Fermi's golden rule

Problem:

Consider spinless non-relativistic free particles of mass M moving in a two dimensional square box.  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.

Solution:

Problem:

Consider a particle in the ground state of a one-dimensional square well of with a and depth U0.
Assume that the well is very deep and
k/k0 = (2m(E + U0)/(2mU0))½ << 1
for the ground state, so that the ground state wave function is nearly identical to that of the infinite square well.  At t = 0, a time dependent perturbation W(t) = Wcosωt is turned on.
(a)  What is the minimum frequency necessary to free the particle from the well? 
(b)  For frequencies greater than this minimum frequency, use perturbation theory to find the transition rate.
You can assume that the free particles will be in a box of size L >> a.

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Solution:

Problem:

A hypothetical particle of mass 2m can decay into two different final states under the action of a perturbation H’.
(a)  2 m --> m + γ,
(b)  2 m --> 1.5 m + γ.
Here m and 1.5 m represent particles of mass m and 1.5 m respectively.  If the matrix elements of H’ are equal, i.e.  <2 m|H'|m> = <2 m|H'|1.5 m>, what will be the relative likelihood of decay to m as opposed to 1.5 m?  You may assume the recoil fragment (m or 1.5 m) receives negligible kinetic energy. 

Solution:

Problem:

A uniform periodic electric field acts upon a hydrogen atom, which at t = 0 is in its ground state.  Determine the minimum frequency of the field necessary to ionize the atom and use perturbation theory to evaluate the probability for ionization per unit time.  For the sake of simplicity, assume the electron in the final state to be free.

Solution:

Problem:

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 wave functions Ψi(r1,r2,..,rz) and Ψf(r1,r2,..,rz), respectively, where ri describes the protons.  The perturbation giving rise to the transition is the proton-electron interaction,

W = -∑i=1z e2/|r - ri|,

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

1/|r - ri| ≈ 1/r + rri/r3.

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

d = <Ψf|∑i=1z rii>.

Solution: