Physics 137B, Fall 2007: Problem Set 4 - Perturbation Theory in Magnetic Fields - Prof. J., Assignments of Quantum Mechanics

Download Physics 137B, Fall 2007: Problem Set 4 - Perturbation Theory in Magnetic Fields - Prof. J. and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 137B, Fall 2007 Problem set 4: perturbation theory Assigned Friday, 21 September. Due in box Friday, 28 September. Starting this week, some of the problems will be assigned from Bransden. Note that the numbering in the second edition is frequently different from the first edition. All numbers here refer to the second edition. 1. Bransden 8.1 2. Bransden 8.11 3. Consider the n = 1 and n = 2 levels of the hydrogen atom (pure Coulomb potential). Ignore fine structure corrections. (a) Make an energy level diagram of how these levels (10 in total, including spin) are split by an applied magnetic field B, assuming g = 2 exactly. (b) What is the energy splitting in an applied magnetic field B between the highest and lowest n = 2 levels? Make sure that your answer has units of energy. 4. What is the numerical value of the Bohr angular frequency ωab for the spin-up and spin-down states in a magnetic field B of 1 Tesla, where H0 = gµBB h̄ Sz. (1) Remember that for an electron, g ≈ 2 and µB = eh̄/2m. (b) Suppose that a perturbation is added for times t > 0, of the form H ′ = gµBB ′ h̄ Sx (2) with B′  B. What is the maximum transition probability in first-order perturbation theory, starting in the spin-down state at t = 0, if B′ = 10−4 Tesla? Plot the transition probability as a function of time. 1