Solution

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

For
$$
H=\frac{p_x^2+p_y^2}{2m}+\frac\kappa2(x^2+y^2),
\qquad x^2+y^2<R^2,
$$
put $a=\kappa R^2/2$. The normalized classical one-particle partition function is
$$
Z_1=\frac{4\pi^2m}{h^2\beta^2\kappa}
\left(1-e^{-\beta a}\right).
$$
Its mean total and potential energies are
$$
\langle E\rangle=\frac2\beta-\frac a{e^{\beta a}-1},
\qquad
\langle V\rangle=\frac1\beta-\frac a{e^{\beta a}-1}.
$$

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 Incoming links: Particle in a finite two-dimensional harmonic trap