4. Quantum mechanics II Winter 2012
Physics 452
Quantum mechanics II
Winter 2012
Karine Chesnel

5. Homework Phys 452 Thursday Mar 22: assignment # 18 10.1, 10.2, 10.10
Tuesday Mar 27: assignment # 19
10.3, 10.4, 10.5, 10.7

6. Adiabatic approximation
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

7. Adiabatic approximation
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

8. Adiabatic approximation
Phys 452
Adiabatic approximation
Characteristic gap
In energy levels
Mathematically:
but
Characteristic time
of evolution
Schrödinger equation:
Final solution
with
Dynamic phase
Geometric phase
The particle stays in the same state, while the Hamiltonian slowly evolves

9. Adiabatic approximation
Phys 452
Adiabatic approximation
Final solution
with
Dynamic phase
Geometric phase
Dynamics
induced by
external change
Internal
dynamics

10. Adiabatic approximation
Phys 452
Adiabatic approximation
Pb 10.1: infinite square well with expanding wall
Proposed solution a w
1. Check that solution verifies Schrödinger equation
4 terms
4 terms
2. Find an expression for the coefficients:
use

11. Adiabatic approximation
Phys 452
Adiabatic approximation
Pb 10.1: infinite square well with expanding wall
Proposed solution a w
Phase factor:
Internal time
Wall motion:
external time
3. Internal/ external time
4. Dynamic phase factor:
Adiabatic approx

12. Adiabatic approximation
Phys 452
Pb 10.2: Spin precession driven by magnetic field
Hamiltonian
Hamiltonian in the space of the Sz spinors
Eigenspinors of H(t)
solution
Check that it verifies the Schrödinger equation

13. Adiabatic approximation
Phys 452
Adiabatic approximation
Pb 10.2: Spin precession driven by magnetic field
Probability of transition up - down
Case of adiabatic transformation
Compare to Pb 9.20
Probability of transition up - down

14. Adiabatic approximation
Phys 452
Adiabatic approximation
Pb 10.10: adiabatic series
Particle initially in nth state
with
(only one term left)
Also
First-order correction to adiabatic theorem

15. Nearly adiabatic approximation
Phys 452
Nearly adiabatic approximation
Evaluate
Pb 10.10: adiabatic series
Application to the driven oscillator
Driving force
eigenfunctions
Evaluate
Using the
ladder operators

16. Nearly adiabatic approximation
Phys 452
Nearly adiabatic approximation
Evaluate
Evaluate
Pb 10.10: adiabatic series
Here
Starting in nth level
Possibility of
Transitions !!!

17. Non- holonomic process
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

18. Quiz 30 Phys 452 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

19. Phys 452
Foucault’s pendulum
Solid angle
pendulum
Earth
rotating

20. Berry’s phase Phys 452 for a closed loop General solution
Adiabatic approx
Geometric phase
with
Dynamic phase
Berry’s phase
(Michael Berry 1984)
for a closed loop

21. Berry’s phase Phys 452 Electromagnetism analogy Berry’s phase
(Michael Berry 1984)
Electromagnetism analogy
Magnetic flux
through loop
Vector “potential”
Magnetic field
Analog “magnetic field”

22. Phys 452
Berry’s phase
Pb 10.3: Application to the infinite square well
The well expands adiabatically
from to
(a) Evaluate the geometric phase: w
1. Calculate
2. Calculate
(integration along x for given w)
3. Calculate
(integration along w)

23. Phys 452
Berry’s phase
Pb 10.3: Application to the infinite square well
The well expands adiabatically
from to
(b) Evaluate the dynamical phase: w
1. Express
2. Integrate with time

24. Phys 452
Berry’s phase
Pb 10.3: Application to the infinite square well
The well expands adiabatically
from to and contracts back
(c) What is Berry’s phase? w
Integrate on closed loop
Reversible process??

25. Berry’s phase Phys 452 Pb 10.4: Case of delta function well Solution
Changing parameter: a
1. Calculate
2. Calculate
(integration along x for given a)
3. Calculate geometric phase
(integration along a)
4. Calculate dynamic phase

26. Berry’s phase Phys 452 When Berry’s phase is zero? Case of real
Pb 10.5: Characteristics of the geometric phase
When Berry’s phase is zero?
Geometric
phase
Case of real
(trick: use the fact that y is normalized)
Case of
Berry’s phase