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Canonical quantization of fields

Yisong Yang

Published Jul 27, 2023• 0 citations
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Abstract

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.

DOI
10.1093/oso/9780192872616.003.0015

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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.
The harmonic oscillator represents as a unique tool in modern physics since it is not only exactly solvable but also relates many phenomena. This paper introduces the relation between Quantum Field Theory and other physical theories and discusses how harmonic oscillator behave in Quantum Mechanics and Quantum Field Theory by the quantization of Klein-Gordon Field. To this end, this paper applies energy-momentum equation given by special relativity to Schrödinger’s Equation to get the Klein-Gordon Field, and then uses canonical method to quantize the field, which is called the second quantization of the field. The comparison is made between the result quantized field, the quantum harmonic oscillator and classical harmonic oscillator. The similarity of two physical systems is demonstrate by the comparison between commutation relations of operators. The similar commutation relations then help people to interpret quantized Klein-Gordon Field as a system that contain a quantized harmonic oscillator in each space coordinate.
: 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.
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.
This chapter begins by developing the relation between the quantum theory and the linear algebra. It discusses the most fundamental postulates of the theory and provides a phenomenological development of the Schrödinger wave equation (SWE) although the results might be gleaned from the classical Lagrangian and Hamiltonian with the Poisson bracket relations. Afterward some simple examples for the infinitely and finitely deep wells help clarify the basic postulates. The simple harmonic oscillator has perhaps the most sweeping implications and applications for the quantum theory. For this reason, we present the most applicable formalism for the Harmonic oscillator that uses the operator approach rather than the classical method of partial differential equations. The angular momentum and spin are presented for the study of the atom and for applications for nanodevices, and quantum computing. Next the representation formalism clarifies the distinction between the dynamical operators and the wave functions and their interrelations. The chapter discusses both timeindependent and time-dependent perturbation theories. The density operator combines the quantum theory with the classical theory for finding the behavior of a system. The density operator represents one of the basic concepts in the quantum theory. The remainder of the chapter introduces the more advanced material meant to introduce the quantum field theory starting with the second quantization and the propagator.
The notion that the electromagnetic field is quantized is usually inferred from observations such as the photoelectric effect and the black-body spectrum. However accounts of the quantization of this field are usually mathematically motivated and begin by introducing a vector potential, followed by the imposition of a gauge that allows the manipulation of the solutions of Maxwell's equations into a form that is amenable for the machinery of canonical quantization. By contrast, here we quantize the electromagnetic field in a less mathematically and more physically motivated way. Starting from a direct description of what one sees in experiments, we show that the usual expressions of the electric and magnetic field observables follow from Heisenberg's equation of motion. In our treatment, there is no need to invoke the vector potential in a specific gauge and we avoid the commonly used notion of a fictitious cavity that applies boundary conditions to the field.