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Classical Waves

Published Apr 13, 2006• 0 citations
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Abstract

Abstract As we initially approach quantum physics through the introduction of wave mechanics, it is useful to begin by recalling some of the standard results obtained from the study of the classical wave equation. For example, using Newton ‘s laws, one can derive the equation of motion for the transverse displacement of a small piece of a stretched one-dimensional string; we obtain the equation

DOI
10.1093/oso/9780198530978.003.0002

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Before plunging into the problems associated with devising a wave description of an electron, let us consider the conceptually simpler one- and two-dimensional waves. A vibrating (guitar or violin) string is a one-dimensional wave. The sounds produced by such a string are caused by the movement of the string in the air surrounding it. There are always at least two fixed points on the string — one at each end, which correspond to the stops which hold the string in place. Thus movements which would require the string to shift at either end are forbidden. Various allowed and forbidden movements (vibrational modes) are shown in Figure 11.1. The requirements that there be zero displacement at each end of the string are known as boundary conditions, and the points of zero displacement are known as nodes. Notice how the boundary conditions severely limit the permitted modes of vibration. The various allowed modes can be described by the relation: (11.1)In the above ‘wave equation’ for the one-dimensional vibrator, ‘n’ is a quantum number; its value controls the wavelength of allowed vibration. If n = 2, the wavelength of allowed motion is one-half that for the mode when n = 1; and so on.

•Jan 1, 1982•1 citation
Abstract As we initially approach quantum physics through the introduction of wave mechanics, it is useful to begin by recalling some of the standard results obtained from the study of the classical wave equation. For example, using Newton ‘s laws, one can derive the equation of motion for the transverse displacement of a small piece of a stretched one-dimensional string; we obtain the equation
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).
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
Before plunging into the problems associated with devising a wave description of an electron, let us consider the conceptually simpler one- and two-dimensional waves. A vibrating (guitar or violin) string is a one-dimensional wave. The sounds produced by such a string are caused by the movement of the string in the air surrounding it. There are always at least two fixed points on the string — one at each end, which correspond to the stops which hold the string in place. Thus movements which would require the string to shift at either end are forbidden. Various allowed and forbidden movements (vibrational modes) are shown in Figure 11.1. The requirements that there be zero displacement at each end of the string are known as boundary conditions, and the points of zero displacement are known as nodes. Notice how the boundary conditions severely limit the permitted modes of vibration. The various allowed modes can be described by the relation: (11.1)In the above ‘wave equation’ for the one-dimensional vibrator, ‘n’ is a quantum number; its value controls the wavelength of allowed vibration. If n = 2, the wavelength of allowed motion is one-half that for the mode when n = 1; and so on.