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Cambridge University Press eBooks

Quantizing Maxwell's Equations

Thomas A. Garrity

Published Jan 19, 2015• 0 citations
Publisher

Abstract

: Our goal is to quantize Maxwell's equations, leading to a natural interpretation of light as being composed of photons. The key is that the quantization of harmonic oscillators (or, more prosaically, the mathematics of springs) is critical to understanding light. In particular, we show that the possible energies of light form a discrete set, linked to the classical frequency, giving us an interpretation for the photoelectric effect. Our Approach From Einstein's explanation of the photoelectric effect, light seems to be made up of photons, which in turn are somewhat particle-like. Can we start with the classical description of light as an electromagnetic wave solution to Maxwell's equations when there are no charges or currents, quantize, and then get something that can be identified with photons? That is the goal of this chapter. (As in the previous chapter, while all of this is standard, we will be following the outline given in chapter 2 of [41].) Maxwell's equations are linear. We will concentrate on the ‘monochromatic’ solutions, those with fixed frequency and direction. We will see that the Hamiltonians of these monochromatic solutions will have the same mathematical structure as a Hamiltonian for a harmonic oscillator. We know from the last chapter how to quantize the harmonic oscillator. More importantly, we saw that there are discrete eigenvalues and eigenvectors for the corresponding Hamiltonian operator. When we quantize the Hamiltonians for the monochromatic waves in an analogous fashion, we will again have discrete eigenvalues and eigenvectors. It is these eigenvectors that we can interpret as photons.

DOI
10.1017/cbo9781139939683.015

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Cambridge University Press eBooks•Jan 3, 2013•0 citations
: Our goal is to quantize Maxwell's equations, leading to a natural interpretation of light as being composed of photons. The key is that the quantization of harmonic oscillators (or, more prosaically, the mathematics of springs) is critical to understanding light. In particular, we show that the possible energies of light form a discrete set, linked to the classical frequency, giving us an interpretation for the photoelectric effect. Our Approach From Einstein's explanation of the photoelectric effect, light seems to be made up of photons, which in turn are somewhat particle-like. Can we start with the classical description of light as an electromagnetic wave solution to Maxwell's equations when there are no charges or currents, quantize, and then get something that can be identified with photons? That is the goal of this chapter. (As in the previous chapter, while all of this is standard, we will be following the outline given in chapter 2 of [41].) Maxwell's equations are linear. We will concentrate on the ‘monochromatic’ solutions, those with fixed frequency and direction. We will see that the Hamiltonians of these monochromatic solutions will have the same mathematical structure as a Hamiltonian for a harmonic oscillator. We know from the last chapter how to quantize the harmonic oscillator. More importantly, we saw that there are discrete eigenvalues and eigenvectors for the corresponding Hamiltonian operator. When we quantize the Hamiltonians for the monochromatic waves in an analogous fashion, we will again have discrete eigenvalues and eigenvectors. It is these eigenvectors that we can interpret as photons.
James Maxwell is one of the most important physicists of all time. He did landmark work in unifying electricity and magnetism and also describing the two with a coherent mathematical framework. In the process, he discovered that light is an electromagnetic wave that can oscillate in a vacuum. He also played critical roles in the development of statistical mechanics, which is why we call e(−E/(kBT)e^(-E/(k_B T)e(−E/(kB​T) ) the Maxwell-Boltzmann distribution. Unfortunately, to fully work with his mathematical formulation of electromagnetism, via the so-called Maxwell equations, requires one to master vector calculus. So, in this chapter, we do not cover the complete derivation of his work, but instead show how it applies directly to light and from that we are then be able to jump into quantum optics. The goal of this chapter is to precisely explain and illustrate for you what a photon really is. Not just a particle of light, nor just a wavepacket, but a specific quantum structure that has the property that it can be measured once and only once. Then we put together all of these new insights to explain how one measures gravitational waves with quantum mechanics.
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
Abstract This chapter begins its exploration of field quantization with an illustration of how a harmonic oscillator may be quantized, which motivates and preludes the subsequent discussion on canonical quantization, also known as the second quantization. Then it uses the formalism to quantize the Klein-Gordon and Schrodinger equations. Afterwards, goes on to present the quantization of the Maxwell equations, which establishes the celebrated photoelectric effect through a derivation of the Planck-Einstein relation and Compton-Debye formula. Finally, the chapter concludes with a comparison study of the thermodynamic properties of a harmonic oscillator with respect to its classical and quantum-mechanical descriptions.
At the conceptual level, the quantization procedure replaces the pure particle description of classical mechanics by a particle-wave treatment satisfying the requirements of the fundamental wave–particle dualism. Technically, one can always start from a generalized wave equation and use it as a basis to discuss wave-front propagation in the form of rays following classical particle trajectories, see Sections 2.2.1 and 3.1. Consequently, any wave theory inherently contains that aspect of the wave–particle dualism. Especially, Maxwell's equations of electrodynamic theory include both wave and particle aspects of electromagnetic fields. Hence, it could be that there are no additional “quantum,” i.e., wave–particle dualistic aspects. However, as we discuss in this chapter, Maxwell's equations can be expressed in the framework of Hamilton's formulation of particle mechanics. This then suggests that in addition to the approximate ray-like behavior, electromagnetic fields do have a supplementary level of particle-like properties. Due to the fundamental axiom of wave–particle dualism, the additional particle-like aspects of electromagnetic fields must then also have a wave counterpart that can be axiomatically introduced via the canonical quantization scheme described in Section 3.2.3. Since the spatial dependency in Maxwell's equation contains wave–particle dualism already at the classical level, it is clear that the new level of quantization cannot be a simple real-space feature but has to involve some other space describing the structure of the electromagnetic fields. In fact, this quantization yields several very concrete implications such as quantization of the light energy, elementary fluctuations in the light-field amplitude and intensity, and more.
We present a succinct way to quantize Maxwell's equations. We begin by discussing the random nature of quantum observables. Then we present the quantum Maxwell's equations and give their physical meanings. Due to the mathematical homomorphism between the classical and quantum cases [1], the derivation of quantum Maxwell's equations can be simplified. First, one needs only to verify that the classical equations of motion are derivable from a Hamiltonian by energy-conservation arguments. Then the quantum equations of motion follow due to homomorphism, which applies to sum-separable Hamiltonians. Hence, we first show the derivation of classical Maxwell's equations using Hamiltonian theory. Then the derivation of the quantum Maxwell's equations follows in an analogous fashion. Finally, we apply this quantization procedure to the dispersive medium case. (This article is written for the classical electromagnetic community assuming little knowledge of quantum theory, an introduction of which can be found in [2, Lecs. 38 and 39].)
In this chapter we explain the procedure for quantizing classical fields, based on the preparatory work we did in the previous chapter. We have seen that classical fields can generally be treated as an (infinite) set of oscillator modes. In fact, for all fields we considered in the previous chapter, we were able to rewrite the Hamiltonian in a way such that it looked identical to the Hamiltonian for a generic (mechanical) oscillator, or a set of such oscillators. We therefore understand that, if we can find an unambiguous way to quantize the mechanical oscillator, we get the quantization of all classical fields for free! We thus start considering the mechanical harmonic oscillator and refresh the quantum mechanics of this system. The new idea is to associate the oscillator with a level for bosonic particles representing the quantized excitations of the oscillator. We then turn to a concrete example and show how the idea can be used to quantize the deformation field of the elastic string discussed in Section 7.2. In this case, the bosonic particles describing the excitations of the field are called phonons: propagating wave-like deformations of the string. As a next example, we revisit the low-lying spinfull excitations in a ferromagnet, already described in Section 4.5.3. Here, we start from a classical field theory for the magnetization of the magnet, and show how the quantization procedure leads to the same bosonic magnons we found in Chapter 4. Finally, we implement the quantization technique for the electromagnetic field, leading us to the concept of photons – the quanta of the electromagnetic field.