Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 32C / b

Table of contents
- Solution b

Solution

Suppose a normalizable solution satisfies

L ψ = - κ^{2} ψ

. Multiplication by

\overline{ψ}

, integration over

R

, and integration by parts give

- κ^{2} \int_{R} ∣ ψ ∣^{2} d x = \int_{R} \overline{ψ} L ψ d x = \int_{R} p^{2} ∣ ψ^{'} ∣^{2} d x \geq 0.

The boundary term vanishes for a normalizable eigenfunction. The left side is negative unless

ψ = 0

, so no nontrivial normalizable solution exists.

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