B2.22

Applications of Quantum Mechanics | Part II, 2003

The Hamiltonian H0H_{0} for a single electron atom has energy eigenstates ψn\left|\psi_{n}\right\rangle with energy eigenvalues EnE_{n}. There is an interaction with an electromagnetic wave of the form

H1=erϵcos(krωt),ω=kc,H_{1}=-e \mathbf{r} \cdot \boldsymbol{\epsilon} \cos (\mathbf{k} \cdot \mathbf{r}-\omega t), \quad \omega=|\mathbf{k}| c,

where ϵ\boldsymbol{\epsilon} is the polarisation vector. At t=0t=0 the atom is in the state ψ0\left|\psi_{0}\right\rangle. Find a formula for the probability amplitude, to first order in ee, to find the atom in the state ψ1\left|\psi_{1}\right\rangle at time tt. If the atom has a size aa and ka1|\mathbf{k}| a \ll 1 what are the selection rules which are relevant? For tt large, under what circumstances will the transition rate be approximately constant?

[You may use the result

sin2λtλ2dλ=πt.]\left.\int_{-\infty}^{\infty} \frac{\sin ^{2} \lambda t}{\lambda^{2}} d \lambda=\pi|t| . \quad\right]

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