60 CHAPTER 3. œPõfü⌃

3.10 õõõfffœœœóóó&&&ÑÑÑqqq���,,,ÅÅÅ˝ppp

%<ÑÙ�⌘ÏÂSõfœÑ^é��,Å<Ñ,Å‚/c§R�Ñ⇥F/⌘Ïv° ‘∆vÄv(��

*,Å< $*Â⌦,Å‚)≈µ⇥Çú˙∞Äv��H›¡,Å‚c§b�)($*�⇢*õfœÑq�

,Å˝p�⌘ÏÔ„≥Äv&eÑÓò⇥

GæA�B/$*õfœÑó&. X(�ÕÔ˝: ÉÏw q�Ñ,Å˝p

A| nli = an| nli, B| nli = bl| nli. (3.219)

| nli\A, BÑq�,Å�. ÇúíP⌅éŸ7Ñ∂�, KœAóan, KœBóbl. KœK�Ñ∂��ÿ. HK

�K‡@◆.

(AB � BA)| nli = (anbl � blan)| nli = 0. (3.220)

ËËË✏✏✏: A, B˘◆◆◆///(((⌦⌦⌦✏✏✏˘˚✏✏✏‚‚‚˝ppp˝⇣⇣⇣ÀÀÀÑÑÑ≈≈≈µµµ↵↵↵. Ÿ�≥ÂÙ÷Ï˘◆�F/Ÿ✏s@A�BàààÔÔÔ˝

///˘◆◆◆ÑÑÑ.

Ÿ/Ô¡�Ñ�M–/| nliÑ⇣å⌥˝p˚, ˚U�˝ÔÂ(ÉU�

| i =X

nl

Cnl| nli, (3.221)

é/

(AB � BA)| i =X

nl

Cnl(AB � BA)| nli =X

nl

Cnl(anbl � blan)| nli = 0. (3.222)

‡d, [A, B] = 0.

Õ«e, ⌘ÏÔ¡�: ÇÇÇúúúÑÑÑ∆∆∆óóó&&&A, B˘◆◆◆, ⇡⇡⇡ÉÉÉÏÏÏ≈≈≈öööXXX(((���˚⌫⌫⌫qqq���,,,ÅÅÅ˝ppp.

¡�: Gæ| ni, n = 1, 2, · · · ,�|�li, l = 1, 2, · · · ,⌃+/ó&A, BÑ�ƒ,Å�, 1éA, B ˝/õfœ�

@Ÿ$ƒ,Å�⌅⌅⌅ÍÍÍ˝/å⌥Ñ.

A| ni = an| ni, n = 1, 2, 3, · · · ,

B|�li = bl|�li, l = 1, 2, 3, · · · .(3.223)

⇤Q0Ô˝ÑÄv, ⌘Ï⌃|�li ,Å<⌃ƒ(bi 6= bj , Çúi 6= j)

B|�i⌫i = bi|�i⌫i, i = 1, 2, 3, · · · ; ⌫ = 1, · · · , fi. (3.224)

fi1/biÑÄv¶⇥

˚✏÷�*AÑ,Å˝p| ni, U�⇣|�i⌫iч†

| ni =X

i⌫

c(n)i⌫

|�i⌫i =X

i

|�nii (3.225)

v-|�nii =P

⌫c(n)i⌫

|�i⌫i, Õ6/BÑ,Å˝p, ^é,Å<bi. ⇤Q

(A� an)| ni =X

i

(A� an)|�nii = 0. (3.226)

1éA�B˘◆, ⌘Ï

B(A� an)|�nii = (A� an)B|�nii = bi(A� an)|�nii. (3.227)

Çú(A� an)|�nii�:ˆ, É1/BÑ,Å�, ^é,Å<bi. F/(3.226)h�Ÿõ�/ø'¯sÑ. �*

Ñ∆ó&Ñ^é��,Å<Ñ,Å˝p/c§Ñ, �Ô˝ø'¯s. @Â

(A� an)|�nii = 0. (3.228)

3.10. õfœó&Ñq�,Å˝p 61

_1/Ù|�nii/AÑ,Å�, ^é,Å<an.

Ûd�⌘Ï¡�Ü|�nii/A,BÑq�,Å�.

1éEq.(3.225)˘˚✏�*AÑ,Å�˝⇣À�sœ�*,Å�˝ÔÂh::A,Bq�,Å�ч†, �

| ni/å⌥Ñ�@Â|�nii___///ååå⌥⌥⌥ÑÑÑ.

�ˆ�⌘Ï ú0—∞|�nii, i = 1, 2, · · · ,˝^éan, �|dc§Ñ, ‡:÷Ï^éBÑ��,Å<bi.

Ÿ�ó˙Àå⌥Ñc§R�˙‚⇣:Ô˝⇥

3.10.1 ÄÄÄvvv���õõõfffœœœåååhhh∆∆∆

• ÇúAÑhË,Å�(�,Å˝p)˝/^ÄvÑ, £HKœAó0Ñ,Å<1ÔÂnö˚flL)0Ñ,

�|ai.

Ÿˆ_Ô˝ ó&B�K˘◆. F/1é^Äv, AÑœ*,Å�˝/BÑ,Å�. vûB≈ö/AÑ˝

p. ãÇ: �Ù˚flÑ®˝T�®œp.

• ÇúAÑ�Ë⌃�hË,Å</ÄvÑ, £HŸöAÑ,Å<v�≥Ânö,Å�. ⌘ÏÔÂ~�A˘

◆ÑÊ�õfœó&B, ÉÏX(q�,Å�, (nöan�, çKB, nöbl�, 1ÔÔÔ˝nö,Å�Ü, ‘

Ç|an, bli.

• ÇúA, BÑq�,Å�(nöan, bl�Õ6 Äv, sÕ6‡’nö,Å�. Ÿˆ⌘Ï�Å~, *õ

fœC.

• �ƒ¯í˘◆»|dÏÀÑõfœ, ÇúÉÏÑq�,Å��çÄv(/�nö), £HŸƒõfœ:

˚flÑõõõfffœœœåååhhh∆∆∆. KœÉÏK�ó0Ñ@ ,Å<1ÔÂåhnö˚flÑ∂�(q�,Å�).

• åh∆-ÑõfœÑ*pvû1/˚flÑÍÍÍ111¶¶¶.

ãP: ˘é�Ù_⇢�íP–®, �,˝œ1ÔÂÑ⇣õfœåh∆. ˘é^_⇢�Ô ®œ:õf

œåh∆. ÙíPÑ–®�Å *õfœÑ⇣åh∆. ˘é^_⇢�Ô px, py, pz. ˘é-√õ_

⇢��Ô ˝œ, “®œsπå“®œ–*⌃œ. Ÿ(�bÑÖπÃ⇢≤0.

ü⇡⌦Çú~0Üõfœåh∆�1ÔÂÂÉÏÑq�,Å�:˙‚˙Àha, Ÿõ˙‚/c§R�

Ñ⇥

ãÇ⇢õfœA, BÑq�,Å�/|lmi, A|lmi = al|lmi, B|lmi = bm|lmi. nöal, bm1ÔÂnö///���

ÑÑÑ q�,Å�|lmi. ⇡A�BÑ⇣õfœåh∆. ⌘ÏÔ«(|lmi:˙‚˙À(A,B)ha. 1éhlm|l0m0i =�ll0�mm0 , ˙‚/c§R�Ñ. ⌘ÏÔŸŸõ˙‚ �öƒ⇡íè|lmi = |ii, i = 1, 2, . . . ,, 1ÔÂfi0⌘Ï

üâÑc§R�b✏hi|ji = �ij .

˚✏∂�↵iÔÂ(|lmiU�.

|↵i =X

lm

clm|lmi (3.229)

˚pclm = hlm|↵i//�nöÑ⇥9nœPõfÑKœü⌃�Kœó0,Å<�˚flÑ∂�L)0˘îÑ,Å�. Çú,Å�Äv�˚fl

L)0¿H∂��

Gæ˚flÑ∂�/

|↵i = c00|00i+ c10|10i+ c11|11i (3.230)

ÇúKœAó0a0 (‡á:|c00|2, c00 = h00|↵i), ˚fl⇢L)0AÑ,Å�|00i.ÇúKœAó0a1, £H˚fl⇢L)0AÑ,Å�|�i = (c10|10i+ c11|11i)/N , v-⌘Ï�eR��‡P

1/N =1p

|c10|2 + |c11|2(3.231)

62 CHAPTER 3. œPõfü⌃

›¡h�|�i = 1. ⌘ÏÔÂ9ôEq.(3.230)

|↵i = c� |�i+ c00|00i (3.232)

°ó˙c� = h�|↵i = N . ÉÑ!π|c10|2 + |c11|21/Kóa1чá.

ÇúKœAó0a1, •@KœB, ó0b0, ⇡˚fl|�i ≈•@L)0BÑ,Å�: |10i. Ÿ*�//�nöÑ,

—�‡á:|h10|�i|2 = |c10|2/N2.

;S��KóA:a1vKB:b0чá:|c10|2. �7�KóA:a1vKB:b1чá:|c11|2.ûE⌦�KAóa1чá|c10|2 + |c11|2/KAóa1vKBób0�ób1чáKå.

3.11. hS“®œ 63

3.11 hhhSSS“““®®®œœœ

3.11.1 ˘◆◆◆sss˚

hS“®œÑöI/

L = r⇥ p (3.233)

⇤Q˚✏∂�|↵i, v‚˝på´L\(��Ñ‚˝pÑs˚:

hr|L|↵i = Lhr|↵i (3.234)

v-L:

L = r⇥ p = �i~r⇥r (3.235)

(Ù“P⌥˚↵, Lx = ypz � zpy

π◆¡�Â↵˘◆s˚:

[Lx, x] = 0, [Lx, px] = 0 (3.236)

 vnll✏.

[Lx, y] = i~z, [Lx, py] = i~pz (3.237)

 vnll✏.

↵b¡�$*ÕÅÑ˘◆s˚.

1. [Lx, Ly] = i~Lz.  vnll✏.

¡�: Œe

2. [L2, Lz] = 0  vnll✏.

¡�: Œe

3.11.2 L2, LzÑÑÑqqq���,,,ÅÅÅ���

“®œ/(-√õÓòÃ8G0Ñõfœ. (-√õ:≈µ↵, «(⇤P⌥/πøÑ.

r = er@

@r+ e✓

@

r@✓+ e�

@

r sin ✓@�(3.238)

Ô¸˙

L2 = �~2( 1

sin ✓

@

@✓sin ✓

@

@✓+

1

sin2 ✓

@2

@'2)

Lz = ee · L = �i~ @

@'

(3.239)

(Â⌦l✏Ñ�*à}Ñ®¸�(±/�⌦œPõf↵DU

æ÷ÏÑq�,Å˝p:Y (✓,').

L2Y (✓,') = �1~2Y (✓,')

LzY (✓,') = �2~Y (✓,')(3.240)

⌃ªÿœ: Y = ⇥(✓)�(').

Lz�(') = �2~�(') (3.241)

�(') = Cei�2'. (3.242)

64 CHAPTER 3. œPõfü⌃

9nÍ6πLaˆ: �('+ 2⇡) = �('), ⌘ÏÂS

�2 = 0,±1,±2, · · · . (3.243)

R��ÅBC = 1p2⇡

. ⇢8ô⇣

�m(') =1p2⇡

eim'

,m = 0,±1,±2, · · · . (3.244)

L2Ñ,Åπ↵ÔÂ�⇣

1

sin ✓

d

d✓sin ✓

d

d✓⇥+ (�1 �

m2

sin2 ✓)⇥ = 0 (3.245)

öI⇠ = cos ✓, �⇣fi&Legendre π↵

d

d⇠((1� ⇠

2)d

d⇠⇥) + (�1 �

m2

1� ⇠2)⇥ = 0. (3.246)

i⌃„:fi&Legendre˝p

⇥ = Pm

l(cos ✓),v-|m| l. (3.247)

˘îÑ,Å<:

�1 = l(l + 1), l = 0, 1, 2, · · · , (3.248)

¸�we, q�,Å˝pô:⇤⇣˝p

Yl,m(✓,') = Nl,mPm

l(cos ✓)eim' (3.249)

˚pNl,m›¡˝pÑc§R�':

(Yl,m, Yl0,m0) = �l,l0�m,m0 . (3.250)

Yl,m/HilbertzÙ‚œ|lmi˘îÑ‚˝p⇥Â⌦s˚_ÔÂô⇣

hlm|l0m0i = �l,l0�m,m0 (3.251)

˚✏Â(✓,�):ÍÿœÑ˝p˝ÔÂ(YlmU��_1/Ùvå⌥

X

l

m=lX

m=�l

|lmihlm| = 1 (3.252)

Äv®∫: l:hS“®œœPp, m:¡œPp(�bÖπ). Ÿöl, m 2l+1ÕÔ˝÷<. �⇧ÙÄv¶

:2l + 1.

‡*y+ÕÅÑ⇤⇣˝p⇢Y00 = 1p4⇡

, Y10 =q

34⇡ cos ✓, Y1± = ⌥

q38⇡ sin ✓e±i�

• ãP: ⇤Q�*´P6(JÑ:aÑ⇤b⌦–®Ñ(œ:µÑíP.

˚flÑHamiltonian:

H =L2

2µa2. (3.253)

˝œ,Åπ↵:

H (✓,�) = E (✓,�). (3.254)

π◆↵˙P

mCmYl,m,m = �l, · · · , l/π↵Ñ„, v-Cm/˚✏8p; ˘î,Å<El = l(l + 1)~2/2µa2.

(Ÿ*Óò-ÂS˝œ,Å<�≥Âåhnö,Å˝p,‡:,Å˝p Äv. ⌘Ï�ÅÊ��*õf

œLz�HÑ⇣õfœåh∆. Çú⌘ÏÂS˝œ:El, vlz:m~, s

Lz (✓,�) = m~ (✓,�). (3.255)

3.11. hS“®œ 65

1ÔÂåhnö‚˝p:Ylm.

Gæ˚flÑ∂�/

|↵i = 1p2|00i+ 1p

3|10i+ 1p

6|11i (3.256)

⌘ÏHKœH, ó00 чá:1/2, ˚fl⇢L)0,Å�|00i. •@KLz, ó0Lz = 0, v‡á/1.

ÇúKœHó0 ~2

µa2 , £H˚fl⇢L)0¿H∂��îÂ/1p3|10i+ 1p

6|11i.

⌃vR���ó0p2( 1p

3|10i+ 1p

6|11i). ‡d

|↵i = 1p2|00i+ 1p

2(

p2p3|10i+

p2p6|11i) (3.257)

_1/Ù˙∞Ÿ*K<чá/1/2. dˆçKœLz�ÔÂó00�⇧~, ‡á⌃+:2/3 å1/3, ⌃+ïq

0|10iå|11i. Ë✏�KœŸõ�Ñ˝œ⌘ÏÕ6ó0 ~2

µa2 . Ÿ/1é[H,Lz] = 0.

66 CHAPTER 3. œPõfü⌃

3.12 KKK���∆∆∆sss˚

⌘ÏÂSÇú$*õfœA,BÑó&˘◆, £HÉÏ �˚⌫q�,Å�. ˚flÇú⌅éŸ7Ñ∂�, ⌘ÏÔ

Â�ˆ∆nÂSA,BÑK<. ‘Ç|a, bi/A,BÑq�,Å�⇥

A|a, bi = a|a, bi; B|a, bi = b|a, bi (3.258)

£HKA�⌘Ï~⌃K~ó0a; ˚flÑ�1é/AÑ,Å��Kœ��L)⇥çKB�⌘Ï~⌃K~ó0b.

†✓zè�≈µ_�7⇥‘Ç(Ylm�↵KœL2, Lz.

F/, ÇúA,BÑó&�˘◆. ≈µ⇢�H7? ”∫/���,,,���˝���ˆKKK∆∆∆. ÉÏÑ�nö¶·≥Ç↵s˚:

�A ·�B � 1

2|h[A, B]i| (3.259)

v-�nö¶:�A = h(A� hAi)2i1/2, /(íPÑ–®∂�(‚˝p)↵°óÑ.

¡�:

æ|↵i/˚✏�*�‚œ. (A+ i⇠B)|↵i/�*Û‚��v˘vÑÊ‚:h↵|(A+ i⇠B)† = h↵|(A� i⇠B). Ÿ

Ã)(ÜA,B/Ñ∆ó&. 1é�*Û‚�˘vÊ‚ÑÖÔ^��@ÂÔÂöI

f(⇠) = h↵|(A� i⇠B) · (A+ i⇠B)|↵i � 0 (3.260)

vû1/

f(⇠) = h↵|(A2 + i⇠(AB �BA) +B2⇠2)|↵i

= hA2i+ (ih[A,B]i)⇠ + hB2i⇠2(3.261)

ÔÂ↵0h[A,B]i≈:ØZp, &⇡ih[A,B]i �/ûp, f(⇠)_1�/ûpÜ. (Ÿ/1éi[A,B] /Ñ∆ó&.

¬¡KM\⇢ò.)

9na⇠2 + b⇠ + c � 0⇣ÀÑaˆ: 4ac � b

2, ⌘Ï

4hA2ihB2i � (ih[A,B]i)2 (3.262)

ÂA� hAi„ˇA, B � hBi„ˇB,

4h(A� hAi)2ih(B � hBi)2i � (ih[A� hAi, B � hBi]i)2 (3.263)

_1/

�A ·�B � 1

2|h[A,B]i| (3.264)

wS0, ‘ÇA = x, B = px, ⌦✏1/⌘ÏüâÑ

�x ·�px � 1

2|h[x, px]i| =

~2. (3.265)

�*ÄUÑãP⇢Çú˚fl⌅éYlmÑ,Å�⇥⌘ÏÂS

�Lx = �Ly =p

(l(l + 1)�m2)~2/2

. vhLzi = m~. 1é|m| l, >6·≥⌦s˚⇥

⇤QüP-Ñ5P. v;®⇤Ù P, xÑœß:10�10s. v®˝Ñ�*⌃œ'¶hp2x/2mi ' 10eV/3. ⇤

Q05P⌅é_⇢�, ⌘ÏÂShpxi = 0, @Â�px ' 2⇥ 103eV/c. (5P(œ:0.5⇥ 106eV/c2) GæMnÑ

�nö¶1/xÑœß, 9n�x�p ' 10�15eV s ' ~. ·≥K�∆s˚.

¯Õ, Çú5P(è¬:¶⌦–®, ‘Ç(òs'✏Ñ“PÖ–®. Gæ⌘ÏÔÂ(Æs:¶⌦nö÷Ñ

Mn,s�x = 10�6m, 9nK�∆s˚, ®œÑ�ÿæ¶/7⇥ 10�10

eV s/m. �®˝:1eVÑ5PÑ®œ'✏

:3⇥ 10�5eV s/m, ÔÂ↵0®œÑ¯˘æ¶à}.