Past exam of the advanced quantum physics course of the University of Cambridge / 2012 / Paper 1 / 4 / Solution

Insert the expansion in the time-dependent Schrödinger equation, cancel the unperturbed evolution, and project onto $⟨{\psi }_n|$. With $V_{nm}(t)=⟨{\psi }_n|V(t)|{\psi }_m⟩$,In first-order time-dependent perturbation theory, put $c_m(t)={\delta }_{m0}$ on the right and use $c_n(-\infty )=0$ for $n\ne 0$. Integration givesThe initial state must be the specified unperturbed eigenstate, the integral must exist under the switching prescription, and total transition probability must stay small. The diagonal term produces a first-order phase; setting $c_0=1$ in the off-diagonal integral changes those amplitudes only at second order. A weak peak perturbation alone is insufficient if its accumulated action invalidates this expansion.

For the infinite square well, retain the source's zero-based labels:An electron in a field along $+x$ has perturbation $V(t)=+e{E}_{0}x{e}^{-t/\tau }$ for $t\ge 0$. About the well centre, $x-w/2$ is odd, whereas ${\psi }_0$ and ${\psi }_2$ are both even. The constant $w/2$ has zero off-diagonal matrix element, so the first-order parity selection in an infinite well gives $⟨{\psi }_2|x|{\psi }_0⟩=0$. Excitation to that state is forbidden at first order, not necessarily at higher orders.

For the first excited state, direct integration or the supplied integral givesThe Fourier amplitude of the exponentially decaying electric-field pulse is ${\int }_0^{\infty }{e}^{-t/\tau +i\omega t}dt=\tau /(1-i\omega \tau )$. ThusThis is the leading, quadratic-in-field transition probability; the total probability of all excitations must remain small.

Starting instead in ${\psi }_1$, which is odd about the centre, the pulse can transfer amplitude to the ground state and to even-labelled excited states ${\psi }_2,{\psi }_4,\dots$. Other odd-labelled states have zero first-order dipole amplitude. Both downward and upward transitions are possible; their relative probabilities depend on dipole matrix elements and the pulse's spectral weight at the corresponding energy gaps. Most population remains in ${\psi }_1$, with a small depletion equal to the summed leading transition probabilities. The common diagonal dipole phase in an infinite well is $-e{E}_{0}w\tau /(2\hslash )$ and has no population effect. After the pulse, the remaining coherent superposition evolves under the unperturbed Hamiltonian; no spontaneous relaxation is implied by this closed-system model.