Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 33A (Principles of Quantum Mechanics)

Insert the stated expansion into the Time-dependent Schrodinger equationWhen the derivative acts on each free phase, its energy factor is cancelled by the action of $H_0$ on the corresponding eigenstate. The remaining terms obeyDivision by $i\hslash$ gives

To first order in $F$, time-dependent perturbation theory replaces the state on the right-hand side by its unperturbed value $e^{-i{E}_{0}t/\hslash }|0⟩$. The bound state is even, while the position operator $x$ is odd. The parity selection rule therefore givesso ${\dot{b}}_k=0$ and, with no initial continuum component,

Projection onto the odd state givesWriting ${\Omega }_k=(E_k-E_0-\hslash \omega )/\hslash$ and integrating from zero to $t$ yields

The required dipole matrix element iswhere differentiating the elementary Laplace integral for $\sin (kx)$ evaluates the last integral. Hence

Put $E_f=E_0+\hslash \omega >0$ and $k_f=\sqrt{2m{E}_f}/\hslash$. The continuum transition probability at long times usesSincethe total escape probability becomesFor the attractive delta potential,Substitution gives the requested leading-order result: