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�