Problem 1:

Suppose you have two non-interacting, spin-less particles in a one-dimensional potential, each in one of two single-particle, orthonormal eigenstates ψ₁(x) and ψ₂(x) of that potential.  Now suppose you measure the position of one of the particles in an ensemble of identically prepared experiments.  What is the expectation value of position if the particles are
(a)  distinguishable particles,
(b)  identical bosons,
(c)  identical fermions?
In any one of these cases, if there is more than one answer to the question, provide all of them.

Problem 2:

(a)  If the spin of the electron were (3/2)ħ,what spectroscopic terms ^2S+1L_J would exist for the electron configuration 2p, 3p?
(b)  If the spin of the electron were (3/2)ħ,what spectroscopic terms ^2S+1L_J would exist for the electron configuration (3p)²?

Problem 3:

Consider N >> 1 non-interacting spin-½ 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)  ³He atoms in liquid ³He, which has an atomic volume of about 0.05 nm³ per atom.

Problem 4:

Two identical spin zero bosons are placed in a one-dimensional infinite square well,  U(x) = ∞,  x < 0 and x > a, U(x) = 0, 0 < x < a.  The bosons interact weakly with one another via the potential energy function  U(x₁,x₂) = -aU₀δ(x₁ - x₂), where U₀ is a constant with the dimensions of energy.
(a)  First, ignoring the interaction between the particles, find the ground state and first excited state wave functions and the associated energies.
(b)  Use first-order perturbation theory to estimate the effect of the particle-particle interaction on the energies of the ground state and the first excited state.

Useful integrals: 
∫₀ ^π sin⁴(x) dx = 3π/8
∫₀ ^π sin²(x)sin²(2x) dx = π/4