Quantum mechanics II Winter 2012 Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

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

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

Adiabatic approximation 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

Adiabatic approximation Phys 452 Adiabatic approximation Characteristic gap In energy levels Mathematically: but Characteristic time of evolution Schrödinger equation: Final solution with Dynamic phase Geometric phase The particle stays in the same state, while the Hamiltonian slowly evolves

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

Adiabatic approximation Phys 452 Adiabatic approximation Pb 10.1: infinite square well with expanding wall Proposed solution a w 1. Check that solution verifies Schrödinger equation 4 terms 4 terms 2. Find an expression for the coefficients: use

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

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

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

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

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

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

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

Quiz 30 Phys 452 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

Phys 452 Foucault’s pendulum Solid angle pendulum Earth rotating

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

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

Phys 452 Berry’s phase Pb 10.3: Application to the infinite square well The well expands adiabatically from to (a) Evaluate the geometric phase: w 1. Calculate 2. Calculate (integration along x for given w) 3. Calculate (integration along w)

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

Phys 452 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? w Integrate on closed loop Reversible process??

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

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