Problems

1. Consider a particle of mass $m$ moving in a three-dimensional isotropic harmonic oscillator potential of force constant $k$. Solve the problem via the separation of variables, and obtain an expression for the allowed values of the total energy of the system (in a stationary state).
2. Repeat the calculation of the Fermi energy of a gas of fermions by assuming that the fermions are massless, so that the energy-momentum relation is $E=p\,c$. [from Gasiorowicz]
3. Calculate the density of states of an electron gas in a cubic box of volume $L^3$, bearing in mind that there are two electrons per energy state. In other words, calculate the number of electron states in the interval $E$ to $E+dE$. This number can be written $dN = \rho(E)\,dE$, where $\rho$ is the density of states. [from Gasiorowicz]
4. Repeat the above calculation for a two-dimensional electron gas in a square box of area $L^2$.
5. Given that the number density of free electrons in copper is $8.5\times 10^{28}\,{\rm m}^{-3}$, calculate the Fermi energy in electron volts, and the velocity of an electron whose kinetic energy is equal to the Fermi energy. [from Gasiorowicz]
6. Obtain an expression for the Fermi energy (in eV) of an electron in a white dwarf star as a function of the stellar mass (in solar masses). At what mass does the Fermi energy equal the rest mass energy?