Semiclassical radiation theory

In this chapter we introduce some of the fundamental concepts needed to understand how light interacts with matter. We start by examining a system of classical charged particles that interacts with a pulse of electromagnetic radiation. We then quantize the particle variables and develop the semiclassical theory of light interacting with quantized particles. The details of the derivations are not required for subsequent chapters. However, the resultant equations [Eqs. (1.50) to (1.52)] form the basis for the theoretical development presented in Chapter 2, which deals with both the interaction of weak lasers with molecules and with photodissociation processes. 

In the semiclassical theory of radiation (1, 2) the probability of a transition between an initial state and a final state is obtained by con-... 

Interest in the polarization correlation of photons goes back to the early measurements of the linear polarization correlation of the two photons produced in the annihilation of para-positronium which were carried out as a result of a suggestion by Wheeler that these photons, when detected, have orthogonal polarizations. Yang subsequently pointed out that such measurements are capable of giving information on the parity state of nuclear particles that decay into two photons. In addition, the polarization correlation observed in the two-photon decay of atoms is considered to be one of the few phenomena where semiclassical theories of radiation are inadequate and it is necessary to invoke a full quantum theory of radiation. The effect has also been used to demonstrate the phenomenon of quantum interference. ... 

In agreement with the semiclassical theory of absorption and emission, Eq. (5.36b) indicates that 06 allows transitions between -ipa and ipb only to the extent that it is parallel to the polarization of the radiation (e). 

W. H. Miller, A classical/semiclassical theory for the interaction of infrared radiation with molecular systems, J. Chem. Phys. 69 2188 (1978). 

We now consider the effect of exposing a system to electromagnetic radiation. Our treatment will involve approximations beyond that of replacing (3.13) with (3.16). A proper treatment of the interaction of radiation with matter must treat both the atom and the radiation field quantum-mechanically this gives what is called quantum field theory (or quantum electrodynamics). However, the quantum theory of radiation is beyond the scope of this book. We will treat the atom quantum-mechanically, but will treat the radiation field as a classical wave, ignoring its photon aspect. Thus our treatment is semiclassical. 

The semiclassical treatment just given has the defect of not predicting spontaneous emission. According to (3.13), if there is no outside perturbation, that is, if // (0 = 0, then dcm/dt = 0 for all m if the atom is in the nth stationary state at / = 0, it will persist in that state forever. However, experimentally we find that unperturbed atoms in excited states spontaneously radiate energy and drop to lower states. Quantum field theory does predict spontaneous emission. Since quantum field theory is beyond us, we shall use an argument given by Einstein in 1917 to find the spontaneous-emission probability. 

Nesbet, R.K. (1971). Where semiclassical radiation theory fails, Phys. Rev. Lett. 27, 553-556. 

In the first part of this introductory section, we summarize the main collective phenomena acquired by the dipolar exciton from the lattice-symmetry collectivization of molecular properties. The crystal is considered as an assembly of electrically neutral systems, the molecules, physically separated from each other and in electromagnetic interaction. This /V-body problem will be treated quantum-mechanically in the limit of low exciton densities. We redemonstrate the complete equivalence of this treatment with the theories of Lorentz and Ewald, as well as with the semiclassical approximation. In Section I.A, in a more compact but still gradual way, we establish the model of the rigid lattice of dipoles and the general theory of low-exciton-density systems in interaction with the radiation field. Coulombic excitons, photons,... 

The interaction between electromagnetic radiation and atoms or molecules is now discussed by empirical methods, then by semiclassical arguments, and finally by quantum theory. 

The first theoretical model of optical activity was proposed by Drude in 1896. It postulates that charged particles (i.e., electrons), if present in a dissymmetric environment, are constrained to move in a helical path. Optical activity was a physical consequence of the interaction between electromagnetic radiation and the helical electronic field. Early theoretical attempts to combine molecular geometric models, such as the tetrahedral carbon atom, with the physical model of Drude were based on the use of coupled oscillators and molecular polarizabilities to explain optical activity. All subsequent quantum mechanical approaches were, and still are, based on perturbation theory. Most theoretical treatments are really semiclassical because quantum theories require so many simplifications and assumptions that their practical applications are limited to the point that there is still no comprehensive theory that allows for the predetermination of the sign and magnitude of molecular optical activity. 

E.T. Jaynes, F.W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51 (1963) 89. 

I.R. Senitzky Semiclassical radiation theory within a quantum mechanical framework . In Progress in Optics 16 (Noith-Holland, Amsterdam 1978) p. 413... 

Unfortunately, Kocher and Commins only made measurements for relative angles of 0° and 90° between the transmission axes of the polarizers and, in addition, the transmission characteristics of their polarizers did not satisfy the criterion illustrated in Figure 4 for a satisfactory test of the BCHSH inequality. However, Clauser was able to use their result to demonstrate the inadequacy of semiclassical radiation theory in this situation and to show that the Schrodinger-Furry hypothesis was not tenable. 

Milonni PW (1976) Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory. Phys Rep 25 1... 

The broadening Fj is proportional to the probability of the excited state k) decaying into any of the other states, and it is related to the lifetime of the excited state as r. = l/Fj . For Fjt = 0, the lifetime is infinite and O Eq. 5.14 is recovered from O Eq. 5.20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (O Eq. 5.4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes caimot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, aU decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions (1) all broadening parameters are assumed to be identical, Fi = F2 = = r, and (2) the value of F is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in O Eq. 5.4 with the substitution to to+iTjl, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. 

The microscopic (hyper)polarizabilities are studied by means of the so-called theory of the response functions which is of importance for aU molecular and cluster entities (Roman et al. 2006). The most commonly used approach in studying the linear and nonlinear optical properties of clusters is the so-caUed semiclassical one. According to this approach a classical treatment is used to describe the response of the cluster to an external field (radiation) while the system itself is treated using the lows and techniques of quantum mechanics. This is done by using a Hamiltonian which combines both of the above treatments ... 

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