3.14. GMó&’ 71

3.14 GGGMMMóóó&&&’’’

3.14.1 BBB„„„���ÙÙÙ⇣⇣⇣///PPPÓÓÓòòò

KM⌘Ï(P⌥haB„Ü�Ù⇣/PÑö�π↵�Æ⌃π↵

H n(x) = En n(x) (3.305)

∞(⌘ÏÀÕ„pπ’⇥

�e

Q = ↵x =

rm!

~ x,

P =p

~↵ =ppm~!

(3.306)

P�Q·≥

[Q,P ] = i, (3.307)

ŸÃ)(Ü1/↵/yÅ�¶�~↵/yÅ®œ⇥v

H =~!2(P 2 +Q

2) (3.308)

öIa ⌘ 1p2(Q+ iP ), a+ ⌘ 1p

2(Q� iP ). Ë✏a, a

+�/Ñ∆Ñ⇥

π◆ó0

[a, a+] = 1, (3.309)

a+a =

1

2(P 2 +Q

2 � 1). (3.310)

é/

H = (a+a+1

2)~!. (3.311)

Gæ⌘Ï�ÂSHÑ,Å</(n+ 12 )~!, _�ÂSHÑ,Å�/ n = Ane�

↵2x2

2 Hn(↵x). ⌘ÏeB„

H|ni = �n|ni (3.312)

1é[a,H] = [a, a+a]~! = a~!,

Ha|ni = (aH � ~!a)|ni = (�n � ~!)a|ni (3.313)

sa|ni_/HÑ,Å��^é,Å<�n � ~!. �⌃, �n � 2~!,�n � 3~!, · · · ,˝/,Å<, ˘î,Å�

/a2|ni, a3|ni, · · · ,.1é

h |H| i = h |a+a| i+ 1

2~! � 0 (3.314)

@Â,,,ÅÅÅ<<<���ööö ***���✏✏✏<<<�0, ˘̆̆îîîÑÑÑ,,,ÅÅÅ���|0i···≥≥≥a|0i = 0.

H|0i = ~!a+a|0i+ 1

2~!|0i = 1

2~!|0i. (3.315)

‡d�0 = 12~!. @ Ñ˝ß_nöÜ⇢�n = (n+ 1

2 )~!.)([a+, H] = �a

+~!, Ô®˙a+|ni/HÑ,Å��,Å<:�n + ~!.

a+|ni⇢�⇢Iéˆ��⇢�⌘ÏÕ¡K. Gæ˘é–*|ni, a+|ni = 0⇥1é

H = (aa+ � 1

2)~! (3.316)

72 CHAPTER 3. œPõfü⌃

@Â

H|ni = ~!aa+|ni � ~!|ni = �1

2~!|ni (3.317)

›ÃÜ(3.314): ˚✏∂�Ñ˝œ��<'éIéˆ. ‡dn = 0, 1, 2, · · · , ° ⌦P⇥Ë✏0

a+a|ni = n|ni (3.318)

@Âa+a/œPpó&⇥

a|ni/˝œ,Å��9nEq.(3.313), a|niI˜é|n� 1i(⇤Q0�Ù_⇢�° Äv.) F/v��ö/

R��Ñ,Å�⇥⌘Ïæ

a|ni = c(n)|n� 1i, (3.319)

£H

hn|a+ = c⇤(n)hn� 1|. (3.320)

�7�a+|ni = d(n)|n+ 1i, @Âhn|a = d

⇤(n)hn+ 1|. °ó

hn|a+a|ni = |c(n)|2 = n (3.321)

hn|aa+|ni = |d(n)|2 = n+ 1 (3.322)

⌘Ïó0

a|ni =pn|n� 1i a

+|ni =pn+ 1|n+ 1i (3.323)

�⇧

a+|n� 1i =

pn|ni (3.324)

⌘ÏÔÂ)(a|0i = 0enö˙�‚˝p⇥9naÑöI�⌘ÏÂS(P⌥ha↵

(i

~↵�i~ddx

+ ↵x) 0(x) = 0 (3.325)

s

(d

dx+ ↵

2x) 0(x) = 0 (3.326)

ó

0(x) = N0e�↵

2x2

2 (3.327)

R���ÔÂö˙N0.

ç)(a+|0i = |1i,

1(x) =1p2(↵x� 1

↵

d

dx) 0(x) = N1↵xe

�↵2x2

2 (3.328)

nM0�

n(x) =1pna+ n�1(x) =

1p2n

(⇠ � d

d⇠) n�1 (3.329)

v-

⇠ = ↵x (3.330)

ù!{®

n(x) =

rN0

2nn!(⇠ � d

d⇠)ne�⇠

2/2 ⌘ NnHn(⇠)e

�⇠2/2 (3.331)

↵Mó&aÑ,Å�^8ÕÅ�\¯̄̄rrr���(coherent state):

a|↵i = ↵|↵i. (3.332)

3.14. GMó&’ 73

v-,Å<`Ô⌦ô:↵�1éa�/Ñsó&�@Â↵:�*�p(Ë✏�/MbÑ↵ =p

m!/~). æ

|↵i =1X

n=0

Cn(↵)|ni (3.333)

@Â

a|↵i =X

n

Cn(↵)pn|n� 1i = ↵

X

n

Cn(↵)|ni (3.334)

‘ÉBå-|n� 1iy˚p�Cn(↵) =

↵pnCn�1(↵) =

↵n

pn!C0(↵) (3.335)

⇤QR��

h↵|↵i = |C0|2X

n

|↵|2

n!= 1 (3.336)

v-Bå:e|↵|2

, @Â

C0 = e�|↵|2/2 (3.337)

|↵i = C0(↵)X

n

↵n

pn!|ni (3.338)

íP⌅é|niчá

|Cn|2 =|↵|2n

n!e�|↵|2 (3.339)

: ~⌃⇤. π◆ó˙h↵|a+a|↵i = |↵|2, a+aÑ⌥∆OÓ:|↵|⇥Çúˆˆ;íP⌅é|↵i, £Htˆ;1/

|ti = C0e�i!t/2

X

n

↵n

pn!|nie�i!nt (3.340)

hx(t)i = x0p2h(a+ a

+)i =p2|↵|x0 cos(!t+ �) (3.341)

(� /�¯�1↵≥ö, x0 =p

~/m!/⇣/PyÅ�¶). ⌥∆OÓ⇢'�:¿HÙ¯r��•—œx∂��

ÈÈÈ555hhh:::

⌘Ïe°ó˝œha↵⌅ó&ÑÈ5b✏

9nEq(3.323),

an�1,n = hn� 1|a|ni =pn (3.342)

a+n+1,n = hn+ 1|a+|ni =

pn+ 1 (3.343)

sÈ5C:an,m =pm�n+1,m, a

+n,m

=pm+ 1�n�1,m. Ë✏�n,m = 0, 1, 2, · · · .

)(

Q =1p2(a+ a

+);P =ip2(a+ � a) (3.344)

π◆ó0

Qn,m =1p2(pm�n+1,m +

pm+ 1�n�1,m) (3.345)

Pn,m =ip2(pm+ 1�n�1,m �

pm�n+1,m) (3.346)

xn,m, pn,m_1ó0Ü⇥

74 CHAPTER 3. œPõfü⌃

3.14.2 “““®®®œœœ

⌘ÏÂSL2, Lz˘◆�vq�,Å�:|l,mi.

L2|l,mi = l(l + 1)~2|l,mi; Lz|l,mi = m~|l,mi (3.347)

(P⌥ha↵�1/

L2Ylm = l(l + 1)~2Ylm; LzYlm = m~Ylm (3.348)

⌘ÏÔÂ(„pπ’ó0˚✏“®œJÑ{<”ú�vs./˚✏“®œ˝·≥“®œ˘◆s˚�

[Jx, Jy] = i~Jz. (3.349)

π◆®¸˙

[J↵, J2] = 0, ↵ = x, y, z (3.350)

£H1ÔÂGæJz, J2Ñq�,Å�:|�,mi.

J2|�,mi = �~2|�,mi (3.351)

Jz|�,mi = m~|�,mi (3.352)

(|�,mi↵°óJ2ó�~2, îÂ'éhJ2

zi = m

2~2, @Â� � m2. ⌘ÏöI

J+ = Jx + iJy, J� = Jx � iJy (3.353)

π◆ÂS

[J+, J2] = 0, [J�, J

2] = 0 (3.354)

[Jz, J+] = ~J+, [Jz, J�] = �~J� (3.355)

J+J� = J2 � J

2z+ ~Jz, J�J+ = J

2 � J2z� ~Jz (3.356)

⌘Ï

J2J+|�,mi = J+J

2|�,mi = �~2J+|�,mi (3.357)

JzJ+|�,mi = J+(Jz + ~)|�,mi = (m+ 1)~J+|�,mi (3.358)

Ÿh�J+|�,mi/J2, JzÑq�,Å��˘î,Å<:�~2å(m+ 1)~. @Â

J+|�,mi = ~a�m|�,m+ 1i (3.359)

a�m/*R��8p⇥{<0�

J�|�,mi = ~b�m|�,m� 1i (3.360)

b�m/8p⇥Õ�⌦bÑ«↵�1é� � m2, @Â≈6X(�*mÑ⌦Pm, å↵Pm

J+|�,mi = 0, J�|�,mi = 0 (3.361)

)((3.356)

J�J+|�mi = (J2 � J2z� ~Jz)|�mi = (� �m

2 �m)~2|�,mi (3.362)

‡d

� = m(m+ 1) (3.363)

{<0�

� = �m(�m+ 1) (3.364)

3.14. GMó&’ 75

@Âm = �m. ⌘Ï∞j ⌘ m,

� = j(j + 1), (3.365)

m = �j,�j + 1, · · · , j � 1, j (3.366)

F/Ë✏j,,,´́́vvv������ööö///tttppp⇥�m�m = 2j≈{/ctp�ˆ�&⇡Œm˙—�‡’†0m = �m. @

ÂÔ˝Ñj:

j = 0,1

2, 1,

3

2, · · · (3.367)

wSjîÂ÷¿H<�Å↵i⌃Óò⇥‘Ç5PÍÀ“®œ�j = 12⇥˙,íPÑj˝/�*yöÑ8�9ÿ

Ñ�:ÍÀ⇥⇡ÀPÍÀ:ˆ�5P1/2, IP1, �íP3/2, �õP2; hS“®œj = 0, 1, 2, · · · .∞(⌘Ïenöajm, bjm.

hjm|J�J+|jmi = ~2|ajm|2 (3.368)

)((3.356),

~2|ajm|2 = (j(j + 1)�m2 �m)~2 (3.369)

@Â

ajm =pj(j + 1)�m(m+ 1) =

p(j +m+ 1)(j �m) (3.370)

{<0

bjm =p

j(j + 1)�m2 +m =p(j �m+ 1)(j +m) (3.371)

s

J+|jmi = ~p(j +m+ 1)(j �m)|j,m+ 1i (3.372)

J�|jmi = ~p

(j �m+ 1)(j +m)|j,m� 1i, (3.373)

ÔÂ↵0J+|j, ji = 0, J�|j,�ji = 0.

76 CHAPTER 3. œPõfü⌃

3.15 Ehrenfestööö⌃⌃⌃

⇤QdÙ˚fl. xi/xÑ�*⌃œ, pi/®œpÑ�*⌃œ. π◆¡�

[xi, F (p)] = i~@F@pi

, (3.374)

[pi, G(x)] = �i~ @G@xi

. (3.375)

ŸÃó&˝/(S-pic↵.

˘éÍ1íP, H = p2/2m. H-Pic ↵

dp(H)i

dt=

1

i~ [p(H)i

, H] = 0 (3.376)

dx(H)i

dt=

1

i~ [x(H)i

, H] =p(S)i

m(3.377)

ŸÃ(0Ü3.374 [pi, H] = 0. B„⌦✏ó

x(H)i

= x(H)i

(0) +p(S)i

mt (3.378)

(Â↵ó&Çú&ˆÙÿœh:H-pic.)

}6

[xi(0), xi(0)] = 0 (3.379)

F/

[xi(t), xi(0)] = [p(S)i

, xi(0)]t

m= � i~t

m(3.380)

)(K�∆s˚�

�xi(t)�xi(0) �~t2m

(3.381)

i⌃⌦h��‚⇧⇢ic�icÑ�¶Rö⇥

ÇúíP(ø:-–®�H = p2

2m + V (x). {<Ñ�(H-pic↵��Â↵�e⌦⌥H

dp

dt=

1

i~ [p, H] = �rV (x) (3.382)

dx

dt=

p

m(3.383)

s

md2x

dt2=

dp

dt= �rV (x) (3.384)

Ÿ�[�π↵b✏⌦/�7Ñ⇥F/�/ó&π↵⇥°ó��<

md2

dt2hxi = �hrVi. (3.385)

Ù���<·≥œx[�π↵�sErenfest ö⌃⇥Â⌦l✏_Ô Ù•)(��<шÙ�l✏(3.271)ó

0⇥

{<Ñ�⌘Ïπ◆ó0d

dthLi = hMi (3.386)

v-

M = r⇥ (�rV) (3.387)

Â⌦{<Erenfest ö⌃

3.16. w¸(HELLMANN)ö⌃åÙÃ(VIRIAL)ö⌃ 77

3.16 www¸̧̧(Hellmann)ööö⌃⌃⌃åååÙÙÙÃÃÃ(Virial)ööö⌃⌃⌃

3.16.1 www¸̧̧ööö⌃⌃⌃   îîî(((

⇤Q�*˝œ,Å�|ni�·≥H|ni = En|ni. (3.388)

æ�/H-Ñ�*¬p(ÔÂ/(œ�⇢Û~). ˘vB¸

(@H

@�� @En

@�)|ni+ (H � En)

@|ni@�

= 0 (3.389)

ŸÃ↵0˘|niÑO¸�'∂�Å L, ÔÂ⌃„:˘⌃œ��‚˝pBO¸.

°óÖÔ

hn|(H � En)@|ni@�

= �hn|(@H@�

� @En

@�)|ni = @En

@�� h@H

@�i (3.390)

I✏ÊπÑHó&⌘Ê\(�ó@En

@�= h@H

@�in (3.391)

Ÿ1/Hellmannö⌃.

ã�. ⇣/P�˘~BO¸�ó0ö�®˝��<.

@H

@~ = � ~m

d2

dx2=

2

~T (3.392)

2

~ hT in =@En

@~ = (n+1

2)! = En/~ (3.393)

s®˝(˝œ,Å�Ñ��</˝œÑ�J⇥�÷~ = m, !_�7Ô ⇥

ãå. 5:-Ñ⇣/P

H =p2

2m+

1

2m!

2x2 � q✏x (3.394)

‰v˝ß:En(✏), à�>En(0) = (n+ 1/2)~!.°ó˘◆✏[p,H] = m!

2x� q✏, ö�↵h[p,H]i = 0, ó0hxi = q✏

m!2 .

h@H@✏

i = @En

@✏= �qhxi (3.395)

˘✏Ô⌃�⌘Ïó0

En = (n+1

2)~! � q

2✏2

2m!2. (3.396)

i⌃„ ⇢V (x) = 12m!

2(x� x0)2 � c. Hú/saπ˚®�ø˝↵M⇥

3.16.2 ÙÙÙÃÃÃööö⌃⌃⌃   îîî(((

œxõfh⇢d

dt(r · p) = p

2

m� r ·rV (3.397)

ÇúíP/_⇢( P⇤ÙÃ�‘Ç(�ÌhS⌦h�–® �r · p/ PÑ�vˆÙ¸pÑ�ˆÙsGIéˆ�&⇡Ô⌃we—c ⇥£H

p2

m= r · (�rV ). (3.398)

œPõfh: (H-pic↵–®π↵(3.397)/�7Ñ⇥˘_⇢ö�÷��<, π↵Êπ�ö/ˆ(�⌥i⌃

œ(ö�↵˝�èˆÙÿ�

h p2

2min =

1

2hr ·rV in (3.399)

78 CHAPTER 3. œPõfü⌃

˘éø˝˝p/P!Ñ≈µ�dö⌃y+ (.

@◆P!˝p1/·≥↵✏Ñ˝p:

V (�x1,�x2, · · · ) = �mV (x1, x2, · · · ) (3.400)

˘�B¸ X

i

xi

@V

@(�xi)= m�

m�1V (3.401)

⌦✏˘˚✏�⇣À�⌘Ï÷� = 1,X

i

xi

@V

@xi

= mV (3.402)

�⇧ô:

r ·rV (r) = mV (r) (3.403)

@Â

hT in =m

2hV in (3.404)

ã�. ⇣/P, ø˝1/P!ø�m = 2: (�x)2 = �2x2. @Â

hT in = hV in = En/2 (3.405)

ãå. "üP, V (r) = �e2/r. P!�m = �1.

hT in = �1

2hV i = �En (3.406)