Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 14B (Quantum Mechanics)

For the one-dimensional quantum harmonic oscillator,This is the Time-dependent Schrodinger equation with harmonic potential.

Fordirect differentiation givesandSubstitution into the Schrodinger equation and equality of the coefficients of $x^2,x,1$ give the necessary and sufficient Riccati equation system

Put $u=a+i\omega t$. ThenThe $G$ equation from part (b) is satisfied whensoWith this value, the $F$ equation follows from ${\tanh }^{2}u=1-{\mathrm{sech}}^{2}u$: both its constant and ${\mathrm{sech}}^{2}u$ coefficients agree. Thus the stated $F,G$ satisfy both shape equations; the remaining scalar equation determines the normalization factor $N(t)$.

The probability density isCompleting the square shows that, when $\mathrm{Re}F>0$,where

Writing $\theta =\omega t$ andthe supplied identities giveUsing $A=m\omega /(2\hslash )$ in part (d),The peak executes simple harmonic motion about the origin with angular frequency $\omega$ and amplitude $\sqrt{\hslash /(m\omega )}/\sinh a$.

At each fixed time, $|\psi (x,t){|}^2$ is a bell-shaped Gaussian centred at $x=h(t)$, with width determined by $\mathrm{Re}F(t)>0$. Completing the square as in part (d) makes the density symmetric about $h(t)$. Therefore its position expectation value isThe sketch is a Gaussian peak whose centre oscillates between the two turning positions found in part (e), while its width varies periodically through $\mathrm{Re}F(t)$.