**Problem 1:**

Determine the energies and the degeneracies of the two lowest levels of a
system composed of three particles with equal masses m, where the particles are

(a) distinguishable
non-interacting spinless particles in a 3-dimensional simple harmonic potential
with spring constants k_{x} = k_{y} = k_{z} = k.

(b) distinguishable
non-interacting spinless particles in a 3-dimensional Coulomb potential

V(x,y,z) = –Ze^{2}/r, where r = (x^{2 }+ y^{2 }+ z^{2})^{1/2}.

(c) indistinguishable
non-interacting spin ˝ particles in a 3-dimensional cubic box (with impenetrable
walls) of dimensions L_{x} × L_{y} × L_{z}, where L_{x}
= L_{y} = L_{z} = L.

**
Problem 2:**

Consider N >> 1 non-interacting
spin-˝ particles with mass M confined to a cubical box of volume V.

(a) Derive an expression for the Fermi energy.

(b) Make some numerical estimates, assuming in each case that the particles in
question are non-interacting, for the Fermi energy of

(i) electrons in a typical metal,

(ii) nucleons in a large nucleus,

(iii) ^{3}He atoms in liquid ^{3}He, which has an atomic volume
of about 0.05 nm^{3} per atom.

**Problem 3:**

Consider a helium atom where both electrons are replaced by
identical charged particles of spin
quantum number s = 1. Ignoring the motion of the nucleus and the
spin-orbit interaction, the Hamiltonian is given by

H = P_{1}^{2}/(2m) + P_{2}^{2}/(2m) - 2e^{2}/r_{1}
- 2e^{2}/r_{2} + e^{2}/|**r**_{1 }-** r**_{2}|.

Construct an energy level diagram (qualitatively) for this "atom", when

(a) both particles are in the n = 1 state, and when

(b) one particle is in the n = 1
state and the other is in the state (n l m) = (2 0 0).

Do this by treating the
e^{2}/|**r**_{1 }-** r**_{2}| term in the
Hamiltonian as a perturbation. Write out the space and spin wave functions for each level
in terms of the single particle hydrogenic wave functions ψ_{nlm}
and spin wave functions χ_{s,ms}. Show the splitting qualitatively,
and state the degeneracy of each level. Don’t forget to include the effect of the
e^{2}/|**r**_{1 }-** r**_{2}| term.

**Problem 4:**

For the Titanium atom (Z = 22) in its ground state find the allowed terms
^{2S+1}L_{J} in the L-S (Russell-Sanders) coupling scheme and
use Hund's rule to find the ground state term.

**Problem 5:**

Consider a system of 4 identical particles. Each particle has 3 possible
eigenvalues, E_{1}, E_{2}, E_{3}, of some observable.

Write down the normalized symmetric wave function in which two of the particles
have eigenvalue E_{1}, one has eigenvalue E_{2}, and one has
eigenvalue E_{3}.