10. The Adiabatic Approximation

1. The Adiabatic Theorem

2. Berry’s Phase

10.1. The Adiabatic Theorem

1. Adiabatic Processes

2. Proof of the Adiabatic Theorem

10.1.1. Adiabatic Processes

Adiabatic processes : Text >> Tint .

Strategy :

1.Solve problem with external parameters held constant.

2.Solutions thus obtained contain external parameters that

are now allowed to vary slowly.

E.g. Pendulum with gradually changing length L.

Period of pendulum with fixed length is 2L

Tg

Period of pendulum with slowly changing length is 2

L tT t

g

Example: H2+ ion

E.g. H2+ ion.

Born-Oppenheimer approximation :

1.Electron stationary states solved for fixed nuclei separation R.

2.Ground state energy E of system obtained as a function of R.

3.Equilibrium separation solved from 0d E

d R

Adiabatic Theorem

If a system is initially in a discrete and nondegenerate eigenstate n of H(0) ,

it’ll be carried to the corresponding eigenstate n of H(t).

1-D Infinite Square Well

E.g. 1-D infinite square well with slowly changing width a.

Ground state for well with fixed a :

1

2sin

xx

a a

Adiabatic approximation :Ground state for well with slowly changing a : 1

2, sin

x

x ta t a t

Diabatic approximation :Ground state for well with suddenly changed a :

1

2,0 sin

x

xa a

10.1.2. Proof of the Adiabatic Theorem

H is time-independent :

n n nH E x x

H is time-dependent :

n m nmt tAt each t, the solutions { n (x, t) } are complete and can be set

Let

, ,i t H tt

x x

, t t x x

, ,i t H t tt

x x

, ,n n nH t t E t t x x ( t here is treated as a constant )

expn n

it E t

1 1di H E

d t

1 di E t

d t

0

expti

t d t E t

Let i te

0

1 t

t d t E t

, ,n n nH t t E t t x x 0

expt

n n

it d t E t

ni te

The above are simply the time-independent H results with E E(t).

Now, n n is NOT a solution to the time-dependent Schrodinger eq.

However { n (x, t) } is complete, so we can write

, ,n n nn

t c t t t x x

, ,i t H t tt

x x n n n n n n n n n n n

n n

i c c i c c H

0

1 t

n n

dt d t E t

d t

1

nE t

0n n n n nn

c c

0m m n m n nn

c c n m nmt t

n mim n m n

n

c c e

n mim n m n

n

c c e

, ,n n nH t t E t t x x n n n n n nH H E E

m n m n n mn n m nH H E E m

m n n mn n m m nH E E E

n mim nm m m m n

n m n m

Hc c c e

E E

m nm n

n m

H

E E

for m n

0

1 t

n nt d t E t

Adiabatic approximation : m m m mc c

0

0 expt

m m m mc t c d t t t

mi te

0

t

m m mt i d t t t

0 mi tm mc t c e

0

t

m m mt i d t t t

, ,n n nn

t c t t t x x

1m mt t *0 m m m m m m m m 2 Re m m

m is real

0 , expn n n nn

c t i t t x

If particles starts out in n , ,0 ,0n x x

, , expn n nt t i t t x x

Example 10.1An electron ( charge e, mass m ) at rest at the origin is subject to a magnetic field

0ˆ ˆ ˆsin cos sin sin cost B t t B i j k

The Hamiltonian is

eH t

m B S 0 sin cos sin sin cos

2 x y z

e Bt t

m

1

cos sin1

2 sin cos

i t

i t

e

e

0 1

1 0x

0

0y

i

i

1 0

0 1z

01

eB

m

with normalized eigenspinors & eigenenergies :

cos

2

sin2

i t

t

e

sin

2

cos2

i tet

See Prob 4.30

1

1

2E

Let

cos20 0

sin2

1 /2 /2coscos sin sin sin

2 2 2i t i tt t t

i e t i e t

1 /2

1 /2

cos sin cos2 2 2

cos sin sin2 2 2

i t

i t

t ti e

tt t

i e

2 21 12 cos

Do Prob 10.2

2

2sin sin

2

tt t

2

2

1

sin sin 02

tt t

adiabatic regime : Text ~ 1/ >> Tint ~ 1/1

1

For nonadiabatic regime : >> 1 , .

22

sin sin2

tt t

10.2. Berry’s Phase

1. Nonholonomic Processes

2. Geometric Phase

3. The Aharonov-Bohm Effect

10.2.1. Nonholonomic Processes

Nonholonomic processes : State of system is path-dependent.

E.g., Parallel transport of pendulum on Earth’s surface.

24

2

RA

2

A

R

Foucault Pendulum

Solid angle subtended by latitude line 0 :

02

0 0

sind d

02 cos 1

Since Earth turns a daily angle 2, the daily precession of the Foucault pendulum is 2 cos0 .

10.2.2. Geometric Phase

,0 ,0n x x , , expn n nt t i t t x xAdiabatic approx.

0

1 t

t d t E t

0

t

n n nt i d t t t

= dynamic phase

= geometric phase

Let H t H R t i.e., R(t) is the prarameter that makes H time-dependent.

n n d R

t R d t

0

R t

nn

R

i d RR

0

tn

n n

d Rt i d t t

R d t

n n d R

t R d t

0

R t

nn n

R

t i d RR

For system with N time-dependent parameters 1 2, , , nR R RR

1

Nin n

i i

d R

t R d t

n

d

d t R

R

0

t

n ni d R

R

R

R 10

t Nin

n ni i

d Rt i d t t

R d t

line integral

in R space

H t T H t n n nT i d RR Berry’s phase

Note : n doesn’t depend on magnitude of T as long as the adiabatic approx is

valid.

But n depends critically on magnitude of T. n is measurable: M.V.Berry, Proc.R.Soc.Lond. A392, 45 (1984).

Example: Splitted Particle Beam

A beam of particles, all in state , is splitted in two.

One beam passes through an adiabatically changing potential,

while the other does not.

When the beams are recombined,

0 0

1 1

2 2ie 0 ~ direct beam

2 2

0

11 1

4i ie e 2

0

11 cos

2 2 2

0 cos2

can be measured from interference pattern.

Furthermore, & can be separated for measurement.

3-D R Space n n nT i d RR

Magnetic flux through surface S :

S

d B aS

d A aC

d A r C bounds S

~n ni R A

~n T

~n ni R R B

n n nT i d R Ra( Stokes’ theorem )

Example 10.2. e in B B e(t)

Let ( see e.g. 10.1 )

0ˆ ˆ ˆsin cos sin sin cost B t t B i j k

1 /2 /2coscos sin sin sin

2 2 2i t i tt t t

t i e t i e t

2 21 12 cos

adiabatic regime : << 1

2

11 1

1 2 cos

11

1 cos

1 cos

1/2 /2

1

cos sin sin sin2 2 2

i t i ttt tt i e t i e t

1 /21

1

exp cos sin sin2 2

i ttit t i e t

1

1

2E

B precesses about z-axis.

1

1

2E

<< 1

Dynamic phase : 0

1 t

t d t E t 1

1

2t

Dynamic phase : t

1

1cos

2t

1cos 1

2t

cos 1T 2T

1 /21

1

exp cos sin sin2 2

i ttit t t i e t

1exp cos2

it t

Solid angle swept out by B : 2

0 0

sin 2 cos 1d d

1

2T

B Sweeps Out Arbitrary Curve on Sphere

cos2

sin2

ie

sin cos , sin cos , cost B t t t t t B

1 1ˆ ˆˆsin

rr r r

10sin

1 12 2 ˆ ˆ1 sin sin

cos 22 2

iir r i e

e

21 1 1 1ˆ ˆcos sin cos sin sin2 2 2 2 2 2 sin 2

ir r

2sin2 ˆ

sini

r

2sin2 ˆ

sini

r

2

ˆ ˆˆ sin

1

sin

sinr

r r r

r r

f r f r f

f

2

2

ˆ ˆˆ sin

1

sin

0 0 sin2

r r r

r r

i

2

sin cos2 2 ˆsin

ir

r

2

ˆ2

ir

r

n n nT i d R Ra

2

1ˆ

2n T d rr

a1

2 2ˆd r r d a

10.2.3. The Aharonov-Bohm Effect

t

A

E B A

E, B unchanged under any gauge transformation :t

A A A

Particle in EM field : 21

2H q q

m p A

21

2q q

m i

A

H is gauge invariant ( see Prob 4.61 )

Classical electrodynamics : No EM effects where E = B = 0.

Aharonov-Bohm : EM effects where E = B = 0 if 0d A r

Can be related to Berry’s phase.

Particle Circling a Long Solenoid

Long solenoid with axis along z, of radius a, and carrying a steady current I.

ˆ

0

B r a

r a

zB

Coulomb’s gauge : 0 A ˆ2 r

A

2

2

r B r a

a B r a

magnetic flux

ˆˆ ˆ

1

0 02

r r z

r r z

rr

A 1ˆ

2z

r r

ˆ

0

B z r a

r a

1r zA A A

r r z

A

02 r

1

2H q q

m i i

A A

= 0 2 2 2 21

2q q

m i

A A A

2 2 2 212

2H q q

m i

A A

A A ANow

in Coulomb gauge A

For r > a,

2ˆ ˆ

2 2

Ba

r r

A

If the particle is confined to a circular orbit of radius b, then

22 2

2 2 2

1

2 2

d q q dH i E

m b d b b d

22 2

2 2 2

1

2 2

d q q di E

m b d b b d

22 2

2 2

2

2

d q q d mE bi

d d

2

22 0

d di

d d

where

2

q

22

2

2

2

mE b q

22

2

2mE b

Ansatz : iA e 2 2 0

2 2b mE

is single-valued n 2

2

22nE nmb

0, 1, 2,n

22

22 2

qn

mb

A-B effect

General Case

Consider particle moving in region where 0 B A 0Abut

21

2H q V i

m i t

A

Leti ge with q

g d r

r A r O

0 B A 0S

d a AC

d r A where C bounds S

g is path independent.O is some reference point.

i ge i g i g qe i

A

i gq e

i i

A

qg A

2i gq q e

i i i

A A

i gq ei i

A

2 2 2i ge i g qi

A

2 2i ge

qg A

2

2

2V i

m t

i get t

i ge qg d

r

r A r O

A-B effect

A-B effect for splitted e beam :

/ /

qg d

r A

r A

0

1

2

qg d r

r

ˆ2 r

A

2

q

Phase shift of combined beam : q

Measured : R.G.Chambers, PRL 5, 3 (1960)

A-B Effect as a Berry’s Phase

2

1

2n n n nH q V Em i

A r R

Leti g

n ne with qg d

r

R

r A r

2

2

2 n n nV Em

r R

Consider a particle outside the solenoid & confined by V near R.

i gn ne R R i g

n

qe i

RA R

n n r R

i g i gn n n n n

qe e i

R RA R

n n n

qi

RA R

n n

qi RA R

n n n n

qi R RA R

n n R r R r Rn n n n R

n

i

p

0 n is a stationary state.

n n

qi R A R

n n nT i d RR qd R A R

S

qd a A

S bounded by contour

S

q A-B effect