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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 1

The classical <canonical partition function> uses one phase-space cell $h^2$ per state. In <polar coordinates>,
$$
\begin{aligned}
Z_1
&=\frac1{h^2}
\int_{\mathbb R^2}e^{-\beta p^2/(2m)}\,d^2p
\int_{r<R}e^{-\beta\kappa r^2/2}\,d^2x\\
&=\frac1{h^2}
\left(\frac{2\pi m}{\beta}\right)
\left(2\pi\int_0^Rr e^{-\beta\kappa r^2/2}\,dr\right)\\
&=\boxed{
\frac{4\pi^2m}{h^2\beta^2\kappa}
\left(1-e^{-\beta\kappa R^2/2}\right)
}.
\end{aligned}
$$
Equivalently, using $h=2\pi\hbar$,
$$
Z_1=\frac{m}{\hbar^2\beta^2\kappa}
\left(1-e^{-\beta\kappa R^2/2}\right).
$$
This is the <particle in a finite two-dimensional harmonic trap> partition function.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 36A / b / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 36A / b / i / Solution

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