Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 1 / A2 / c / iv

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- Solution iv

Solution

Integration by parts and the zero endpoint values turn the Rayleigh quotient into a positive kinetic energy integral:

E = \frac{ℏ ^{2}}{2 m} \int_{0}^{a} ∣ ψ^{'} (x) ∣^{2} d x = \frac{ℏ ^{2}}{2 m} \frac{30}{a ^{5}} \int_{0}^{a} (a - 2 x)^{2} d x = \frac{5 ℏ ^{2}}{m a ^{2}} .

This is the parabolic variational trial for a particle in a box. The potential energy vanishes inside the box.

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