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Physics 452

Physics 452. Quantum mechanics II Winter 2011. Karine Chesnel. Phys 452. Homework. Friday Mar 23 : assignment # 18 10.1, 10.2, 10.10 Tuesday Mar 27 : assignment # 19 10.3, 10.4, 10.5, 10.7. Phys 452. Homework. W April 6 & April 8 assignment # 24 Research &QM presentations.

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Physics 452

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  1. Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

  2. Phys 452 Homework Friday Mar 23: assignment # 18 10.1, 10.2, 10.10 Tuesday Mar 27: assignment # 19 10.3, 10.4, 10.5, 10.7

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

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

  6. Berry’s phase Phys 452 (a) Evaluate the geometric phase: 1. Calculate 2. Calculate (integration along x for given w) (integration along w) 3. Calculate Pb 10.3: Application to the case of infinite square well Easier way to solve Pb 10.1!! The well expands adiabatically from to 0 w

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

  8. Phys 452 Integrate on closed loop 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? 0 w Reversible process??

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

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

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

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

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

  14. Phys 452 Aharonov-Bohm effect

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

  16. Phys 452 Aharonov-Bohm effect Experimental proof

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

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

  19. 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

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

  21. 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. Recent observations PRL 98, 176802 (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

  22. Aharonov Bohm effect Phys 452 B(T) PRL 98, 176802 (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 Experiment Theory

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

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