Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A1 / b / iv

Table of contents
- Solution iv

Solution

For this quantum harmonic oscillator,

\hat{H}^{(1)} = x^{4}

. The Gaussian integral gives

E_{0}^{(1)} = \frac{1}{π} \int_{- \infty}^{\infty} x^{4} e^{- x^{2}} d x = \frac{1}{π} \frac{3 π}{4} = \frac{3}{4} .

The distinction between the coefficient and the actual first-order shift matters: the energy shift is $3 λ /4$ , so

E_{0} = 1/2 + 3 λ /4 + O (λ^{2})

in the stated atomic units.

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