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Physics 452

Physics 452. Quantum mechanics II Winter 2011. Karine Chesnel. Announcements. Preparing for GRE Tips, advices, and refreshments…. Today 4pm N 209. Phys 452. Homework. Tuesday Mar 22 : assignment # 18 10.1, 10.2, 10.10 Friday Mar 25 : assignment # 19 10.3, 10.4, 10.5, 10.7.

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Physics 452

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  1. Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

  2. Announcements Preparing for GRE Tips, advices, and refreshments… Today 4pm N 209

  3. Phys 452 Homework 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. Phys 452 Adiabatic approximation Internal process Very small / slow Energy exchange With outside Classical meaning in thermodynamic In adiabatic process, the system does not exchange energy with the outside environment

  5. 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. Phys 452 Adiabatic approximation Characteristic gap In energy levels Characteristic time of evolution Schrödinger equation: Final solution with Dynamic phase Geometric phase Mathematically: but 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. Phys 452 Adiabatic approximation Final solution with Dynamics induced by external change Internal dynamics Dynamic phase Geometric phase

  9. 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:

  10. 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

  11. Phys 452 Hamiltonian in the space of the Sz spinors Eigenspinors of H(t) solution Check that it verifies the Schrödinger equation Adiabatic approximation Pb 10.2: Spin precession driven by magnetic field Hamiltonian Probability of transition up - down

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

  13. Phys 452 with (only one term left) Also First-order correction to adiabatic theorem Adiabatic approximation Pb 10.10: adiabatic series Particle initially in nth state

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

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

  16. 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

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

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

  19. Phys 452 Berry’s phase Electromagnetism analogy Magnetic flux through loop Vector “potential” Magnetic field Analog “magnetic field” Berry’s phase (Michael Berry 1984)

  20. 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 The well expands adiabatically from to 0 w

  21. 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??

  22. 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

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

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