Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 2 / 17B (Quantum Mechanics)

For the quantum harmonic oscillator, the Time-independent Schrodinger equation iswith $m,\omega ,\hslash >0$. The given dimensionless coordinate yields ${\psi }_{\xi \xi }+(2E/(\hslash \omega )-{\xi }^2)\psi =0$. Substituting the Gaussian factor givesBy the supplied normalizability fact, $f$ is a nonzero polynomial. If its degree is $n$, comparison of the leading coefficient forces $2E/(\hslash \omega )-1=2n$. Conversely, at this value the coefficient recurrenceproduces a degree-$n$ polynomial with parity $n$: start with a nonzero constant for even $n$ or a nonzero linear coefficient for odd $n$, and the recurrence terminates at degree $n$. This is a multiple of the Hermite polynomial. Its product with the Gaussian is square integrable and can be normalized. Therefore all and only the allowed energies are

Set $d=qE$. With the stated unit mass and oscillator frequency, complete the square:Translation by $d$ preserves the spectrum of the quantum harmonic oscillator, and the last term is a constant energy shift. The displaced quantum harmonic oscillator therefore hasIts normalized eigenfunctions are the original oscillator eigenfunctions evaluated at $x-d$, in particular the ground state is $(\pi \hslash {)}^{-1/4}{e}^{-(x-d{)}^2/(2\hslash )}$. The energy gaps remain $\hslash$.

The sudden approximation keeps the wavefunction unchanged at the instant of switching: a finite step of potential creates no delta-function impulse in the Schrodinger equation. Just before switching the old ground state differs from the printed real wavefunction only by an overall phase, which does not affect probabilities.

Let ${\tilde{\psi }}_0(x)=(\pi \hslash {)}^{-1/4}{e}^{-(x-d{)}^2/(2\hslash )}$, $d=qE$, be the new ground state. Its overlap with the initial state is, up to that phase,Since $x^2+(x-d{)}^2=2(x-d/2{)}^2+d^2/2$, the Gaussian integral gives $A_0=e^{-d^2/(4\hslash )}$. The Born rule and the ground-state overlap after sudden oscillator displacement thus giveThis is an overlap probability, not a claim that the original wavefunction instantaneously relaxes to the new ground state. It becomes one when $qE=0$.