 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 1

The one-particle partition <function> is
$$
z_1=\frac{A}{h^3}\int_{\mathbb R^3}e^{-\beta p^2/(2m)}\,d^3p
\int_0^\infty e^{-\beta mgz}\,dz
=\frac{A}{\lambda^3\beta mg},
$$
where $\lambda=(2\pi\hbar^2\beta/m)^{1/2}$. Thus
$$
Z_N=\frac{z_1^N}{N!},\qquad
\log z_1=\text{constant}-\frac52\log\beta.
$$
Canonical <differentiation> now gives
$$
\langle E\rangle=-\partial_\beta\log Z_N=\frac{5N}{2\beta}
=\frac52Nk_BT
$$
and
$$
(\Delta E)^2=\partial_\beta^2\log Z_N
=\frac{5N}{2\beta^2}.
$$
Therefore
$$
\frac{\Delta E}{\langle E\rangle}=\sqrt{\frac{2}{5N}}.
$$
The factor $5/2$ consists of $3/2$ from <kinetic energy> and $1$ from gravitational <potential energy>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 36C / b / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 36C / b / i / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 36C / b / i / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)