Problem 1:

Nuclei sometimes decay from excited states to the ground state by internal conversion, a process in which an atomic electron is emitted instead of a photon.  Let the initial and final nuclear states have wavefunctions F_i(r₁,r₂,..,r_z) and F_f(r₁,r₂,..,r_z), respectively, where r_i describes the protons. The perturbation giving rise to the transition is the proton-electron interaction,

,

where r is the electron coordinate.

(a)  Write down the matrix element for the process in lowest-order perturbation theory, assuming that the electron is initially in a state characterized by the quantum numbers nlm, and that its energy, after it has been emitted, is large enough so that its final state may be described by a plane wave.  Neglect spin.

(b)  Write down an expression for the internal conversion rate.

(c)  For light nuclei, the nuclear radius is much smaller than the Bohr radius for the given Z, and we can use the expansion

.

Use this approximation to express the transition rate in terms of the dipole matrix element

.

Problem 2:

An experimenter has carefully prepared a particle of mass m in the first excited state of a one dimensional harmonic oscillator, when he sneezes and knocks the center of the potential a small distance a to one side.  It takes him a time T to blow his nose. and when he has done so he immediately puts the center back where it was.  Find, to lowest order in a, the probabilities P₀ and P₂ that the oscillator will now be in its ground state and its second excited state.

Problem 3:

Consider the system described by the Hamiltonian

.

(a)  Calculate an approximate value for the ground state energy using first-order perturbation theory, perturbing the harmonic oscillator Hamiltonian

.

(b)  Calculate an approximate value for the ground state energy using the variational method.