Quantization of the Electromagnetic Field

Since the energy of the electromagnetic field fluctuates during one cycle, it is the 1 cycle-averaged energy HR that represents the energy of the field after many cycles. Hence, this is the quantity that we now proceed to quantize. We do so, in the 4 coordinate representation, by replacing the classical momenta with the operators r7  

As a result, the radiative Hamiltonian UIt [Eq. (12.3)] assumes the quantized formff 

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Quantization of the Electromagnetic Field.—Instead of proceeding as in the previous discussion of spin 0 and spin particles, we shall here adopt essentially the opposite point of view. Namely, instead of formulating the quantum theory of a system of many photons in terms of operators and showing the equivalence of this formalism to the imposition of quantum rules on classical electrodynamics, we shall take as our point of departure certain commutation rules which we assume the field operators to satisfy. We shall then show that a... 

Dirac s density matrix, 422 Dirac s quantization of the electromagnetic field, 485... 

It is possible to go beyond the dipole approximation in the length gauge and treat the interactions between higher multipoles with the field derivatives, which is relevant when the variation of the field with ry- cannot be neglected [3], However, we do not pursue these extensions here because, in all the applications discussed below, the dipole approximation will be found to suffice. Equations (1.50), (1.51), and (1.52) are the central expressions used below to describe molecule-light interactions. Extensions of this approach to include quantization of the electromagnetic field are described in Chapter 12. 

The Hamiltonian (16.88) describes a system of classical harmonic oscillators. It is reasonable to assume that it represents a classical approximation to a more fundamental quantum theory. Such a theory may be obtained in analogy to the quantization of the electromagnetic field carried in Section 3.2.2, by quantizing ... 

The further developments of quantum mechanics, including the discussion of maximal measurements consisting not of the accurate determination of the values of a minimum number of independent dynamical functions but of the approximate measurement of a larger number, the use of the theory of groups, the formulation of a relativistically invariant theory, the quantization of the electromagnetic field, etc., are beyond the scope of this book. 

In order to limit the size of the book, we have omitted from discussion such advanced topics as transformation theory and general quantum mechanics (aside from brief mention in the last chapter), the Dirac theory of the electron, quantization of the electromagnetic field, etc. We have also omitted several subjects which are ordinarily considered as part of elementary quantum mechanics, but which are of minor importance to the chemist, such as the Zeeman effect and magnetic interactions in general, the dispersion of light and allied phenomena, and most of the theory of aperiodic processes. 

The basic expression for the quantization of the electromagnetic field is the expansion Eq(54). In the quantized theory the numbers Ck,, C x become operators of the creation C x and the annihilation Ck,x of photons. These operators are acting on the state vector < ) that is defined in the Fock space (occupation number space). The C xt Ck, operators satisfy the commutation relations ... 

While there are serious difficulties to define a relativistic many-electron Hamiltonian, its non-relativistic limit is, fortunately, well-defined, because in this limit the magnetic interaction of the moving electrons and the retardation of the Coulomb interaction vanish, and there is no quantization of the electromagnetic field (and, of course, no absorption or emission of light). So if relativistic effects are small, one is close to a well-defined situation, namely the non-relativistic limit, and one need not worry much about the appropriate choice of a relativistic many-electron Hamiltonian. 

The formulation of a relativistic n-electron Hamiltonian is much more problematic, since one must combine the theory of the Dirac field with the full electromagnetic field, and quantize both fields. The construction of an n-electron Hamiltonian in this context is outside of the scope of this chapter. However, we can consider approximations to the exact electron interaction, which are valid to low orders in c and the derivation of which does not require the quantization of the electromagnetic field. 

So far we have treated absorption and stimulated emission of radiation. However, it is well known that an atom can emit radiation even when it is not externally perturbed, i.e. spontaneous emission. It is not possible to treat this process fully here, since consideration of the quantization of the electromagnetic field as described by Quantum ElectroDynamics (QED) is necessary. According to QED a coupling between the atom and the "vacuum state" of the field is responsible for the emission. 

Structure resulting from the examination of different samples. It should be noted that only one part in 10 of the electrons in a solid directly participate in the isomer shift the nuclear parameter (r -r ) is of the order of 10" cm. The isomer shift is four orders of magnitude smaller than the Lamb shift caused by quantization of the electromagnetic field. 

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