Physics 512 Homework Set #10: Problems on Atomic Physics and Quantum Mechanics, Assignments of Physics

Download Physics 512 Homework Set #10: Problems on Atomic Physics and Quantum Mechanics and more Assignments Physics in PDF only on Docsity! Physics 512 Winter 2003 Homework Set #10 – Due Friday, March 28 1. This is based on Sakurai, Chapter 5, Problem 28. A hydrogen atom is initially in its ground state (1s). At time t = 0 we turn on a spatially uniform electric field as follows:
E(t) = { 0 t < 0 E0e−t/τ ẑ t ≥ 0 a) Using first-order time dependent perturbation theory, compute the probability for the atom to be found in each of the three 2p states at time t  τ . You need not evaluate the radial integrals, but perform all other integrations. b) What would happen if instead the atom was in the 2s state to begin with? 2. Consider a two-photon transition from an initial state of angular momentum 1 = 0 to a lower energy intermediate state of angular momentum 2 = 1 to the ground state of the system with angular momentum 3 = 0. Both photons are emitted via electric dipole transitions. a) Using second order time-dependent perturbation theory, show that the amplitude for this transition is proportional to ̂1 · ̂2 where ̂1 and ̂2 are the polarization vectors of the two photons (ignore identical particle effects). b) Averaging over photon polarizations, show that the probability distribution for the angle θ between the two photons has the form P (θ) ∼ 1 + cos2 θ. 3. A hydrogen atom initially in its ground state is exposed to a harmonic perturbation:
E(t) = E0 cos(ωt)ẑ t ≥ 0 Calculate the rate of ionization of the atom as a function of ω. 4. Given two distinguishable spin-1 particles with vanishing orbital angular momenta, one can form states of total angular momentum j = 0, 1 and 2. Now suppose the two particles are identical. What restrictions do we get? What about two spin-2 particles? What is the general rule for allowed values of j? [See Sakurai, Chapter 6, Problem 2]. 1