I'm struggling with solving this question:
Show that the entropy per particle ($\ S/N$) is only a function of $\mu /KT$ for an ideal gas of Fermions and Bosons.
Assume that the gas is not relativistic, with known fixed values of $\ V \ (volume)\ ,\ T \ (teperature)\ ,\ m \ (mass)\ ,\ N \ (number\ of\ particles) $.
You don't need to solve explicitly.
Can someone help me with this questions?
Thank you!
The Hamiltonian of an ideal bose/fermion gas: $$\hat{H}=\frac{\hat{P}}{2m}$$ Then you can compute the great partition function: $$Z = Tr \ {e^{-\beta(\hat{H}-\mu \hat{N})}}$$ and after that the grand potential: $$\Omega = -kT\ln Z$$ When you have this you can easly compute the entropy with: $$S=-(\frac{\partial \Omega}{\partial T})_{V,\mu}$$ I hope this helps, there are intermediate steps but this is only an outline. If you want more details I could recommend reading eather David Tong's lecture notes (a bit easier) or more advanced Feynman's Statistical Mechanics.
