Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 15A (Quantum Mechanics)

For $\psi =A{e}^{-B{x}^2}$,Substitution into the Schrodinger equation and comparison of the constant and $x^2$ coefficients gives

Put $B=\xi \tan (\varphi +\alpha t)$. Matching the constant and ${\tan }^2$ coefficients in the Riccati equation requiresOne convenient choice is thereforeso

Since $A^{\prime }/A=-i\hslash B=-\frac{i}{2}\tan (\varphi -it)$ andintegration giveswith the branch chosen continuously from the initial value. The constant $A_0$ fixes normalization.

Normalizability requires $\mathrm{Re}B>0$. Sincethe given Gaussian integral givesAlso ${\psi }_x=-2Bx\psi$, so integration by parts yieldsThereforeThese are the second moments of a complex Gaussian wave packet.