Quantum Theory of Electromagnetic Radiation

The Schrodinger equation was first applied to electromagnetic radiation in 1927 by Paul Dirac [26]. The notion of a quantized radiation field that emerged from this work reconciled some of the apparent contradictions between earlier wave and particle theories of light, and as we will see in Chap. 5, led to a consistent explanation of the spontaneous fluorescence of excited molecules. 

To put the energy of the radiation field in Hamiltonian form, we now define two 

A little algegra wiU show that Qj obeys a classical wave equation homologous to Eqs. (3.13) and (B3.1.7), and that Qj and Pj have the formal properties of a time-dependent position Qj) and its conjugate momentum Pj) in Hamiltonian s classical equations of motion  

In addition, Dirac noted that Eq. (3.48) is identical to the classical expression for the energy of a harmonic oscillator with unit mass (Eq. 2.28). The first term in the braces corresponds formally to the kinetic energy of the oscillator the second, to the potential energy. It follows that if we replace Pj and Qj by momentum and position operators Pj and Q, respectively, the eigenstates of the Schrodinger equation for electromagnetic radiation will be the same as those for harmonic oscillators. In particular, each oscillation mode will have a ladder of states with wavefunctions r y and energies 

The transformation of the time-dependent function Pj into a momentum operator is consistent with Einstein s description of light in terms of particles (photons), each of which has momentum hv (Sect. 1.6 and Box 2.3). We can interpret the quantum number rij in Eq. (3.50) either as the particular excited state occupied by oscillator j, or as the number of photons with frequency Vj. The oscillating electric and magnetic fields associated with a photon can stiU be described by Eqs. (3.44) and (3.45) if the amplitude factor is scaled appropriately. However, we will be less concerned with the spatial properties of photon wavefunctions themselves than with the matrix elements of the position operator Q. These matrix elements play a central role in the quantum theory of absorption and emission, as we ll discuss in Chap. 5. 

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To develop the quantum theory of electromagnetic radiation, it is useful to reformulate the harmonic-oscillator problem in terms of creation and annihilation operators, following a derivation due to Dirac. The Schrodinger equation (5.9) can be written... 

This dual wave/particle nature is the basis of the quantum theory of electromagnetic radiation, which states that radiant energy can be absorbed or emitted only in discrete packets called quanta or photons. The energy E of each photon is given by... 

A full explanation of the properties of light requires both the wave theory of electromagnetic radiation and the quantum theory. Most photochemical processes are best understood in terms of the quantum theory, which says that light is made up of discrete particles called quanta or photons. Each quantum carries an amount of energy, S, determined by the wavelength of the light, A. Equation 13.1, in which h is Planck s constant and c is the speed of light in a vacuum,... 

We will use this expression in Sect. 3.4 when we consider the quantum mechanical theory of electromagnetic radiation. 

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. 

There are several reasons for starting this account with a discussion of electromagnetic radiation. Historically, it was in this area that the quantum theory first developed. It is easier here to understand the evidence for the theory, and to appreciate some of its paradoxical consequences, than it is in the quantum theory of matter. The applications of the light-quantum hypothesis, as it was first called, also provide key pieces of evidence for the quantization of energy in atoms and molecules. Studies of the absorption and emission of radiation—the field of spectroscopy—and of the effect of light on chemical reactions—photochemistry—are very important areas of modem chemistry, in which the quantum nature of radiation is crucial. 

It turns out that electromagnetic waves exhibit properties of both waves and particles, or equally valid, electromagnetic waves are neither waves nor particles. This fundamental paradox is at the heart of quantum theory. You can perform experiments that unequivocally demonstrate light is definitely a wave. You can also perform experiments that unequivocally demonstrate light is definitely a particle. Nonetheless, there is one important relationship that allows the energy of electromagnetic radiation to be calculated if the frequency or wavelength is known ... 

This is reminiscent of Planck s formula for the energy of a photon. It comes as no surprise then that the quantum theory of radiation has the structure of an assembly of oscillators, with each oscillator representing a mode of electromagnetic waves of a specified frequency. 

The Rayleigh-Jeans picture of the radiation field as an ensemble of different modes of vibration confined to an enclosure was applied to the blackbody problem in Chapter 1. The quantum theory of radiation develops this correspondence more explicitly, identifying each mode of the electromagnetic field with an abstract harmonic oscillator of frequency coa- The Hamiltonian for the entire radiation field can be written... 

Max Planck in 1900 derived the correct form of the blackbody radiation law by introducing a bold postulate. He proposed that energies involved in absorption and emission of electromagnetic radiation did not belong to a continuum, as implied by Maxwell s theory, but were actually made up of discrete bundles—which he called quanta. Planck s idea is traditionally regarded as the birth of quantum theory. A quantum associated with radiation of frequency v has the energy... 

All the considerations that follow are only valid for radiation that is stimulated thermally. Radiation is released from all bodies and is dependent on their material properties and temperature. This is known as heat or thermal radiation. Two theories are available for the description of the emission, transfer and absorption of radiative energy the classical theory of electromagnetic waves and the quantum theory of photons. These theories are not exclusive of each other but instead supplement each other by the fact that each describes individual aspects of thermal radiation very well. 

In this chapter we have reviewed some results concerning the quantum multipole radiation. Although the representation of quantum electromagnetic radiation in terms of spherical waves of photons known since the first edition in 1936 of the famous book by Heitler on quantum theory of radiation [2], where this subject is discussed in the Appendix, this representation is not a widespread one. The spherical waves of photons are considered in very few advanced monographs on quantum optics [26]. The brilliant encyclopedic monographs [14,15] just touch on the subject. 

Planck s revolutionary idea about energy provided the basis for Einstein s explanation of the photoelectric effect in 1906 and for the Danish physicist Niels Bohr s atomic model of the hydrogen atom in 1913. Their success, in turn, lent support to Planck s theories, for which he received the Nobel Prize in physics in 1918. In the mid-1920s the combination of Planck s ideas about the particle-like nature of electromagnetic radiation and Erench physicist Louis de Broglie s hypothesis of the wavelike nature of electrons led to the formulation of quantum mechanics, which is still the accepted theory for the behavior of matter at atomic and subatomic levels. 

First comes the question of the crude charge distribution within the molecule, which largely determines the nature of its interaction with radiation. Then there comes the quantum theory of molecular emission and absorption, and finally the electromagnetic theory of radiation itself. 

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