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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 1

The normalized one-particle spatial density gives the local number density
$$
\boxed{
n(x)=\frac{N}{4\pi V(k_BT)^{3/(2n)}I_n}
\exp\!\left[-\frac{|x|^{2n}}
 {V^{2n/3}k_BT}\right].}
$$
The radial probability density is proportional to
$$
r^2\exp\!\left[-\frac{r^{2n}}{V^{2n/3}k_BT}\right].
$$
Its logarithmic <derivative> vanishes when
$$
\frac2r-\frac{2n r^{2n-1}}{V^{2n/3}k_BT}=0,
$$
so the most likely radius is
$$
\boxed{r_{\rm mode}=V^{1/3}\left(\frac{k_BT}{n}\right)^{1/(2n)}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 36B / c / iv / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 36B / c / iv / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 36B / c / iv / Solution

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