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Quantization of waves: the stretched string

Geoffrey Brooker

Published Jul 16, 2021• 0 citations
Publisher

Abstract

Abstract “Quantization of waves: the stretched string” discusses waves in one dimension, in order to display the quantization procedure without the complication of three dimensions and of two polarization possibilities. Quantization goes via classical Lagrangian mechanics. The waves travel in both directions along the string, and we face up to disentangling these. The quantization procedure yields raising and lowering operators, their commutation rules, and their matrix elements.

DOI
10.1093/oso/9780198857242.003.0031

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Abstract “Quantization of waves: the stretched string” discusses waves in one dimension, in order to display the quantization procedure without the complication of three dimensions and of two polarization possibilities. Quantization goes via classical Lagrangian mechanics. The waves travel in both directions along the string, and we face up to disentangling these. The quantization procedure yields raising and lowering operators, their commutation rules, and their matrix elements.
Quantum mechanics was initially formulated as a type of matrix theory by Heisenberg and collaborators. Soon after, it was alternately expressed by Schrödinger as a wave theory through what is now referred to as the Schrödinger wave equation. Schrödinger and Eckart showed that these two approaches were equivalent manifestations of the same underlying formalism, viz., quantum theory. Quantum theory is formulated in this book (Chapter 5) via six postulates. In order that this postulational approach not be too abstract, it is helpful first to analyze a system that is easily treated by quantum theory. We do so in this chapter, in which the so-called quantal box is studied. The analysis is carried out using the Schrödinger equation, which is introduced in this chapter but without the formal statements of the postulates. Because the quantal-box problem is similar mathematically to the problem of describing the clamped, vibrating string of classical physics, the string is also analyzed in this chapter. This provides not only a comparison system for the box, but also a brief review of classical waves and wave theory. We also include a section on Hermitian differential operators, Sturm–Liouville systems and the Dirac delta function. All of this material serves as an introduction to Chapters 4 and 5. We begin with the string. Because many readers will already have seen aspects of the following analysis, we put all of Section 3.1 in small print. This section may be omitted by the knowledgeable reader; the crucial result is Eq. (3.3).
Introduction The main theme of this survey is the quantization of classical mechanical systems in terms of algebraic geometry. The aim is thus to relate questions of mathematics and theoretical physics. We first recall briefly the main problems and methods which turn us to study the new subject. Quantization itself is the main problem of theoretical physics. The need to introduce and develop it was dictated by the creators of quantum theory. According to the ‘Copenhagen philosophy’, the physical predictions of a quantum theory must be formulated in terms of classical concepts (the first sentence of Woodhouse; here we quote the beginning of this survey). Thus in addition to the usual structures (Hilbert space, unitary transformations, selfadjoint operators …, see, for example, Landau and Lifschitz) any reasonable quantum theory has to admit an appropriate passage to a classical limit under which quantum observables are transferred to their classical analogues. However, as Dirac pointed out at the beginning of the quantum age, the correspondence between quantum theory and classical theory should be based not just on numerical coincidences taking place in the limit h → ∞, but on an analogy between their mathematical structures. Classical theory does approximate the quantum theory, but it does more – it supplies a framework for some interpretation of quantum theory. Using this idea, we can in general understand a quantization procedure as a correspondence between classical and quantum theories. In this sense, quantization of the classical mechanical systems is a movement in one direction, while taking the quasiclassical limit goes in the opposite direction.
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.
The study of the behaviour of microscopic systems using Schrodinger's wave equation is a branch of quantum mechanics known as wave mechanics. This approach is inadequate to deal with all problems of interest, but a remarkably wide range of processes can successfully be investigated solely with the theory. The physics of a particle which moves in a static potential is of particular interest. Classically, the energy of such a particle is conserved. The wave function is reminiscent of the classical problem of standing waves on a stretched string with fixed endpoints, with the stationary states corresponding to the normal modes of vibration. Just as there are discrete frequencies of vibration on the string, so there are discrete energy levels. The energy, which is a continuous parameter in classical mechanics, is 'quantized' to certain discrete energy levels.