Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

Homework Phys 452 Tuesday Mar 20: assignment # , 9.20, 9.21 Thursday Mar 22: assignment # , 10.2, 10.10

Adiabatic approximation Phys 452 Classical meaning, in thermodynamic In adiabatic process, the net energy transfer between the system and the outside environment is zero Internal process Very small / slow Energy exchange With outside

Adiabatic approximation Phys 452 Example in mechanics & thermodynamic A pendulum slowly carried out in space A diesel engine thermodynamics

Adiabatic approximation Phys 452 Example in mechanics & thermodynamic A frigorific system (Carnot cycle)…

Adiabatic approximation Phys 452 Example in quantum mechanics Hydrogen molecule ion H 2 + Phys 452 Born- Oppenheimer approximation Evidence of bonding Equilibrium separation distance:

Adiabatic approximation Phys 452 Adiabatic theorem If the Hamiltonian changes SLOWLY in time, a particle in the N th state of initial Hamiltonian H i will be carried into the N th state of the final Hamiltonian H f

Quiz 29a Phys 452 A. Only (a) B. Only (b) C. Only (c) D. Only (a) and (c) E. All of them Which one of these transformations can be treated in the adiabatic approximation under certain conditions ? 0 a Brick in the well (a)(a) 0 a (b)(b) Floor of well moving 0 a (c)(c) Wall of the well extending

Adiabatic approximation Phys 452 Mathematically: but Characteristic gap In energy levels Characteristic time of evolution Schrödinger equation: Proposed solution: where

Adiabatic approximation Phys 452 Exact solution: Schrödinger equation: Approximated solutionAdiabatic approximation

Phys 452 Approximated solution Final solution withGeometric phase Dynamic phasewith The particle stays in the same state, while the Hamiltonian slowly evolves

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

Quiz 29b Phys 452 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 For a quantum system subject to an adiabatic transformation, where, in the wave function, can we best evaluate the timescale of the external transformation?

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

Adiabatic approximation Phys a Pb 10.1: infinite square well with expanding wall Proposed solution  1. Check that solution verifies Schrödinger equation4 terms 2. Find an expression for the coefficients: use

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

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

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

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

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