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An Introduction to Wave Mechanics

David A. Robinson, John McK. Woollard

Published Jan 1, 1982• 1 citation
Publisher

Abstract

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.

DOI
10.1007/978-1-349-04540-2_11

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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.
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.
Abstract Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.
Simple Harmonic Oscillation Massona Spring Simple Harmonic Oscillator Equation LC Circuits Simple Pendula Compound Pendula Damped and Driven Harmonic Oscillation Damped Harmonic Oscillation Quality Factor LCR Circuits Driven Damped Harmonic Oscillation Driven LCR Circuits Transient Oscillator Response Coupled Oscillations Two Spring-Coupled Masses Two Coupled LC Circuits Three Spring-Coupled Masses Transverse Standing Waves Normal Modes of a Beaded String Normal Modes of a Uniform String General Time Evolution of a Uniform String Longitudinal Standing Waves Spring-Coupled Masses Longitudinal Waves on a Thin Elastic Rod Sound Waves in an Ideal Gas Fourier Analysis Traveling Waves Standing Waves in a Finite Continuous Medium Traveling Waves in an Infinite Continuous Medium Wave Interference Energy Conservation Transmission Lines Normal Reflection and Transmission at Interfaces Electromagnetic Waves Doppler Effect Wave Propagation in Inhomogeneous Media Multi-Dimensional Waves Plane Waves Three-Dimensional Wave Equation Cylindrical Waves Spherical Waves Oscillation of an Elastic Sheet Polarization of Electromagnetic Waves Laws of Geometric Optics Fresnel Relations Total Internal Reflection Sound Waves in Fluids Wave Pulses Fourier Transforms General Solution of One-Dimensional Wave Equation Bandwidth Dispersive Waves Pulse Propagation Electromagnetic Waves in Unmagnetized Plasmas Faraday Rotation Electromagnetic Wave Propagation in Conductors Waveguides Pulse Propagation in Two Dimensions Gravity Waves Wave Dragon Ships Ship Wakes Capillary Waves Wave Optics Introduction Two-Slit Interference Coherence Multi-Slit Interference Thin Film Interference One-Dimensional Fourier Optics Single-Slit Diffraction Multi-Slit Diffraction Two-Dimensional Fourier Optics Wave Mechanics Introduction Photoelectric Effect Electron Diffraction Representation of Waves via Complex Numbers Schrodinger's Equation Probability Interpretation of Wavefunction Wave Packets Heisenberg's Uncertainty Principle Wavefunction Collapse Stationary States Three-Dimensional Wave Mechanics Particle in Finite Square Potential Well Square Potential Barrier Appendix A: Physical Constants Appendix B: Useful Mathematics Appendix C: Electromagnetic Theory Bibliography Index Exercises appear at the end of each chapter.
This paper thinks that the conclusion will disagree with the Bohr angular momentum quantized condition,if the standing wave condition of linear string vibration is used directly to deal with angular momentum of the motion of electron going round nucleus in a circle in the textbook.The paper holds that if one employs the character of the standing wave and the symmetry satisfied by standing wave condition of circinal string vibration,he can reach that the condition of the standing wave of circinal string vibration is just the circumference of the circle equal to even times of half of the wavelength.Through that condition,one can obtain the quantized angular momentum consistent with the stationary state of Bohr Theory.At the end of the paper the authors explain the quantized angular momentum and the quantized energy with the character of the standing wave.
According to classical physics (i.e., physics prior to the 20th century), particles and waves are distinct classes of physical entity that possess markedly different properties. For instance, particles are discrete, which means that they cannot be arbitrarily divided. In other words, it makes sense to talk about one electron, or two electrons, but not about a third of an electron. Waves, on the other hand, are continuous, which means that they can be arbitrarily divided. In other words, given a wave whose amplitude has a certain value, it makes sense to talk about a similar wave whose amplitude is one third, or any other fraction whatsoever, of this value. Particles are also highly localized in space. For example, atomic nuclei have very small radii of order 10−15 m, whereas electrons act like point particles (i.e., they have no discernible spatial extent). Waves, on the other hand, are nonlocalized in space. In fact, a wave is defined as a disturbance that is periodic in space, with some finite periodicity length (i.e., wavelength). Hence, it is fairly meaningless to talk about a disturbance being a wave unless it extends over a region of space that is at least a few wavelengths in size.