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 Incoming links: Radial Dirichlet modes of a spherical cavity

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 10 / b / ii / Solution 2026-10-18

The radial Rayleigh quotient is

k_{trial}^{2} = \frac{\int _{0}^{a} r ^{2} ( ψ _{trial}^{'} ) ^{2} d r}{\int _{0}^{a} r ^{2} ψ _{trial}^{2} d r} .

For the given trial profile, the numerator is

4 a^{- 4} \int_{0}^{a} r^{4} d r = 4 a /5

. The denominator is

\int_{0}^{a} r^{2} (1 - r^{2} / a^{2})^{2} d r = a^{3} (\frac{1}{3} - \frac{2}{5} + \frac{1}{7}) = \frac{8 a ^{3}}{105} .

Therefore

k_{0}^{2} \leq k_{trial}^{2} = \frac{21}{2 a ^{2}} = \frac{10.5}{a ^{2}} .

The profile is regular at the origin and vanishes at the wall, so it belongs to the admissible trial space. This is the variational estimate for the radial Dirichlet modes of a spherical cavity.

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