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

2. Homework Phys 452 Today Mar 23: assignment # 18 10.1, 10.2, 10.10
Friday Mar 25: assignment # 19
10.3, 10.4, 10.5, 10.7

3. Research &QM presentations
Phys 452
Homework
W April 6 & April 8
assignment # 24
Research &QM presentations
Briefly describe your research project and how Quantum Mechanics
can help you or can be connected to your research field
If no direct connection between your research and QM, mention one
topic of QM that could potentially be useful or that you particularly liked
2-3 minutes / student (suggested 2-3 transparencies)

4. 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

5. 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:

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

7. Phys 452
Berry’s phase
Pb 10.3: Application to the case of infinite square well
Easier way to solve Pb 10.1!!
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)

8. 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??

9. 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

10. 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

11. Quiz 28 Phys 452 The Berry’s phase has no physical effect
on actual measurable quantities
since it is, by nature, just a phase in the wave function
A. True
B. False

12. Berry’s phase Phys 452 Electromagnetism analogy Berry’s phase
Magnetic flux
through loop
Vector “potential”
Magnetic field
Analog “magnetic field”

13. Phys 452
Aharonov-Bohm effect

14. Aharonov-Bohm effect Phys 452 B The proposed experiment B=0
Long solenoid B
B=0

15. Phys 452
Aharonov-Bohm effect
Experimental proof

16. Aharonov-Bohm effect Phys 452 B A Potential field outside the solenoid
Electrical field
Magnetic field A
Potential field outside the solenoid
B=0
Magnetic flux inside
The solenoid:

17. Aharonov-Bohm effect Phys 452 Pb 10.7 Hamiltonian Solution
where is solution to the Hamiltonian without A
and
The vector potential A can affect the physical state of the particle!

18. at the interference region
Phys 452
Aharonov-Bohm effect
Interference effect
For particle rotating same direction
than the current in the solenoid
For particle rotating opposite direction
Phase difference
at the interference region

19. Aharonov-Bohm effect Phys 452 Geometric phase in presence of potential
Connection with Berry’s phase

20. Aharonov-Bohm Conductance Modulation in Ballistic Carbon Nanotubes
Aharonov Bohm effect
Phys 452
Recent observations
PRL 98, (2007)
Aharonov-Bohm Conductance Modulation in Ballistic Carbon Nanotubes
B. Lassagne, J-P. Cleuziou, S. Nanot, W. Escoffier, R. Avriller, S. Roche, L. Forro´, B. Raquet, and J.-M Broto
1Laboratoire National des Champs Magnetiques Pulses, UMR5147, Toulouse, France
FIG. 1. Left-hand panel: Magnetoconductance G(Vg, B) at 100 K with magnetic field parallel to the tube axis. Selected gate voltages (in volts) are shown. Right-hand panel:3D representation of GB; Vg at 100 K.

21. Aharonov-Bohm Conductance Modulation in Ballistic Carbon Nanotubes
Aharonov Bohm effect
Phys 452
PRL 98, (2007)
Aharonov-Bohm Conductance Modulation in Ballistic Carbon Nanotubes
B. Lassagne, J-P. Cleuziou, S. Nanot, W. Escoffier, R. Avriller, S. Roche, L. Forro´, B. Raquet, and J.-M Broto
1Laboratoire National des Champs Magnetiques Pulses, UMR5147, Toulouse, France
Flux dependence of the conductance B
B(T)
Experiment
Theory

22. Aharonov Bohm effect Phys 452 Production of high magnetic fields
Generators (14 MJoules)
Used to generate 30 to 70T long impulsion (>100ms)
Slice of
a coil
Pulsed fields