Physics 452
Quantum mechanics II
Winter 2012
Karine Chesnel
Phys 452
Homework
Thursday Mar 22: assignment # 18
10.1, 10.2, 10.10
Tuesday Mar 27: assignment # 19
10.3, 10.4, 10.5, 10.7
Phys 452
Adiabatic approximation
Classical meaning in thermodynamic
Internal
process
Very small / slow
Energy exchange
With outside
In adiabatic process,
the system does not exchange energy
with the outside environment
Phys 452
Adiabatic approximation
Adiabatic theorem
If the Hamiltonian changes SLOWLY in time,
a particle in the Nth state of initial Hamiltonian Hi
will be carried into
the Nth state of the final Hamiltonian Hf
Phys 452
Adiabatic approximation
Mathematically:
H t 
Schrödinger equation:
Final solution
with
but
Characteristic gap
In energy levels
E
t
dH
dt
Characteristic time
of evolution

i
 H  t   t 
t
  t   cn  0  ei t ei t  n  t 
n
n  t   
1
t
 En t 'dt '
0
Dynamic phase
n
 m t   i
t

0
m
d m
dt '
dt
Geometric phase
The particle stays in the same state, while the Hamiltonian slowly evolves
Phys 452
Adiabatic approximation
Final solution
with
  t   cn  0  ei t ei t  n  t 
n
n  t   
1
t
 En t 'dt '
0
Dynamic phase
Internal
dynamics
n
 m t   i
t

0
m
d m
dt '
dt
Geometric phase
Dynamics
induced by
external change
Phys 452
Adiabatic approximation
Pb 10.1: infinite square well with expanding wall
Proposed solution
  x, t    cn  n
n
 n
 n  x, t  
sin 


2
a
0
 i mvx2  2 Eni at  /2
xe



1. Check that solution verifies Schrödinger equation i

 H  t   t 
t
4 terms

2
 i z 2
c

sin(
nz
)sin(
z
)
e
dz
2. Find an expression for the coefficients: n

0
use
cn  n  x,0   x,0
4 terms
Phys 452
Adiabatic approximation
Pb 10.1: infinite square well with expanding wall
Proposed solution
  x, t    cn  n
n
 n
 n  x, t  
sin 


2
a
0
 i mvx2  2 Eni at  /2
xe



e
3. Internal/ external time
 iEt /
 (t )  a  vt
Phase factor:
Internal time
4. Dynamic phase factor:
n  
1
t
Wall motion:
external time
t
 E (t ')dt '   a  vt '
1
0
0
Adiabatic approx
dt '
2
Ti
Te
Phys 452
Adiabatic approximation
Pb 10.2: Spin precession driven by magnetic field
B  B0 cos  kˆ  sin  cos(t )iˆ  sin  sin(t ) ˆj 


Hamiltonian
e
H   MB 
SB
me
Hamiltonian in the space of the Sz spinors
Eigenspinors of H(t)
solution
z
z

,

  
  t  ,   t 
 t   c  t   c  t 
Check that it verifies the Schrödinger equation

i
 H  t 
t
Phys 452
Adiabatic approximation
Pb 10.2: Spin precession driven by magnetic field
B  B0 cos  kˆ  sin  cos(t )iˆ  sin  sin(t ) ˆj 


• Probability of transition up - down
 t   t 
2

 t  
  sin  sin   
 2 

2
• Case of adiabatic transformation

E  E
 1
Probability of transition up - down
 t   t 
2
0
Compare to Pb 9.20
Phys 452
Adiabatic approximation
Pb 10.10: adiabatic series
Particle initially in nth state
  t    cm  t  m  t  ei
m
t 
with
cm  t    nmei n t 
m
(only one term left)
dcm
i  
  ck  t   m  k e  k m 
Also
dt
k
First-order correction to adiabatic theorem
t
cm  t   cm  0    e 
0
i n m  i n  t '
e
 m  n dt '
Phys 452
Nearly adiabatic approximation
t
cm  t   cm  0    e 
Pb 10.10: adiabatic series
i n m  i n  t '
e
 m  n dt '
0
Application to the driven oscillator
2 1
2 2
2
H 

m

x

m

xf  t 
2
2m x
2
2
Evaluate
eigenfunctions
Driving force
m n
 n  x, t    n  x  f 
 n
i
 n  x, t   f
  f  pˆ n 
f
Evaluate
m pn
Using the
ladder operators
pi
m
2
 a  a 
Phys 452
Nearly adiabatic approximation
t
cm  t   cm  0    e 
Pb 10.10: adiabatic series
i n m  i n  t '
e
0
Evaluate
 n t   i
t

0
Evaluate
n
d n
dt ' 
dt
m  t   n  t   
1
i
t
  n p  n dt '
0
t
  E t '  E t 'dt '
m
n
0
Here
1
1

En  t    n     m 2 xf
2
2

Starting in nth level
cn1  t   0
cn1  t   0
Possibility of
Transitions !!!
 m  n dt '
Phys 452
Non- holonomic process
A process is “non-holonomic”
when the system does not return to the original state
after completing a closed loop
irreversibility
pendulum
Earth
Example in Mechanics
After one
Complete
Hysteresis
loop
Example in magnetism
Phys 452
Quiz 30
Which one of these process does not fall
under the non-holonomic category?
A. The motion of a vehicle after one engine cycle
B. The daily precession of Foucault’s pendulum
C. The circular motion of a skater on frictionless ice
D. The circular motion of cars on racing ring
E. The circular motion of a skiing-boat on a lake
Phys 452
Foucault’s pendulum
Solid angle
pendulum
0
2
0
0
   sin  d  d
  2 1  cos0 
Earth
rotating
Phys 452
Berry’s phase
  t   cn  0  ei t ei t  n  t 
General solution
Adiabatic approx
n
 m t   i
with
t

m
0
 m
dt '
t
 m t   i 
Ri
 m t   i

n  t   
1
t
 E t 'dt '
n
0
Dynamic phase
Geometric phase
Rf
n
 m
m
dR
R
m
 R m
for a closed loop
dR
Berry’s phase
(Michael Berry 1984)
Phys 452
Berry’s phase
 m t   i

 m  R m
Berry’s phase
(Michael Berry 1984)
dR
Electromagnetism analogy
Magnetic flux
through loop
   A.dR   B.dS
S
Vector “potential”
Analog “magnetic field”
Magnetic field
B  i R   m  R m
Phys 452
Berry’s phase
Pb 10.3: Application to the infinite square well
The well expands adiabatically
from 1 to 2
(a) Evaluate the geometric phase:
0

1
2
2
 n 1  2   i 
1
1. Calculate
d n
d
2. Calculate
n
d n
d
d n
n
d
d
(integration along x for given )
3. Calculate  n 1  2  (integration along )
Phys 452
Berry’s phase
Pb 10.3: Application to the infinite square well
The well expands adiabatically
from 1 to 2
(b) Evaluate the dynamical phase:
0

1
2
n  t   
1
t
 E t 'dt '
n
0
1. Express
En  t '
2. Integrate with time
n  
1
t2
 En t 'dt '  
t1
1
2
 E  d
n
1
Phys 452
Berry’s phase
Pb 10.3: Application to the infinite square well
The well expands adiabatically
from 1 to 2 and contracts back
(c) What is Berry’s phase?
0

1
2
Integrate on closed loop
2
 n 1  2   i   n
1
d n
d
d
Reversible process??
Phys 452
Berry’s phase
Pb 10.4: Case of delta function well
Solution

1. Calculate
d
d
2. Calculate

m
e
 m x /
m 2
E 2
2
2
Changing parameter: 
d
d
(integration along x for given )
3. Calculate geometric phase  n 1   2 
4. Calculate dynamic phase
n  
1
(integration along )
t2
1
2
 E  t 'dt '    E  d
n
t1
n
1
Phys 452
Berry’s phase
Pb 10.5: Characteristics of the geometric phase
When Berry’s phase is zero?
n  0
n
• Case of real
Geometric
phase
(trick: use the fact that  is normalized)
• Case of
 n   ne
'
i n
 n (loop)  0
Berry’s phase