Assignment 12

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 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)1/2.
(c)  indistinguishable non-interacting spin ˝ particles in a 3-dimensional cubic box (with impenetrable walls) of dimensions Lx × Ly × Lz, where Lx = Ly = Lz = 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)  3He atoms in liquid 3He, which has an atomic volume of about 0.05 nm3 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 = P12/(2m) + P22/(2m) - 2e2/r1 - 2e2/r2 + e2/|r- r2|.
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 e2/|r- r2| 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 ψnlm and spin wave functions χs,ms.  Show the splitting qualitatively, and state the degeneracy of each level.  Don’t forget to include the effect of the e2/|r- r2| term.

Problem 4:

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.

Problem 5:

Consider a system of 4 identical particles.  Each particle has 3 possible eigenvalues, E1, E2, E3, of some observable.
Write down the normalized symmetric wave function in which two of the particles have eigenvalue E1, one has eigenvalue E2, and one has eigenvalue E3.