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

Table of contents
- Solution ii

Solution

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.

 Ancestors (13)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version