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(0Ü3.374Â [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)
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