Two identical spin-1 particles
obeying Bose-Einstein statistics are placed in a 3D isotropic harmonic
(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 AS1∙S2. 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
L3. Assume periodic boundary conditions.
(a) Show that in the limit L --> ∞ the number of states with momentum p in the range d3p = dpxdpydpz (that is px between px and px + dpx etc.) is L3d3p/(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 ∝ n5/3.
Consider a system of two non-interacting particles (1, 2) and two orthonormal
energy eigenstates (α, β) with energies Eα and Eβ
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 kx = ky = kz = k.
(b) distinguishable non-interacting spinless particles in a 3-dimensional Coulomb potential
V(x,y,z) = -Ze2/r, where r = (x2 + y2 + z2)½.
(c) indistinguishable non-interacting spin ½ particles in a 3-dimensional cubic box (with impenetrable walls) of dimensions Lx × Ly × Lz, where Lx = Ly = Lz = L.
For the Titanium atom (Z = 22) in its ground state find the allowed terms 2S+1LJ in the L-S (Russell-Sanders) coupling scheme and use Hund's rule to find the ground state term.