8. Atoms in Electromagnetic Fields Our goal in this chapter is to understand how atoms interact with electromagnetic fields. There will be several stages to our understanding. We start by looking at atoms in constant, background electromagnetic fields. Because these fields break various symmetries of the problem, we expect to see a splitting in the degeneracies of states. The splitting of the atomic spectrum due to an electric field is called the Stark effect. The splitting due to a magnetic field is called the Zeeman effect. We deal with each in turn. We then move on to look at what happens when we shine light on atoms. Here the physics is more dramatic: the atom can absorb a photon, causing the electron to jump from one state to a higher one. Alternatively the electron can decay to lower state, emitting a photon as it falls. We will begin with a classical treatment of the light but, ultimately, we will need to treat both light and atoms in a quantum framework. 8.1 The Stark Effect Consider the hydrogen atom, where the electron also experience a constant, background electric field. We'll take the electric field to lie in the z direction, E = E z. The Hamiltonian is H = ?2 2m 2 (8.1) 4000r e2 + eEz The total potential energy, V (z) = eEz-e2/40lor is sketched in the diagram. The first thing to note is that the potential is unbounded below as z -> -0. This means that all electron bound states, with wavefunctions lo- calised near the origin, are now unstable. Any electron can tunnel through the barrier to the 4 V left, and then be accelerated by the electric field to z -> -oo. However, we know from our WKB analysis in Section 6.2.5 that the probability rate Figure 81: for tunnelling is exponentially suppressed by the height of the barrier (see, for exam- ple, (6.30)). This means that the lowest lying energy levels will have an extremely long lifetime. /V(z) z - 217 -
If you want some numbers, the strength of a typical electric field is around E ~ 10 eV cm11. We know that the ground state of hydrogen is Eo ~ - 13.6 eV and the Bohr radius is do ~ 5 x 1009 cm, which suggests that the typical electric field inside the atom is around Eatom ~ 109 eV cm01, which is eight orders of magnitude greater than the applied electric field. On general, ground we expect that the tunnelling probability is suppressed by a factor of e010°. At this point is doesn't really matter what our units are, this is going to be a very small number. The states which are well bound are stable for a very long time. Only those states very close to threshold are in danger of being destabilised by the electric field. For this reason, we'll proceed by ignoring the instability. 8.1.1 The Linear Stark Effect We're going to work in perturbation theory. Before we look