Consider three (non-interacting) particles in thermal equilibrium, in a
one-dimensional harmonic oscillator, with a total energy E = (7/2)ћω.
(Reminder: the energies of the one-particle states are En = ½ћω + nћω, with
n = 0, 1, 2, ..., ).
(a) If they are distinguishable particles (all with the same ω), what are the
possible occupation-number configurations, and how many distinct (three
particle) states are there for each one? If you picked a particle at
random and measured the energy, what values might you get, and what is the
probability of each one? Check that the sum of probabilities is 1.
(b) Repeat for identical fermions, ignoring spin.
(c) Repeat for identical bosons, also ignoring spin.
Consider a system of 4 identical particles. Each particle has 3 possible
eigenvalues, E1, E2, and 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.
Let us consider a carbon atom whose electrons are in the following configuration (1s)2 (2s)2 2p 3p. List all the expected states on the basis of the L-S (Russell-Sanders) coupling scheme.