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?

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.

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.

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).

Consider a spinless independent electron in the average spherically symmetric
potential due to the nucleus and the other electrons in an atom.

Assume the electron is interacting with a monochromatic plane wave

**E**(r,t)
= (**z**/z)E_{0}cos(ky
- ωt), **B**(r
t) = (**x**/x
)B_{0}cos(ky
- ωt), B_{0}/E_{0} = ω/k
= c.

**E** = -∂**A**/∂t - **∇**Φ,
**B** =
**∇**×**A**.

Choose the Coulomb gauge,
**∇**·**A** = 0, and let Φ = 0,

**A**(r,t)
=
(**z**/z)[A_{0}exp(i(ky
- ωt)) + A_{0}*exp(-i(ky
- ωt))],

with iωA_{0}
= E_{0}/2 and ikA_{0} = B_{0}/2.

The Hamiltonian of the electron interacting with this plane wave is

H = (1/(2m)) (**p** - q_{e}**A**(**r**,t))^{2} +
U(r) = H_{0} + W,

with W = -(q/m)**p**∙**A** + (q_{e}^{2}/(2m))**A**^{2}.

Assume that that the intensity of the wave is low enough so that the term
containing A_{0}^{2} can be neglected compared to terms
containing A_{0}.

(a) Expand
exp(iky) in a Taylor series expansion and evaluate W to zeroth order in ky.

(Since ky is on the order of a_{0}/λ, it is much smaller than 1 in the
visible region of the EM spectrum.)

Write the zeroth order W^{(0)}
in terms of E_{0}. This is the electric dipole Hamiltionian W_{DE}.

(b) Evaluate the commutator [z, H_{0}], and show that the
transition matrix element <Φ_{f}|p_{z}|Φ_{i}>
can be written in terms of the matrix
element <Φ_{f}|z|Φ_{i}>, thus showing that W_{DE}
is equivalent to the form we would get starting with the energy of an electric
dipole in an electric field.

(c) Show that the first order term in the expansion of W is W^{(1)}
= -(q/m)B_{0}cos(ωt)
p_{z}y.

Using
p_{z}y = ½(yp_{z} - zp_{y})
+ ½(yp_{z} + zp_{y})
= ½L_{z}
+ ½(yp_{z} + zp_{y})
write W(1) = W_{DM} + W_{QM}, where W_{DM} is called the
magnetic dipole Hamiltonian and W_{QM} is called the electric
quadrupole Hamiltonian.

For initial and final states for which all transitions are allowed, estimate the
order of magnitude of the ratio of the magnetic
dipole and the electric quadrupole transition probabilities to the electric
dipole transition probability.

(d)
Show that for electric dipole transitions Δl = ±1.