Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

Homework Phys 452 Friday Mar 18: assignment # , 9.20, 9.21 Tuesday Mar 22: assignment # , 10.2, Friday Mar 25: assignment # , 10.4, 10.5, 10.7

Chapter 9 summary: Two- level systems time-dependent perturbation Phys 452 Solving Schrödinger equation

Chapter 9 summary: Generalization to N - levels Phys 452 Solving Schrödinger equation (Pb 9.15)

Time- dependant perturbation Phys 452 Varying ground in infinite square wellPb 9.17 This perturbation does not induce any transition 0 a

Two- level systems Time- dependent perturbation Phys 452 Probability of transition:  P(  ) Rabi flopping frequency Sinusoidal perturbation

Emission and absorption of a radiation Phys 452 Magnetic resonancePb 9.20 Build SpinorSolve Special frequency:

Emission and absorption of a radiation Phys 452 Expression for the perturbation If the wavelength of the light is comparable to the atom Pb 9.21 Electric dipole transition Magnetic dipole transition

Emission and absorption of a radiation Phys 452 Expression for the perturbation Case of Harmonic oscillator Pb 9.21 Case of atomic transition 2S – 1S

Adiabatic approximation Phys 452 Classical meaning in thermodynamic In adiabatic process, the system does not exchange energy with the outside environment 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 locomotion engine

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

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 with Final solution Geometric phase Dynamic phasewith The particle stays in the same state, while the Hamiltonian slowly evolves

Quiz 28 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) 0 a (b) Floor of well moving 0 a (c) Wall of the well extending