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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 3

At zero temperature all states up to $E_F$ are occupied. Writing
$$
g=\frac{Am}{\pi\hbar^2},
$$
we find
$$
N=\int_0^{E_F}g\,dE=gE_F
$$
and
$$
E_{\rm tot}=\int_0^{E_F}Eg\,dE
=\frac12gE_F^2
=\boxed{\frac12NE_F}.
$$

Because $g=CA$ with $C=m/(\pi\hbar^2)$, fixed particle number gives $E_F=N/(CA)$ and
$$
E_{\rm tot}=\frac{N^2}{2CA}.
$$
The two-dimensional <degeneracy pressure of a two-dimensional Fermi gas> is therefore
$$
\boxed{p=-\left(\frac{\partial E_{\rm tot}}{\partial A}\right)_N
=\frac{N^2}{2CA^2}
=\frac{NE_F}{2A}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 35C / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 35C / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 35C / b / Solution

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