˚�Ô/‚˝pÑL)9ÿÜ{z›ªK�ÑíPÑ∂

�. ‡d�<N�¯˘∫€˛. Ÿ1/Einstein, Podolsky, Rosen–˙ÑEPRo,, ˙À(˚UqÕÑ ≠�

¶�˝'éI�Ñ˙@⌦. ÷Ï€�§:œœœPPPõõõfff///‘‘‘���ååå⌥⌥⌥’’’ÑÑÑ. ‚˝pv�/@ �:::ÜÜÜåååhhhœœœ˚

flflflÑÑÑ∂∂∂���������ÅÅÅ–––***ùùù���ÑÑѬ¬¬œœœ�, :::êêêÿÿÿœœœ.

êÿœ⌃∫/&cn�Ù°’å¡. Ù01964t�J. Bell¡�Üêÿœ⌃∫ÅBKœ”ú≈{uŒBell�

I✏��œPõf⌃∫° Ÿ*ÅB. Ÿ:’åå¡–õÜ˙@.

Bell˙ÆKœP (~n1,~n2), s$*ÍÀXÔÑsG<. êÿœ⌃∫ÅB”ú≈{·≥

|P (~n1,~n2)� P (~n1,~n3)| 1 + P (~n2,~n3), (6.100)

Ÿ1/WWW���ÑÑÑBell���III✏✏✏.

œP⌃∫�>�·≥Ÿ��I✏: ÷~n1,~n2,~n3(�*sbÖ, ~n1,~n2⇣⇡/2, ~n3�ÉÏ⇣⇡/4. é/9nœP

õf”ú(6.98)

P (~n1,~n2) = 0, (6.101)

114 CHAPTER 6. “®œ&�

P (~n1,~n3) = � 1p2= P (~n2,~n3) (6.102)

Bell�I✏:p22 1�

p22 , �⇣À�

6.6.1 Bell���III✏✏✏ÑÑÑ¡¡¡���

êÿœ⌃∫§:‚˝p° åhœ˚fl�:Üåhœ˚flÑ∂���Å–*ù�Ѭœ�.

Gæåh(complete)Ñ�1�Ÿ˙, �(œ*⇡ÀPpÿˆ˝��7�⌘Ï‚�⌃„_°’ß6É. b�

K�$*íPÑ��/nöÑ(6.94).

€�e�⌘ÏGæÊπ5PÑKœ�Ûπc5PÑ“¶~n0åh‡s. �ÔÂ(⇢ÅKœ5PKM�1c

5P�Ôûå⇧æn~n0, �ó° ·oÔÂ æ05P�Ô .

ûå”ú�öÔÂô⇣–Õ˝p

A(~n,�) = ±1, (6.103)

�⌃�˘éÛπÑ’å”ú��öÔÂô⇣Ê�*˝p

B(~n0,�) = ±1 (6.104)

Ë✏�/œt*∂�Ñ. ⌘Ï�ˆÂSûå”úÅBA,BÕsL�‡d˘@ Ñ�, ·≥

A(~n1,�) = �B(~n1,�) (6.105)

FÅπ/: ŸŸŸ���///⇢⇢⇢«««‚‚‚˝pppLLL)))eee›››¡¡¡ÑÑÑ������///111KKKMMM⇡ÀÀÀPPPpppÿÿÿˆÑÑÑÿÿÿœœœ�›››¡¡¡ÑÑÑ.

⌘Ïe°óKœÑsG<

P (~n1,~n2) =

Z⇢(�)A(~n1,�)B(~n2,�)d�, (6.106)

v-⇢(�) /œœœxxx‡‡‡ááá∆∆∆¶¶¶: ·≥R���^�'.

‡d

P (~n1,~n2) = �Z⇢(�)A(~n1,�)A(~n2,�)d� (6.107)

Ë✏Çú~n1 = ~n2, £H⌦✏:-1, /⌘Ï’˛)(Ñûåãû.

∞(⌘Ï⇤Q, *π⌘~n3, É/˚✏Ñ

P (~n1,~n2)� P (~n1,~n3) = �Z⇢(�)[A(~n1,�)A(~n2,�)�A(~n1,�)A(~n3,�)]d� (6.108)

1é(A(~n,�))2 = 1.�‡:Kœ<:±1

P (~n1,~n2)� P (~n1,~n3) = �Z⇢(�)[1�A(~n2,�)A(~n3,�)]A(~n1,�)A(~n2,�)d�

1é(6.103),

�1 A(~n1,�)A(~n2,�) 1, (6.109)

⇢(�)[1�A(~n2,�)A(~n3,�)] � 0 (6.110)

›˘<✏é

|P (~n1,~n2)� P (~n1,~n3)| Z⇢(�)[1�A(~n2,�)A(~n3,�)]d�

= 1�Z⇢(�)A(~n2,�)(�B(~n3,�))d�

= 1 + P (~n2,~n3)

Ÿ1/Bell�I✏Ü.

6.6. EPRo,�BELL�I✏ 115

⌘ÏÑ˙—π/(6.106),s$*π⌘ÑsT/⇢«ãHX(Ñêÿœ≥öÑ���/‚˝pÑL).

ÓM�Ú 'œûåD�œPõfÑ��›ÃBell�I✏�‚˝pѨÙL)/„ ûå@≈�Ñ.

F/‚˝pÑL)⇢�⇢&e‡ú'Ñ~æ�ûE⌦�*Õ\Êπ5PKœÑ∫�ÔÂ9nÍÒÑûå

∞UÂSÛπc5PÑûå∞U�F/t° ˚Uπ’ÔÂ)(÷ÑKœ˘Ûπc5PKœÑ∫—˙�*

˝�ß��úÑ·˜, ‡dv�›Ã¯˘∫. �˜'∂¬⇤Gri�thsf12.2Ç.

116 CHAPTER 6. “®œ&�

Chapter 7

hhh���íííPPPååå‚‚‚˝pppÑÑѧ§§bbb˘'''

7.1 ⇢⇢⇢íííPPP‚‚‚˝ppp

˘éUíP‚˝p (~r, t)��, | (~r, t)|2 /~r⌅íP˙∞чá∆¶. Çú⌘ÏvÑ˚flÃb $*íP�

£HS˚Ñ‚˝pîÂô: (~r1,~r2, t). | (~r1,~r2, t)|2/(~r1D—d~r1—∞íP1�(~r2 D—d~r2Ö—∞íP2Ñ

‡á∆¶. ·≥R�� Z| (~r1,~r2, t)|2d~r2d~r2 = 1 (7.1)

åS � eq:

i~@ (~r1,~r2, t)@t

= H (~r1,~r2, t) (7.2)

v-

H =p21

2m1+ V (~r1) +

p22

2m2+ V (~r2) + V (~r1,~r2) (7.3)

V (~r1), V (~r2)⌃+/$*íP(�:-Ñø˝, V (~r1,~r2)/íPKÙѯí\(ø˝⇥p1 åp2⌃+/í

P1å2Ñ®œó&⇢

p1 = �i~ @

@~r1, p2 = �i~ @

@~r2(7.4)

⌃ªÿœ„:

(~r1,~r2, t) = E(~r1,~r2)e�iEt

~ , (7.5)

v-zÙ‚˝p E·≥

(p21

2m1+ V (~r1) +

p22

2m2+ V (~r2) + V (~r1,~r2)) E(~r1,~r2) = E E(~r1,~r2) (7.6)

⌘Ïev�*ãP⇢�Ù‡PÒø1�Ωa - $*íP

H =p21

2m1+ V (x1) +

p22

2m2+ V (x2) + V (x1, x2). (7.7)

V (x1), V (x2)/1�V (x1, x2)/¯í\(�⌘Ï˝e¯í\(. ⇡

H = h1(x1) + h2(x2), (7.8)

h1, h2⌃+/$*íPUÏ(1-ÑHamiltonian�,Åπ↵(7.6)- E(x1, x2) Ôç⌃ªÿœ

E(x1, x2) = E1(x1) E2(x2). (7.9)

117

118 CHAPTER 7. h�íPå‚˝pѧb˘'

s

H E1(x1) E2(x2) = h1(x1) E1(x1) E2(x2) + h2(x2) E1(x1) E2(x2)

= E E1(x1) E2(x2)

$πd E1(x1) E2(x2),

h1(x1) E1(x1) = E1 E1(x1), (7.10)

h2(x2) E2(x2) = E2 E2(x2), (7.11)

E = E1 + E2. (7.12)

‚˝på˝ß⌘ÏÂS

E1(x1) =

r2

asin

n1⇡

ax1, E1 =

~22m1

n21⇡

2

a2

E2(x2) =

r2

asin

n2⇡

ax2, E2 =

~22m2

n22⇡

2

a2

E1(x1) E2(x2) ÔÄô: n1(x1) n2(x2) , �⇧)(Dirac∞˜�ô:|Ei = |n1i1|n2i2,

n1(x1) n2(x2) =2 hx2|1hx1|Ei. (7.13)

Ωa0Ù�ŸÕ≈µ1/

(~r1,~r2) = a(~r1) b(~r2) (7.14)

íP1⌅éa��íP2⌅éb�.

7.2 §§§bbb˘���ÕÕÕ˘������ÔÔÔ⌃⌃⌃®®®ÑÑÑhhh���íííPPP

Çú$*íP/��ÑíP⇢(œ�ÍÀ�5w-Û⌘�*���Â⌦®∫° Óò.

Çú$*íP/h�íP�Â⌦®∫ûûûEEE⌦⌦⌦:::⌃⌃⌃ÜÜÜÉÉÉÏÏÏ⇢íP1(n1��íP2(n2���Ÿ/�Ô˝

Ñ�° û’Ÿ$*íP“⌦r”�ƬíPÑyÅ“^8⌘”�

œœœPPPõõõfff˙,,,üüü⌃⌃⌃⇢⇢⇢hhh���íííPPP���ÔÔÔ⌃⌃⌃®®®!

‡d

| (~r1,~r2)|2 = | (~r2,~r1)|2 (7.15)

,�*Íÿœ:,�*íPÑMn�,å*Íÿœ/,å*íPÑMn: Êπ:íP1(~r1�íP2(~r2�Û

π:íP1(~r2�íP2(~r1. £H

c (~r1,~r2) = (~r2,~r1), (7.16)

v-|c| = 1.

�e§bó&P12:

P12 (~r1,~r2) = (~r2,~r1) = c (~r1,~r2) (7.17)

çe�!⇢

P12P12 (~r1,~r2) = c2 (~r1,~r2) (7.18)

F/P12P12 = 1, ‡:$!§bIé°b, @Â

c2 = 1, c $ÕÔ˝{

1 Bosons §b˘

�1 Fermions Õ˘

1/Ù�h�íP‚˝p≈{·≥

(~r1,~r2) = ± (~r2,~r1)

Ÿ/œPõfÑü⌃.

7.2. §b˘�Õ˘��Ô⌃®Ñh�íP 119

• @ ÍÀ:~tp�ÑãP:‚rP�⇧Ï��íP�üP�ÀP�IP

• @ ÍÀ:~Jtp�ÑíP:9sP�(P�5P�8K

• ÍÀ�§b˘'Ñs˚/¯˘∫Ñ”∫�⌘ϟÕ◊

fi0MbÑãP�Â$*íPh�⇢m1 = m2,(n1 6= n2Ñ≈µ↵�˝œ,Å�î:⇢

Bosons : E(x1, x2) =1p2[ n1(x1) n2(x2) + n2(x1) n1(x2)]

Fermions : E(x1, x2) =1p2[ n1(x1) n2(x2)� n2(x1) n1(x2)]

�⇧ô⇣

| Ei =1p2(|n1i1|n2i2 + |n2i1|n1i2),Boson (7.19)

| Ei =1p2(|n1i1|n2i2 � |n2i1|n1i2),Fermion (7.20)

✏�/⇢���***íííPPP⌅⌅⌅ééén1������ÊÊÊ������***⌅⌅⌅ééén2���.

Sn1 = n2ˆ�

Boson : E(x1, x2) = n1(x1) n2(x2), (7.21)

Fermion : E(x1, x2) = n1(x1) n2(x2)� n1(x2) n2(x1) = 0 (7.22)

$*ªrPÔÂ⌅é�*∂���$*9sP�L, Ÿ1/···)))���¯πππüüü⌃⌃⌃.

ÔÂ↵0, P12_Ô⌃„:$*íP§bÜ∂�.

1éíPh��»∆�ó&/§b�ÿÑ:

P12H(1, 2) = H(2, 1) = H(1, 2) (7.23)

‡d

[P12, H(1, 2)] (r1, r2) = P12(H(1, 2) (r1, r2))�H(1, 2)P12 (r1, r2)

= H(2, 1) (r2, r1)�H(1, 2) (r2, r1) = 0(7.24)

ŸÙ�P12�H(1, 2) q�,Å�, vû1/Mb~0Ñ˘�Õ˘b✏. �ŸÕ˘'�Õ˘'/àR

Ñ��èˆÙÿ�Ñ.

6�MbÑ®∫° ⇤QÍÀ∂�⇥Çú⇤QÍÀ∂�, íPÑ∂�îÂ/ n(~r)�ms, çÅB§b�‚˝

p˘�Õ˘.

‘Ç5P�(7.20)ÔÂô:

| Ei =1p2(|n1msi1|n2m

0si2 � |n2m

0si1|n1msi2) (7.25)

ô⇣‚˝p

E(x1, x2) = 2hx2|1hx1| Ei

=1p2( n1(x1)�ms

(1) n2(x2)�m0s(2)

� n2(x1)�m0s(1) n1(x2)�ms

(2)) (7.26)

120 CHAPTER 7. h�íPå‚˝pѧb˘'

Çú⌘ÏäœPp(n,ms)fl�∞:k, £H⌦✏ÔÂô:

| Ei =1p2(|ki1|k0i2 � |k0i1|ki2) (7.27)

�⇧Äô:‚˝p

E(1, 2) =1p2( k(1) k0(2)� k0(1) k(2)) (7.28)

Sn1 = n2ˆ, Çúms1 = 1/2 6= ms2 = �1/2, ˚fl∂�/

| Ei =1p2(|n1, 1/2i1|n1,�1/2i2 � |n1,�1/2i1|n1, 1/2i2) (7.29)

ô⇣‚˝p

E(x1, x2) =2 hx2|1hx1| Ei = n1(x1) n1(x2)[� 12(1)�� 1

2(2)� �� 1

2(1)� 1

2(2)] (7.30)

⌘Ï↵0dˆzÙ‚˝p§b˘�ÍÀ‚˝pÕ˘�t*‚˝p·≥§bÕ˘⇥vû(n1 6= n2ˆ_

ÔÂÑ Ÿ7Ñ‚˝p�‘Ç:

1p2[ n1(x1) n2(x2) + n2(x1) n1(x2)]�00 (7.31)

É/(n1, 1/2;n2,�1/2)�(n2, 1/2;n1,�1/2)ч†�⇥

�⌃�(n1 6= n2ˆ�§bÕ˘ÿÔÂ/zÙ‚˝pÕ˘�ÍÀ˘.

1p2[ n1(x1) n2(x2)� n2(x1) n1(x2)]

8>>><

>>>:

�11

�10

�1�1

(7.32)

7.2.1 ���,,,'''®®®∫∫∫

⌘Ï∞(⇤Q˚✏$*‡¯í\(h�íPÑ⇣Ñ˚fl: H = h(1)+ h(2), h(1)�h(2)b✏¯��v,Åπ↵

:

h'k = ✏k'k, (7.33)

k:�ƒ}œPp�>6

H'k1(1)'k2(2) = (✏k1 + ✏k2)'k1(1)'k2(2) (7.34)

9nMbÑ®∫�˚flÑ˝œ,Å�îÂ/˘�Õ˘Ñ.

• Bosons:

(S)k1k2

= {1p2['k1(~r1)'k2(~r2) + 'k2(~r1)'k1(~r2)], k1 6= k2

'k1(~r1)'k1(~r2) k1 = k2

• Fermions

(A)k1k2

=1p2['k1(~r1)'k2(~r2)� 'k2(~r1)'k1(~r2)]

=1p2

�����'k1(~r1) 'k1(~r2)

'k2(~r1) 'k2(~r2)

�����

≈{k1 6= k2. pauli�¯πü⌃�

7.2. §b˘�Õ˘��Ô⌃®Ñh�íP 121

Ô®�0N*íPÑ≈µ

Pij (x1, · · · , xi, · · · , xj , · · · , xN )

= (x1, · · · , xj , · · · , xi, · · · , xN )

= ± (x1, · · · , xi, · · · , xj , · · · , xN )

ãÇ�3*Bosons⌅é��Ñk1, k2, k3�⇢

(s)k1k2k3

=1p3!( k1(1) k2(2) k3(3) + k2(1) k1(2) k3(3) + · · · )

3!/k1, k2, k3Ñ@ ��í⌫ÑpÓ.

$*⌅ék1, �*⌅ék2 6= k1:

(s)k1k1k3

=

p2!p3![ k1(1) k1(2) k2(3) + k2(1) k1(2) k1(3) + k1(1) k2(2) k1(3)]

2!/¯�Ñ$*�k1, k2Ñí⌫p.

*˝⌅é��*k1:

(s)k1k1k1

= k1(1) k1(2) k1(3)

;Ñyp/hí⌫pd¯��KÙÑí⌫p⇥‡dR��‡P :yp⌃K�ç�π.

3*Fermion≈{(��Ñ�

(A)k1k2k3

=1p3!

�������

k1(1) k1(2) k1(3)

k2(1) k2(2) k2(3)

k3(1) k3(2) k3(3)

�������

122 CHAPTER 7. h�íPå‚˝pѧb˘'

7.3 œœœPPPflflfl°°°

⇤Q�*˝ß✏/ ÕÄvÑ⇢k1, k2, k3, Dz7.1@:, $*íP`n�E = 2✏), Ô ‡Õ��Ñ∂��

Figure 7.1: �* ÕÄvÑ˝ß.

˛7.2Ÿ˙ÜíP/ªrP�9sP�⇧Ô⌃® Õ≈µ↵ÑƬ∂�p.

Figure 7.2: $*íP`n ÕÄvÑ˝ß˘îÑœP�.

Çú/N*íP`nfÕÄvÑ˝ß�⇢ ⇢⌘Õ≈µ�

Çú�*˝ßÑÄv¶✏é˚flÑíPp�dˆ9sP⌃�ó�˚~v÷˝ßª`n⇥

⌘Ï↵0‚˝pѧb˘'ÔÂP6�*˚fl@A∏ÑƬ∂�pÓ�S6�⌘ϟî∫Ñ/°

¯í\(Ñ⇢íP˚fl,(Ÿö;˝œÑ≈µ. ŸÕP6ÔÂ%Õ9ÿ�*è¬i⌃˚flÑ'(�œPõf

�/Í Æ¬Hî�