In this chapter, the Lagrangian obtained in the last chapter is used to show how atomic decay is explained by field theory. Then the famous Lamb shift is calculated and discussed. The Lamb shift is the splitting between atomic levels with the same total angular momentum but different orbital angular momentum and cannot be explained without the use of field theory. The largest such splitting is between the 2Sx/2 and 2Pi/2 levels and is a noticeable feature of the hydrogen atom spectrum. Finally, we calculate the photodisintegration of the deuteron, one of the first examples of the conversion of energy to mass. To set the stage for these calculations, the chapter begins with a discussion of how to determine the time evolution operator in a case when the Hamiltonian depends on time.

Since the interaction Hamiltonian is, in general, time-dependent, we will calculate the interaction between nonrelativistic systems and the quantized EM radiation field using time-dependent perturbation theory.* For definiteness, the nonrelativistic system will be taken to be a heavy atomic nucleus with charge Z at rest at the origin and a single electron of mass m with a negative point charge located at re (other systems will be discussed in Sec. 3.5). The charge distribution for these two particles, in the language of Eq. (2.27), is therefore

p„(\r\) = Ze63(r) , and the only particle coordinates we need to consider are those of the electron. The Hamiltonian given in Eq. (2.31) can therefore be broken up into three parts:

*The particles in this chapter are treated nonrelativistically, but the derivation of the time evolution operator is completely general, and the results we obtain here will be applied, in Chapter 9, to relativistic systems.

where

where a = e2/4n is the fine structure constant. The first two terms, Ha + Hem. will be considered the unperturbed Hamiltonian, with H\ the perturbation. Note that we have included the Coulomb interaction term [the third term in Eq. (2.31)] in HA because we intend to develop the perturbation theory in terms of atomic wave functions, which include the (nonrelativistic) Coulomb interaction to all orders (exactly). We have omitted the Coulomb self-energies of the atomic nucleus and the electron; for point particles these are infinite constants which may be subtracted by a convenient definition of the energy [as discussed following Eq. (2.31)]. The interaction term H'j is the expansion of the familiar (p-eA)2 factor and includes a term which is first order in the electron charge e and linear in A and a second order term proportional to A2.

First, consider the case when the interaction term is zero. Then the EM field coordinates, which are the vector potential operators A, are contained only in Hem, and the electron coordinates, r, are contained only in Ha, which is the usual Schrodinger Hamiltonian. We found the quantum mechanical eigenstates for the free EM field in Chapter 2; the solutions are a Fock space of photon states vhich are time-independent. The eigenstates of Ha are also known from previous studies of nonrelativistic quantum mechanics; the bound states of hydrogen-like atoms can be described by wave functions where a labels the quantum numbers of the bound state. These states evolve in time by a phase factor only, in the sense that

This expression is similar to Eq. (1.31) with the choice tQ = 0. [Any time to could be chosen, but this choice corresponds to the usual phase convention in which atomic states are real when t = 0.]

In Chapters 1 and 2 we used the Heisenberg representation for the fields, while the atomic wave functions are usually given in the Schrodinger representation. It is more convenient to choose a common representation for all fields and operators, and in the remainder of this book we will use the interaction representation. In this representation the time dependence of the free, non-interacting Hamiltonian

3.1 TIME EVOLUTION AND THE S-MATRIX 59

is in the operators, and under the influence of the free Hamiltonian, the states will not evolve in time. However, under the full Hamiltonian, which includes an interaction term, the states will evolve in time, and the principal goal of this section is to calculate this evolution. But before we proceed with this calculation, we must give the electron operators the time dependence associated with the free Hamiltonian. This means that, instead of using H' given above, we will use

Since Ha commutes with itself and the EM field operators A, Ha and Hem are unaffected by this transformation, but H\ becomes H], where

H = Ha + Hem + Hj(t) = H0 + H^t) Hj{t) = UX11^ (pe •A(re,t) + A(re,t) -pe) + |-A2(re,i)} UA

The solutions to the free Hamiltonian Hq — Ha + Hem are just direct products of atomic wave functions and photon states, which we will write

where a labels the atomic states and | n) the photon Fock states as described (for the string) in Eq. (1.20). The scalar product of the atomic states requires an ntegration over the coordinate re,

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