Physics 452

1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Phys 452 Homework Thursday Mar 22: assignment # 18 10.1, 10.2, 10.10 Tuesday Mar 27: assignment # 19 10.3, 10.4, 10.5, 10.7

3. 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

4. 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

5. 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

6. Phys 452 Adiabatic approximation Final solution with Dynamics induced by external change Internal dynamics Dynamic phase Geometric phase

7. 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:

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

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

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

11. 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

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

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

14. 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

15. Phys 452 Quiz 30 Which one of these process does not fall under the non-holonomic category? A. The motion of a vehicle after one engine cycle B. The daily precession of Foucault’s pendulum C. The circular motion of a skater on frictionless ice D. The circular motion of cars on racing ring E. The circular motion of a skiing-boat on a lake

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

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

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

19. Phys 452 (a) Evaluate the geometric 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 infinite square well The well expands adiabatically from to 0 w

20. 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

21. 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??

22. 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

23. 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