Problem 1:

A system makes transitions between eigenstates of H₀ under the action of the time dependent Hamiltonian H₀ + Wcoswt, W << H₀.  Assume that at t = 0 the system is in the state |y_i>.
(a)  Find an expression for the probability of transition from |y_i> to |y_f>, where |y_i> and |y_f> are eigenstates of H₀ with eigenvalues E_i and E_f.
(b)  Find an expression for the probability of transition from |y_i> to |y_f>, for a constant perturbation W.

Problem 2:

Consider a particle in the ground state of a one-dimensional square well of with a and depth V₀.
Assume that the well is very deep and
k/k₀ = (2m(E+V₀)/(2mV₀))^1/2 << 1
for the ground state, so that the ground state wavefunction is nearly identical to that of the infinite square well.  At t = 0, a time dependent perturbation W(t) = Wcoswt is turned on.  What is the minimum frequency necessary to free the particle from the well?  For frequencies greater than this minimum frequency, use perturbation theory to find the transition rate.

Problem 3:

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 4:

Consider a one-dimensional oscillator in its ground state.
The unperturbed Hamiltonian is H₀ = p²/(2m) + (1/2)mw₀²x².
At t = 0 the Hamiltonian becomes H = H₀ + H₁, where H₁ = (1/2)mw₁²x²cosft, w₁ << w₀.
Calculate the transition probability to the second excited state.
Can there be a transition to any other excited state?

Problem 5:

The electron in a hydrogen atom is in a 3d state.  Neglect the fine structure.
(a)  To what state or states (i.e. 1s etc.) can it go by radiating a photon in an allowed transition?
(b)  What is the degeneracy of the electron (include spin, but ignore spin-orbit interaction) in a 3d state?

Problem 6:

Consider a 2-dimensional system containing a gas of electrons completely free to move in the x direction but confined by a square-well potential of infinite depth and total width w in the y direction.  In such a system, the electrons can often be approximated as non-interacting provided that the mass of the electron is replaced by an effective mass.  Assume for this problem that the effective mass of the electrons is about 1/10 the mass of free electrons. 
(a)  This system contains states that involve quantum-mechanical motion in both the x and y directions; describe qualitatively the nature of the spectrum that you expect. 
(b)  Write or derive a formula for the discrete levels expected for quantized motion in the y direction in terms of the width w. 
(c)  How small does the width w have to be before the transition energy between the first two discrete levels found in b is larger than the average energy available from thermal excitation at room temperature?

Problem 7:

The correctly normalized hydrogen ground state wavefunction in 3D is given by

y₀(r) = (pa₀³)^-1/2exp(-r/a₀),

where a₀ = h²/(m_ee²) is the Bohr radius, which is numerically ~0.529Ǻ.

(a)  Confirm that this does indeed satisfy the radial Schroedinger equation for hydrogen, and that the wavefunction is normalized to ò d³r |y(r)|² = 1
(b)  Two identical ions are introduced on the z-axis at locations z = +d and -d.  Assuming that the effect of each ion on the electron can be treated as a point interaction,

U_{e – ion} = n₀ d(r – r_ion),