Quantum mechanics II Winter 2011 Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

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Homework Phys 452 Tuesday Mar 22: assignment # 18 10.1, 10.2, 10.10 Friday Mar 25: 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

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

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 approximation Phys 452 Adiabatic approximation 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 Probability of transition up - down

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

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 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

Phys 452 Foucault’s pendulum Solid angle pendulum Earth rotating

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

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 case of infinite square well The well expands adiabatically from to Evaluate the Berry’s 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 case of infinite square well The well expands adiabatically from to Evaluate the dynamical phase: w 1. Express 2. Integrate with time 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 Berry’s phase (integration along a) 3. Calculate dynamic phase

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