Two identical spin-1 particles
obeying Bose-Einstein statistics are placed in a 3D isotropic harmonic
potential.

(a) If the particles are non-interacting, give the energy and degeneracy of the
ground state of the two-particle system.

(b) Now assume that the particles have a magnetic moment and interact through a
term in the Hamiltonian of the form A**S**_{1}∙**S**_{2}.
How are the energies and degeneracies of the states in (a) changed by this
interaction?

Solve the Schroedinger equation for a particle of mass M in a cubical box of volume
L^{3}. Assume periodic boundary conditions.

(a) Show that in the limit L --> ∞ the number of
states with momentum p in the range d^{3}p = dp_{x}dp_{y}dp_{z}
(that is p_{x} between p_{x} and p_{x }+ dp_{x}
etc.) is L^{3}d^{3}p/(2πħ)^{3}.

(b) Assume that the lowest energy levels in the box are filled with N
electrons, taking due account of the Pauli exclusion principle. Show that the
energy per unit volume, u, is related to the number of particles per unit
volume n by u ∝ n^{5/3}.

Consider a system of two non-interacting particles (1, 2) and two orthonormal
energy eigenstates (α, β) with energies E_{α} and E_{β}
= 3E_{α}.

Determine the factor by which the probability for finding both particles in the
same state for bosons
exceeds that for classical particles.

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})^{½}.

(c) indistinguishable
non-interacting spin ½ 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.

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.