 Solution

ID: 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

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

\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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