Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S3 / 2 / b / Solution

For the Gaussian, ${\varphi }_0^{\prime \prime }=(x^2/{\sigma }^4-1/{\sigma }^2){\varphi }_0$. Using $\hslash /(m\omega )={\sigma }^2$, the $x^2$ terms cancel and leave $E_0=\hslash \omega /2$. For ${\varphi }_1\propto x{\varphi }_0$, differentiation gives ${\varphi }_1^{\prime \prime }=(x^2/{\sigma }^4-3/{\sigma }^2){\varphi }_1$, and the same cancellation gives $\boxed{E_1=3\hslash \omega /2}$. The normalization constants do not affect these eigenvalue equations.