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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/10/b/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 10 / b / Solution by

Codex  0 2026-10-18

Use the normalized eigenfunctions and their weighted inner product

\int_{a}^{b} w \overline{ψ_{m}} ψ_{n} d x = δ_{mn}

. Orthogonality gives

G [ψ] = \sum_{n \geq 0} ∣ a_{n} ∣^{2}, F [ψ] = \int_{a}^{b} \overline{ψ} L ψ d x = \sum_{n \geq 0} λ_{n} ∣ a_{n} ∣^{2} .

For complex trials,

F

means the corresponding Hermitian quadratic form with

∣ ψ^{'} ∣^{2}, ∣ ψ ∣^{2}

; it agrees with the displayed functional on real trials. If

a_{0} \neq = 0

∣ a_{0} ∣^{2}

yields

Λ [ψ] = \frac{λ _{0} + \sum _{n \geq 1} ∣ b _{n} ∣ ^{2} λ _{n}}{1 + \sum _{n \geq 1} ∣ b _{n} ∣ ^{2}}, b_{n} = \frac{a _{n}}{a _{0}} .

Since

λ_{n} \geq λ_{0}

, this is at least

λ_{0}

. The Rayleigh-Ritz variational principle therefore gives an upper bound from any admissible trial, improved by minimizing over trial parameters or enlarging the trial space. Equality holds when the trial is entirely in the lowest eigenspace.

The displayed ratio using

b_{n}

presupposes

a_{0} \neq = 0

. If

a_{0} = 0

, use the undivided quotient

\sum λ_{n} ∣ a_{n} ∣^{2} / \sum ∣ a_{n} ∣^{2}

instead; the same upper-bound argument still applies. No ground-state coefficient can be divided out in that case.

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