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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 3

The number of <momentum> states in a shell is
$$
\frac{g_sV}{(2\pi\hbar)^d}S_{d-1}p^{d-1}dp.
$$
With $E=p^2/(2m)$ this gives
$$
g(E)=g_sBVE^{(d-2)/2},
\qquad
B=\frac{S_{d-1}m(2m)^{(d-2)/2}}{(2\pi\hbar)^d}.
$$
At zero <temperature> all states up to $E_F$ are occupied, hence
$$
N=\frac{2g_sBV}{d}E_F^{d/2},
\qquad
E_F=\left(\frac{dN}{2g_sBV}\right)^{2/d}.
$$
The energy is $U=dNE_F/(d+2)$. Since a nonrelativistic <ideal> gas in $d$ dimensions obeys $pV=2U/d$,
$$
\boxed{pV=\frac{2}{d+2}NE_F.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 35B / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 35B / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 35B / b / Solution

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