The energy eigenvalues of an \(s\)-dimensional harmonic oscillator can be written as

\[
\varepsilon_{j}=(j+s / 2) \hbar \omega ; \quad j=0,1,2, \ldots
\]

Show that the \(j\) th energy level has a multiplicity \((j+s-1) ! / j !(s-1) !\). Evaluate the partition function, and the major thermodynamic properties, of a system of \(N\) such oscillators, and compare your results with a corresponding system of \(s N\) one-dimensional oscillators. Show, in particular, that the chemical potential \(\mu_{s}=s \mu_{1}\).