36 - Physics 6210/Spring 2007/Lecture 36 Lecture 36...

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Unformatted text preview: Physics 6210/Spring 2007/Lecture 36 Lecture 36 Relevant sections in text: § 5.7 What happens when you shine light on an atom? You will have noticed that up to this point in our discussion of time-dependent pertur- bation theory I have assiduously avoided much in the way of concrete, specific applications of our results. This was partially because you will do the relatively easy applications in your homework, while the more involved applications would distract too much from the general features of the perturbative approach. Also, I wanted to use the perturbative point of view not just to solve this or that example situation, but rather to give a general point of view on quantum dynamics that is available whenever the perturbative approximation is valid. Now, though, we should have a look at a good example where this technology is useful. We shall study the perturbative dynamics of an atomic electron when exposed to a (weak) plane electromagnetic wave. This is a simple model which can be used to answer (at least in part) the question posed in the title of this section. The story is a little long, but I think it is useful and instructive. We are principally interested in the case where the electron is, in the absence of the perturbation, in a “bound stationary state” of some potential, with the electromagnetic field serving to stimulate transitions between the bound or ionized states. Our strategy is as follows. We model the atom as a spinless particle bound by some potential V . Of course, more sophisticated models of the atom are available. We could specialize V to be a central potential, or even the Coulomb potential, but we won’t have to make any such choices for a while. Thus, the unperturbed Hamiltonian is of the form H = P 2 2 m + V ( ~ R ) . The unperturbed stationary states are just the energy eigenstates of H — these are the “energy levels” of the atom. Now we want to introduce the electromagnetic radiation. The total electromagnetic field that the particle interacts with will then be the superposition of the field represented by V ( e.g., an electrostatic field) with the field of the incident radiation. Let us describe the radiation using the vector potentials ( φ ( ~ r,t ) , ~ A ( ~ r,t )). It is a standard result of elec- tromagnetic theory that, in regions of space free of electromagnetic sources, it is always possible to choose the electromagnetic potentials so that they are in the “radiation gauge”: φ = 0 , ∇ · ~ A = 0 ....
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36 - Physics 6210/Spring 2007/Lecture 36 Lecture 36...

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