Post on 03-Jan-2016
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