1. Quantum mechanics II Winter 2011
Physics 452
Quantum mechanics II
Winter 2011
Karine Chesnel

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3. Homework Phys 452 Tuesday Mar 22: assignment # 18 10.1, 10.2, 10.10
Friday Mar 25: assignment # 19
10.3, 10.4, 10.5, 10.7

4. 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

5. 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

6. 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

7. Phys 452
Quiz 27
For a quantum system subject to an adiabatic transformation,
where, in the wave function, can we best evaluate
the timescale of the external transformation?
A. In the amplitude
B. In the dynamic phase factor
C. In the geometric phase factor
D. In all the terms
E. In none of the terms

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

9. 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

10. 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:

11. Adiabatic approximation
Phys 452
Adiabatic approximation
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
Probability of transition up - down

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

13. 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

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

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

16. 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

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

18. Berry’s phase Phys 452 General solution Adiabatic approx with
Dynamic phase
with
Geometric phase
Berry’s phase
(Michael Berry 1984)

19. 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”

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

21. Phys 452
Berry’s phase
Pb 10.3: Application to the case of infinite square well
The well expands adiabatically
from to
Evaluate the dynamical phase: w
1. Express
2. Integrate with time
Reversible process??

22. 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 Berry’s phase
(integration along a)
3. Calculate dynamic phase

23. Berry’s phase Phys 452 When the geometric phase is zero? Case of real
Pb 10.5: Characteristics of the geometric phase
When the geometric phase is zero?
Case of real
Case of