Quantization of electromagnetic field

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  

The longitudinal polarization j = 3 is added to Eq(67) for convenience. Now it is possible to introduce the spin operator for a single photon  

The operators Si satisfy the standard commutation relation for the angular momentum operators  

Unlike the electron spin, the photon spin cannot be interpreted as the total angular momentum in the rest reference frame there is no such a frame for the photon. Thus the photon spin cannot be separated from the orbital angular momentum. However the photon spin projection quantum number /i may have 

The coefficient by the operator (7 in the quantized expansion Eq(54) can be interpreted as the photon wave function  

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In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

In 1900 Rayleigh introduced density of electromagnetic modes in the theory of equilibrium electromagnetic radiation [16]. In 1916 Einstein showed that the ratio of spontaneous to stimulated emission coefficients was /zta3/ji2c3. Then in 1927 Dirac [18] introduced the quantization of electromagnetic field and showed that for the Einstein relationship to be fulfilled the spontaneous emission rate should be proportional to the number of modes available for light quanta to be emitted. Later, in solid state theory concept of the density of modes was developed with respect to electrons and other elementary excitations and evolved towards a consistent density of states (DOS) inherent in every quantum particle of matter. The notion of local density of states was introduced in complex solids. 

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

It is very well known that Einstein, developing Planck s ideas, quantized the electromagnetic field by introducing a quantum particle named the photon. Consequently, each mode or state of a classical electromagnetic field is characterized by an angular frequency, co, and a wave vector,... 

To quantize the dynamics of the particles first requires that we express the velocities of the particles in terms of canonical momenta. In the presence of electromagnetic fields, the canonical momenta are not merely m dx-Jdt). Rather, in order to incorporate Lorentz s velocity-dependent forces into Hamilton s formulation of classical mechanics, the canonical momenta are given by [2]... 

The radiation field is an ensemble of photons with an energy of ho) (o) = c A ) and a momentum of hk. By quantizing the electromagnetic field we have been able to introduce the concept of a photon (number). The vector potential can be rewritten as Eq. (1.108) by using creation and annihilation operators. 

Since we are interested mainly in the roots of spontaneous emission, we shall quantize the electromagnetic field because we know that the semi-classical description where the atom is quantized and the field is classical, does not provide aity spontaneous emission it is introduced phenomenologically by a detailed balance of the population of the two-states atom and comparison with Planck s law. This procedure introduced by Einstein gave the well-known relationship between induced absorption (or emission), and spontaneous emission probabilities, the B12, B21 and A21 coefficients, respectively, but caimot produce the coherent aspect and its link with spontaneous emission. 

Reference [7] provides the detailed analysis of the peculiarities of excited nucleus spontaneous gamma decay both for various types of environment (screen and the system of excited nucleus atom electrons) and accounting for the multimode structure and phase characteristics of the quantized electromagnetic field. It was shown for the first time that the form, distance, and spectral characteristics of the screen most essentially influence the nuclei decay rate. It was also shown that the spontaneous decay character is considerably influenced by the degree of correlation and synchronization of the modes of electromagnetic field surrounding the excited nucleus. Such effect is possible only for the system of resonant Mossbauer nuclei. 

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... 

It should be stressed, however, that the introduction of the operator 2(k) in the present context is purely for mathematical convenience. All the subsequent development could also be carried out without its introduction. It is only when we consider the interaction of the quantized electromagnetic field with charged particles that the potentials assume new importance—at least in the usual formulation with its particular way of fixing the phase factors in the operators of the charged fields—since the potentials themselves then appear in the equations of motion of the interacting electromagnetic and matter fields. 

Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

For a discussion of the quantized electromagnetic field interacting with a given (prescribed) external current, see ... 

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

Eigenstates of a crystal, 725 Eigenvalues of quantum mechanical angular momentum, 396 Electrical filter response, 180 Electrical oscillatory circuit, 380 Electric charge operator, total, 542 Electrodynamics, quantum (see Quantum electrodynamics) Electromagnetic field, quantization of, 486, 560... 

Statistical properties of light are described within the framework of quantum optics which is based on a quantized description of the electromagnetic field. In section 21.2 we will depict specific experimenfs which have been performed fo show fhaf a quanfum description is necessary in some cases. We will describe in Section 21.3 fhe sfandard fools for fhe analysis of fhe sfafisfical properties of lighf and give fhe resulfs obfained for a number of sources. 

The authors of Ref. [12] reconsidered the problem of magnetic field in quark matter taking into account the rotated electromagnetism . They came to the conclusion that magnetic field can exist in superconducting quark matter in any case, although it does not form a quantized vortex lattice, because it obeys sourceless Maxwell equations and there is no Meissner effect. In our opinion this latter result is incorrect, since the equations for gauge fields were not taken into account and the boundary conditions were not posed correctly. 

As a first step, the treatment in this chapter is limited to electromagnetic field theory in orthogonal coordinate systems. Subsequent steps would include more advanced tensor representations and a complete quantization of the extended field equations. 

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