Other problems with identical particles

Problem:

A particle of mass m is in a cubical well if side L, corresponding to the potential energy function
U(r) = 0 if |x|, |y| and |z| < L/2,  U(r) = ∞ otherwise.
(a)  What is its ground-state energy?
(b)  What is the energy of the first excited state?  Is this energy level degenerate or non-degenerate?  Explain!
(c)  Suppose 20 identical, non-interacting particles of mass m and spin ½ are in this well.  What is the ground state energy of this system?  Is this ground state degenerate or non-degenerate?  Explain!

Problem:

Consider a "box" that is a two dimensional square well with sides of length L x L.  Inside the well the potential is U = 0, outside the well the potential is infinite. 
(a)  Let the box contain one particle of mass M.  Determine (or write down) the general set of wave functions Φ(x,y), find the allowed energies for these states and identify the eigenvalues kx and ky for the particle.
(b)  Write down the three lowest energy eigenvalues and note the degeneracies, if any.
(c)  Make a plot, as a function of the eigenvalues kx and ky, showing the allowed states.  On this plot, consider a circle of radius k = (kx2 + ky2)½, with k large.  What is the area of this circle?  Approximately how many states are contained in this circle?
(d)  Now suppose that the same box contains N identical, non-interacting Fermi particles, each with mass M and spin ½.  We have N >> 1.  Find the energy of the highest occupied state, if the system is in its ground state.
(e)  For the N particle system in part d, calculate its total energy

Solution:

Problem:

Consider an ideal 3D Fermi gas comprising N non-interacting fermions, each of mass M, in a container of volume V at T = 0
(a)  Find the Fermi energy.
(b)  Find ρ(E)dE  the number of quantum states whose energy lies in the range
E to E + dE.
(c)  Find the average energy per fermion at absolute zero by making a direct calculation of U(0)/N, where U(0) is energy at T = 0 for N fermions. Express this in terms of the Fermi energy, EF.

Solution: