Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

Homework Phys 452 Friday Mar 23: assignment # , 10.2, Tuesday Mar 27: assignment # , 10.4, 10.5, 10.7

Homework Phys 452 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)

Non- holonomic process Phys 452 A process is “non-holonomic” when the system does not return to the original state after completing a closed loop pendulum Earth Example in Mechanics After one Complete Hysteresis loop irreversibility Example in magnetism

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

Berry’s phase Phys  The well expands adiabatically from to (a) Evaluate the geometric phase: 1. Calculate 2. Calculate (integration along x for given  ) 3. Calculate (integration along  ) Pb 10.3: Application to the case of infinite square well Easier way to solve Pb 10.1!!

Berry’s phase Phys 452 Pb 10.3: Application to the infinite square well 0  The well expands adiabatically from to (b) Evaluate the dynamical phase: 1. Express 2. Integrate with time

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

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

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

Quiz 31 Phys 452 A. True B. False Berry’s phase has no physical impact on actual measurable quantities since it is, by nature, just a phase in the wave function

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

Electrical field Magnetic field Electromagnetic potentials in quantum mechanics Phys 452 Maxwell’s equations Hamiltonian

Aharonov-Bohm effect Phys 452

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

Aharonov-Bohm effect Phys 452 Experimental proof

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

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

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

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

Aharonov Bohm effect Phys 452 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. 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 Recent observations

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 Theory Experiment B B(T)

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