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Physics 452

Physics 452. Quantum mechanics II Winter 2011. Karine Chesnel. Phys 452. Homework. Today Mar 23 : assignment # 18 10.1, 10.2, 10.10 Friday Mar 25 : 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 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. 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 Proposed solution 1. Check that solution verifies Schrödinger equation 4 terms 4 terms use Adiabatic approximation Pb 10.1: infinite square well with expanding wall a 0 w 2. Find an expression for the coefficients:

  5. Phys 452 Phase factor: Internal time Wall motion: external time 4. Dynamic phase factor: Adiabatic approximation Pb 10.1: infinite square well with expanding wall Proposed solution a 0 w 3. Internal/ external time

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

  7. Phys 452 Evaluate the Berry’s phase: 1. Calculate 2. Calculate (integration along x for given w) (integration along w) 3. Calculate 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 0 w

  8. Phys 452 1. Express 2. Integrate with time Berry’s phase Pb 10.3: Application to the case of infinite square well The well expands adiabatically from to Evaluate the dynamical phase: 0 w Reversible process??

  9. Phys 452 1. Calculate 2. Calculate (integration along x for given a) (integration along a) 3. Calculate Berry’s phase 3. 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 the geometric phase is zero?

  11. Phys 452 Quiz 28 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. Phys 452 Berry’s phase Electromagnetism analogy Magnetic flux through loop Vector “potential” Magnetic field Analog “magnetic field” Berry’s phase

  13. Phys 452 Aharonov-Bohm effect

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

  15. Phys 452 Aharonov-Bohm effect Experimental proof

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

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

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

  20. 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

  21. 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

  22. 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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