**Problem 1:**

The quantum numbers l_{1} and l_{2} of the
orbital momenta of particle A and particle B are 1 and 2, respectively.
Find the 15 possible ‘kets' in the coupled representation (notation |l_{1},l_{2};L,M_{L}>)
where L represents the quantum number of the total orbital momentum.

**Problem 2:**

A pair of magnetic ions with individual spins **s**_{1} and **s**_{2}
interact through the scaled Hamiltonian H =** s**_{1}∙**s**_{2}.
Let s(s + 1) be the eigenvalue of s^{2}, where **s **=** s**_{1}
+ **s**_{2} is the total angular momentum. Note ħ = 1.

(a) Show that the total angular momentum is a conserved quantity.

(b) Find the ground state energy E_{0} and the energy of the highest
state E_{max}.

(c) Now suppose a magnetic field is applied and the new Hamiltonian is H',
where

H' = H - b(s_{1z} + s_{2z}). What is the residual symmetry of
the new Hamiltonian H', and what are the associated "good quantum numbers"?

**Problem 3:**

Consider two distinguishable spin ½^{ }systems that can be repeatedly
prepared in the state

|ψ> = 2^{-½}(|++> + |-->) = 2^{-½}(|+>^{(1)}|+>^{(2)}
+ |->^{(1)}|->^{(2)}).

Observer 1 measure the X^{(1)} = σ_{x}^{(1)} or Y^{(1)}
= σ_{y}^{(1)}, i.e. the x- or y-component of the spin in units
of ħ/2, of system 1, and observer 2 measures X^{(2)} or Y^{(2)}
of system 2.

(a) Show that the product operators X^{(1)}Y^{(2)} and Y^{(1)}X^{(2)}
commute and therefore form a complete set of commuting operators for the system.

(b) What are the possible results of measuring X^{(1)}, X^{(2)},
Y^{(1)}, or Y^{(2)} for the given |ψ>?

(c) What are the possible outcomes of measuring X^{(1)}Y^{(2)}
or Y^{(1)}X^{(2)}?

**Problem 4:**

A system has a wave function ψ(x,y,z) = N*(x + y + z)*exp(-r^{2}/α^{2})
with α real. If L_{z} and L^{2} are
measured, what are the probabilities of finding 0 and 2ħ^{2}?

**Problem 5: **

Some organic molecules have a triplet (S = 1) excited state
that is located at an energy Δ above the
singlet (S = 0) ground state. Consider an ensemble of N such molecules where N
is of the order of Avogadro's number

(a) Find the average magnetic moment <μ>
per molecule in the presence of a magnetic field **B. **Assume Boltzmann
statistics. You may also assume that Δ
is large compared to the field-induced level splittings.

(b) Show that the magnetic susceptibility χ = N d<μ>/dB is approximately
independent of Δ when k_{B}T >> Δ.