previousnextindexPhotoelectric Effect in Hydrogen Michael Fowler 3/26/10 Introduction In the photoelectric effect, incoming light causes an atom to eject an electron. We consider the simplest possible scenario: that the atom is hydrogen in its ground state. The interesting question is: for an ingoing light wave of definite frequency and amplitude, what is the probability of ionization of a hydrogen atom in a given time? In other words, assuming we can use time-dependent perturbation theory, what is the ionization rate? Formally, we know what to do. We must find the interaction Hamiltonian 1H, then use Fermi’s Golden Rule for the transition rate with a periodic perturbation: ()212iffiRf HiEEπδω→=−−But it’s not that easy! For one thing, the outgoing electron will be in some kind of plane wave state, so whatever convention we adopt for normalizing such states appears in the rate. But also the δfunction is tricky for excitation into the continuum: just how many of these plane wave states satisfy fiEEω=+? We shall discover that with a consistent formalism, these two difficulties cancel each other. The Interaction Hamiltonian Taking the incoming wave to be an electromagnetic field having vector potential ()()0,cosA r tAk rtω=⋅−The interaction Hamiltonian is given by replacing the electron kinetic energy term 2/ 2pmwith ()2// 2pqA cm−. The relevant new term is ()()()()1/ 2//mq cp AA pe mc A p−⋅+⋅=since q= −eand 0A∇⋅=in our gauge. Therefore ()()()()100cos.2i k rti k rteHk rtApmceeeApmcωωω⋅ −−⋅ −=⋅−⋅=+
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