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

Phys 452 Homework Friday Mar 9: assignment # 14 Pb 9.1, 9.2, 9.5, 9.6 Tuesday Mar 13: assignment # 15 Pb 9.7, 9.18, 9.19

Two- level systems Time- dependent perturbation Phys 452 Two- level systems Time- dependent perturbation In case of off-diagonal perturbation

Two- level systems Time- dependent perturbation Phys 452 Two- level systems Time- dependent perturbation Perturbed state:

Two- level systems Time- dependent perturbation Phys 452 Two- level systems Time- dependent perturbation Goal: solve for ca and cb Initial conditions: First order:

Two- level systems Time- dependent perturbation Phys 452 Two- level systems Time- dependent perturbation Zero order: First order: Second order:

Two- level systems Time- dependent perturbation Phys 452 Two- level systems Time- dependent perturbation Application: sinusoidal perturbation Initial conditions First order Probability of transition:

Two- level systems Time- dependent perturbation Phys 452 Two- level systems Time- dependent perturbation Sinusoidal perturbation Probability of transition: P(t) for a given frequency w t

Two- level systems Time- dependent perturbation Phys 452 Two- level systems Time- dependent perturbation Sinusoidal perturbation Probability of transition: P(w) for a given time t w

Two- level systems Time- dependent perturbation Phys 452 Two- level systems Time- dependent perturbation Pb 9.7 Solving exactly for Probability of transition: w P(w) Rabi flopping frequency

Time- dependent perturbation Phys 452 Time- dependent perturbation Pb 9.1 Electric field perturbation n=2 n=1 Simulation of atomic dipole transition www.falstad.com/qmatomrad/

Time- dependent perturbation Phys 452 Time- dependent perturbation Apply this theory for time-independent perturbation Pb 9.2 Second order differential equation Transition between two states still possible

Time- dependent perturbation Phys 452 Time- dependent perturbation Apply this theory for time-independent perturbation Pb 9.6 Use the perturbation theory And integrate at second order Compare with the exact answer of problem 9.2

Time- dependent perturbation Phys 452 Time- dependent perturbation Two levels system with different initial conditions Pb 9.5 Initial conditions