Assignment 6

Problem 1:

(a)  Determine the energy levels and normalized wave functions ψ(r) of a particle with zero angular momentum in a spherical "potential well" U(r) = 0, (r < a), U(r) = ∞, (r > a).
(b)  If the particle is initially in the ground state of this well and the U(r) is suddenly changed so that U(r) = 0, (r < 3a), U(r) = ∞, (r > 3a), what is the probability that after that change an energy measurement will find that the energy of the particle has changed?

Problem 2:

The Lα line of the characteristic X-ray spectra of heavy atoms consists of several components of different frequencies corresponding to the various allowed transitions from levels with n = 3 to levels with n = 2.  Predict the number of different frequencies to be observed, on the basis of the selection rules  Δl = ±1, Δj = 0, ±1 (except ji = jf = 0).

Problem 3:

A hydrogen atom with Hamiltonian H0(r) is placed in a time-dependent electric field E = E(t) k. The perturbed Hamiltonian is H(r,t) = H0(r) + H'(r,t).
(a)  Show that H'(r,t) = qeE(t) r cos(θ).
(b)  Assuming the electron is initially in the ground state, and recalling that the first excited state of hydrogen is quadruply degenerate, to which state of the quadruply degenerate first excited states is a dipole transition from the ground state possible?  Prove this.
(c)  If the electron is in the ground state at t = 0, find the probability (to first order in perturbation theory) that at time t the electron will have made the transition to the state determined in (b), as a function of E(t).

Hydrogen atom energy eigenfunctions:

Φ100(r,θ,φ)  =  πa0-3/2 exp(-r/a0).
Φ200(r,θ,φ)  = (4π) (2a0)-3/2 (2 - r/a0) exp(-r/(2a0)),
Φ211(r,θ,φ)  = (8π) (2a0)-3/2 (r/a0) exp(-r/(2a0)) sinθ e,
Φ210(r,θ,φ)  = (4π) (2a0)-3/2 (r/a0) exp(-r/(2a0)) cosθ,
Φ21-1(r,θ,φ) = (8π) (2a0)-3/2 (r/a0) exp(-r/(2a0)) sinθ e-iφ.

Problem 4:

Consider spinless non-relativistic free particles of mass M moving in a three dimensional cubical box of side length L, with L very large.  Find an expression for the density of states ρ(E).
Remember ρ(E)dE is defined as the number of energy levels per unit volume between E and E + dE.

Problem 5:

A one-dimensional harmonic oscillator is in its ground state for t < 0.
The unperturbed Hamiltonian is H0 = p2/(2m) + ½mω02x2.
For  t > 0 it is subjected to a time-dependent but spatially uniform force in the x-direction, 
F = F0cos(ωt), ω << ω0.
Using time-dependent perturbation theory to first order, obtain the probability of finding the oscillator in an excited state for t > 0, t << 2π/ω.