**
Problem 1:**

Consider a composite system made of two spin ½ particles. For t < 0 the
Hamiltonian does not depend on time and can be taken to be zero. For t > 0
the Hamiltonian is given by H = (4∆/ħ^{2})**S**_{1}∙**S**_{2}, where ∆ is a constant. Suppose that the system is in the state |+ -> for t
≤ 0. Find, as a function of time, the probability for being in each of the
states |+ +>, |+ ->, |- +>, and |- ->,

(a) by solving the problem exactly, using |Ψ(t)> = U(t, t_{0})Ψ(t_{0})>
and

(b) by solving the problem assuming the validity of first-order
time-dependent perturbation theory with H as a perturbation which is switched on
a t = 0.

(b) Under what conditions does the perturbation calculation disagree with
the exact solution and why?

**Problem 2:**

A hydrogen atom with Hamiltonian H_{0}(r) is placed in a
time-dependent electric field **E** = E(t) **k**. The perturbed
Hamiltonian is H(**r**,t) = H_{0}(r) + H’(**r**,t).

(a) Show that H’(**r**,t) = q_{e}E(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).

**Problem 3:**

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
absorption 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 4:**

In a one-dimensional structure a particle of mass m is in the ground state of a
potential energy function U(x) = ½kx^{2}. A phase transition occurs,
and the effective spring constant suddenly doubles. What is the probability that
the particle will be found in an exited state after the phase transition? Give a
numerical answer.

**Problem 5:**

In one dimension, consider a spinless particle trapped in a delta-function potential U(x) = -Cδ(x), C > 0.

At t = 0, a time dependent perturbation W(t) = Wcosωt is turned on.

Assume ω >> mC^{2}/(2ħ^{3}) so that the particle can be ejected
from the trap. Use perturbation theory to find the transition rate.

You can assume that the free particles will be in a box of size L, L =
very large.