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)