138 CHAPTER 8. œP√¡
8.4 ›››ÌÌÌööö⌃⌃⌃���Berry phase
œP√¡—�(˚flHamitonianèˆÙÿ�Ñ≈µ↵⇥⌘Ïev$ÕÅÔ≈µ.
1 Å—«↵: ¨Ù�ˬp9ÿ�Te ! 0.
ã⇢�Ù‡PÒø1�t = 0ˆΩ¶Å61a ! 2a. ÂíP��:˙��Ω¶:a �Bt > 0ˆ˚fl⌅é¿H
��
ˆˆ;Ñ˙�:
(x, 0) = {
q2asin ⇡x
a, 0 < x < a
0, x < 0, x > a
˝œ:E1 = ~2⇡2
2ma2 . t > 0, H-V ! V0, F/�e� 9ÿ��Õ6/
(x, 0+) = {
q2asin ⇡x
a0 < x < a
0 x < 0, x > a
(8.52)
£HŸ*�>6�/“∞”1Ñ˙��⌅é“∞”˙�чá�H°ó�
“∞”,Å�îÂ/⇢
�n(x) = {
q1asin
n⇡
2a , 0 < x < 2a
0, x < 0, x > 2a
(�nªU�(8.52),
c1 = h�1| (0)i =Z
a
0
r1
asin
⇡x
2a
r2
asin
⇡x
adx
|c1|2 =32
9⇡2
K��H��
(x, t) =X
n
cn�ne�i
E0n
~ t, (8.53)
v-
E0n=
~2n2⇡2
2m(2a)2, cn = h�n| (0)i
∞(⌘Ïe®∫Ê��*ÅÔ≈µ⇢
2 ›Ì«↵⇢�Ëaˆ◆bÿ��Te � Ti, Ti:‘Öˈْ⇥
ã⇢�Ù⇣/P�w5q��Àt = �1ˆ⌅é˙� 0. ◆b†⌦xπ⌘5:�
H0 = �q✏xe�
t2
⌧2
⌧1/�ˈÙ�°ót = 1ˆ�/P⌅é 1чá.
)(Æpπ’
c10(t = 1) =1
i~
Z 1
�1H
010e
i!10tdt (8.54)
v-√¡È5C
H010 = �q✏e
� t2
⌧2 h1|x|0i
= �q✏e� t
2
⌧2
r~
2m!
8.4. ›Ìö⌃�BERRY PHASE 139
�+�˝œÓ
!10 =E1 � E0
~ = ! !⇣/Pëá
Ÿ˙ÖˈÙ
Ti =1
!
(8.54)Ñ°ó”ú:
c10 =�q✏
i~
r~
2m!
Z 1
�1e� t
2
⌧2 +i!tdt
=iq✏p2m~!
p⇡⌧e
�!2⌧2
4
Ô⌃)(ÜMπ⇢� t2
⌧2 + i!t = �( t
⌧� i!⌧
2 )2 � !2⌧2
4 .
Ÿ7√¡‡á:
P10(1) =q2✏2⇡2⌧2
2m~! e�!
2⌧2
2 (8.55)
®∫:
1 . ⌧! ! 1, P10 ! 0, Ÿ1/›Ì«↵�√¡�—�⇥
2 . -Ù˚✏ˆÙt,‘Çt = 0, ≈µ�7�⌘Ï�Å›Ìö⌃
›Ìö⌃⇢GæH(0) 0H(t)Ñ«↵‡P◆b. ÇúíP�À⌅éH(0)Ñ,n*,Å� n, Å�
0H(t)Ñ,n*,Å�(Gæ˝ß⌃À�Äv�(Ô‘fl*’,Å�ˆ�_⇥(éÄv≈b .
¡�⇢æH(t)Ñ,Å��,Å<ÚÂ
H(t) (t)n
= E(t)n (t)n
(8.56)
ŸÃ⌘(⌦⌥⌥h:H(t)Ñ‘¨ˆ’,Å�å¯î,Å<⇥�,´v�+ˆÙ⇥ÉÏÕ6Ñ⇣c§R�˙
‚
h (t)n
| (t)m
i = �nm.
‡d�õöπ↵
i~ @@t (t) = H(t) (t) (8.57)
Ñ„ (t)ÔGæ⇣
(t) =X
n
cn(t) (t)n
ei✓n(t) (8.58)
Ñb✏�v-
✓n(t) = �1
~
Zt
0E
(t0)n
dt0 (8.59)
ÇúH(t)ûE�ÿ�⌦✏vû1/ (t) =Pn
cn neiEn
~ t⇥
„fi(8.57)
i~X
n
[cn (t)n
+ cn (t)n
+ icn (t)n✓n]e
i✓n =X
n
cnH (t)n
ei✓n (8.60)
v-)((8.56)å(8.59), X
n
cn (t)n
ei✓n = �X
n
cn (t)n
ei✓n (8.61)
�h (t)m |ÖÔ⇢
˙cm = �X
n
cnh (t)m
| (t)n
iei(✓n�✓m) (8.62)
140 CHAPTER 8. œP√¡
)((8.56)✏шٸp
H (t)n
+H (t)n
= E(t)n (t)n
+ E(t)n (t)n
(8.63)
⌘Ï
h (t)m
|H| (t)n
i+ h (t)m
|H| (t)n
i = E(t)n�mn + E
(t)n
h (t)m
| (t)n
i (8.64)
˘ém 6= n,
h (t)m
|H| (t)n
i = (E(t)n
� E(t)m
)h (t)m
| ˙ (t)n i (8.65)
˝ß^Äv�Gæ �„e(8.62)
˙cm(t) = �cm(t)h (t)m
| ˙ (t)m i �
X
n 6=m
cn(t)Hmn
E(t)n � E
(t)m
ei(✓n�✓m) (8.66)
›Ì—<H ⌧ 1 :
cm(t) = �cm(t)h (t)m
| (t)m
i (8.67)
$πdÂcm(t), Ô⌃
cm(t) = cm(0)ei�m(t) (8.68)
v-
�m(t) = i
Zt
0h (t0)
m| (t0)
midt0 (8.69)
)(‚˝pR�'åh↵|�i = h�|↵i⇤, Ô¡��:ûp⇥⇤Q0��: n =
(0)n , @Âcn(0) = 1, cm 6=n(0) = 0, ⌘Ï
(t) = (t)n
ei✓n(t)e
i�n(t) (8.70)
Õ6(,n*¨ˆ,Å��✓n®õf¯‡P�˘E(t)n �èˆÙÿ�Ñ≈µ�✓n(t) = �i
En
~ t. ˜Ë✏⇢Ü�
*¯‡P�n(t). �n(t)@ ∫˝§:°✏I�Ù01984.
8.4.1 Berry phase
Mb⌘Ï¡�Ü��:H(0)Ñ,n*,Å� (0)n �›Ì«↵0H(t)ˆ�íPÕ⌅é,n*,Å� (t)
n . F/û
†Ü$*¯‡P⇥i⌃⌦H(t)Ñ�/1é–õ¬p~R(t)èˆÙÿ�¸ÙÑ. ‡d�(8.69)ÔÂ9ô:
�n(t) = ih (t)n
| @@t (t)n
i
= ih (t)n
| @@ ~R
(t)n
i · @~R
@t
Ô⌃
�n(t) = i
Zt
0h (t0)
n| @@t0
(t0)n
idt0 = i
Z ~R(t)
~R(0)h n(~R)|@ n(~R)
@ ~R
i · d~R (8.71)
Ë✏
ih n(~R)| @@ ~R
n(~R)i = ih n(~R)| n(~R+�~R)i � h n(~R)| n(~R)i
�~R
(8.72)
:‚œ, :Berry connection(�ÃT‹ .
M∫ÍË✏0~R/�ÙÑ≈µ�dˆ,Çú˚fl�¬œfi0�À≈µ, R(t) = R(0), (8.71)-Ô⌃R:0.
Berry(1984tñHË✏0~R(åÙÂ⌦Ì�ÔÑÑÔ⌃
�n(T ) = i
Ih n(~R)|rR n(~R)i · d~R (8.73)
8.4. ›Ìö⌃�BERRY PHASE 141
ÔÂ�:ˆ�‡d:Berry Phase.
⌃ÔÑÔ⌃ô⇣bÔ⌃�˘~R⇧ÙÑbÔ~S
�n(T ) = i
ZZ
S
rR ⇥ h n(~R)|rR n(~R)i · d~S (8.74)
v-
irR ⇥ h n(~R)|rR n(~R)i (8.75)
:�ÃÚá.
�ÃT‹��ÃÚáÑs˚{<é¡�î:¶ ~B�‚ø ~AÑs˚:
ih n(~R)|r~R n(~R)i ! ~A
irR ⇥ h n(~R)|r~R n(~R)i ! ~B
�Berry phaseÔÂ↵\¡⇢�
�n(T ) = � =
ZZ
S
~B · d~S !¡⇢
Berry phaseœxãP⇢
⇤QÂRö“�¶!’ztÀlÑ¡: ~B(t) -Ñ�*5P. ¡:l�‡P◆b⇥
H(t) = �~µ · ~B =e~2mc
~� · ~B,
=~!0
2
cos ✓, sin ✓e�i!t
sin ✓ei!t, � cos ✓
!
¡:ÀlÑ“�¶:!,‡d¡:π⌘Ñ'“Ié' = !t. ~!0/˝ß˝œÓ�1B'✏≥ö⇥
⌘ÏÂS(8.76)Ñ$*,Å�:
�(t)+ =
cos ✓
2
sin ✓
2ei!t
!,�
(t)� =
sin ✓
2e�i!t
� cos ✓
2
!(8.76)
⌘Ï(t = 0ˆ�‰5P⌅é
�(0) = �(0)+ =
cos ✓
2
sin ✓
2
!
9n›Ìö⌃, Àl�h�(t! = 2⇡)
�(t) = �(t)+ e
i✓+(t)+i�+(t) (8.77)
142 CHAPTER 8. œP√¡
v-
✓+(t) = �1
~
Zt
0E+(t
0)dt0 = �1
~~!0
2t = �!0t
21é˝œ�Ù�ÿ�
�+(t) = i
Ih�+( ~B)|rB�+( ~B)i · d ~B
¡�î:¶‚œ1/⌘ÏÑèˆÙÿ�¬œ, »∆�œÑ¨ˆ,Å‚ûE/É≥ö. ˘�ÑضûE/˘,Å
�(8.76)-Ñ¡:¯s¬p€L�
9n⇤P⌥↵ضl✏⇢
rB�+ =@
@B�+B +
1
B
@�+
@✓✓ +
1
B sin ✓
@�+
@''
=1
B
� 1
2 sin✓
212 cos
✓
2ei'
!✓ +
1
B sin ✓
0
i sin ✓
2ei'
!'
ç°óÖÔ⇢
�†+rB�+ =
1
2B(� sin
✓
2cos
✓
2+ sin
✓
2cos
✓
2)✓ +
i
B sin ✓sin2
✓
2'
=i sin2 ✓
2
B sin ✓'
Berry connection
A = i�†+rB�+ = �
sin2 ✓
2
B sin ✓' (8.78)
Ù•\ØÔÔ⌃�ÔÂó0
�+ = �1
2⌦ = ⇡(cos ✓ � 1) (8.79)
_ÔÂBBerryÚá�\ØÔ⇧ÙÑÚbÔ⌃⇥BÀ¶
r⇥ h�+|r�+i =1
B sin ✓
@
@✓[sin ✓
i sin2 ✓
2
B sin ✓]B =
i
2B2B
BerryÚá
~B = � 1
2B2B (8.80)
8.4. ›Ìö⌃�BERRY PHASE 143
‡d
�+(T ) = �I
B
2B2· d~S
÷⇤bÔ⌃⇢
�+ = �1
2⌦ = (cos ✓ � 1)
2⇡
2= ⇡(cos ✓ � 1)
⌦/¡:(⇤b⌦k«�h@⇧ÙÑbÔ˘îÑÀS“⇥Ô¡��°�7æ°Ì�fiÔ(¡:)��»
ÑBerry phaseIé¡:‚œk«ÀS“Ñ(�)�J�Ÿ_/:¿HBerry Phase»‡U¯‡P�
Â⌦l✏_ÔÂ9n�(t)Ñ%<„eå¡.
(˜¬⇤Gri�thsf⌦Öπ).
æt = 0,ÍÀ�(0) = �+(0)
ænB„S � eq:
�(t) =
(cos �t
2 � i!0�!
�sin �t
2 ) cos✓
2e� i!t
2
(cos�t
2 � i!0�!
�sin
�t
2 ) sin✓
2ei!t
2
!
� =q!2 + !
20 � 2!!0 cos ✓
√¡‡á⇢
|h�(t)|��(t)i|2 = (!
�sin✓sin
�t
2)2 (8.81)
Ë✏: Te =1!, Ti =
1!0s! ⌧ !0. dˆ�! !0 (8.81)ÿ:
|c�+|2 = (!
!0sin✓sin
!0t
2)2 ! 0 (8.82)
Ÿc/›Ìö⌃”∫.
¯Õ�STe ⌧ Ti,s!0 ⌧ !, ⌘Ï �! !, √¡‡á:
|c�+|2 = |sin✓sin!2t|2 (8.83)