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Oxford University Press eBooks

Translational motion

Peter Joshua Atkins, Julio de Paula, James Keeler

Published Dec 5, 2022• 0 citations
Publisher

Abstract

This chapter evaluates translational motion, motion through space, which is one of the fundamental types of motion treated by quantum mechanics. According to quantum theory, a particle constrained to move in a finite region of space is described by only certain wavefunctions and can possess only certain energies. That is, quantization emerges as a natural consequence of solving the Schrödinger equation and the conditions imposed on it. The solutions also expose a number of non-classical features of particles, especially their ability to tunnel into and through regions where classical physics would forbid them to be found. Light particles are more able to tunnel through barriers than heavy ones. The chapter also considers the occurrence of degeneracy, which is a consequence of the symmetry of the system.

DOI
10.1093/hesc/9780198847816.003.0036

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This chapter evaluates translational motion, motion through space, which is one of the fundamental types of motion treated by quantum mechanics. According to quantum theory, a particle constrained to move in a finite region of space is described by only certain wavefunctions and can possess only certain energies. That is, quantization emerges as a natural consequence of solving the Schrödinger equation and the conditions imposed on it. The solutions also expose a number of non-classical features of particles, especially their ability to tunnel into and through regions where classical physics would forbid them to be found. Light particles are more able to tunnel through barriers than heavy ones. The chapter also considers the occurrence of degeneracy, which is a consequence of the symmetry of the system.
This chapter describes a particle confined to a finite region of space that can possess only certain discrete energies. It also looks at the corresponding wavefunctions. It analyzes the quantization of energy that emerges as a natural consequence of solving the Schrödinger equation and the conditions imposed on it. It also introduces a striking non-classical feature of small particles, which describes their ability to penetrate into and through regions where classical physics would forbid them to be found. The chapter talks about the translational energy levels of a particle which are confined to a finite region of space and are quantized. It explores how quantum mechanics reveal the ability of a particle to penetrate into and through regions where classical physics would forbid it to be found.
This chapter considers the quantum mechanics of translation and vibration. Both types of motion can be solved in certain cases, and both are important not only in their own right but also because they form a basis for the description of more complicated types of motion. It mentions translational motion, which has the advantage of introducing in a simple way many of the striking features of quantum mechanics. The chapter describes certain features of wavefunctions that are common to all the problems. The combination of features of wavefunctions with the solution of the Schrödinger equation results in one of the most characteristic features of quantum mechanics, the quantization of energy.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.
This chapter discusses the central concepts and equations of quantum theory which demonstrates how particles spread through space like waves and are described mathematically by a wavefunction. It examines how the wavefunction is interpreted and introduces the uncertainty principle, which is one of the most profound departures of quantum mechanics from classical mechanics. It also highlights the dynamical properties of a system that are contained in the wavefunction and are obtained by solving the Schrödinger equation. The chapter reconciles the facts that atoms and molecules can possess only certain energies, that waves exhibit the properties of particles, and that particles exhibit the properties of waves. It reviews the classical concept that have been accommodated by the development of quantum mechanics, in which equations are set up to treat a particle as spread through space like a wave.
Abstract A distinguishing feature of quantum mechanics, as compared with classical mechanics, is the phenomenon of matter exhibiting wave-like properties. A direct consequence of this wave property of particles is the tunneling of particles through potential energy barriers.-3 The classical particle is forbidden from penetrating the barrier region. In this chapter, the problem of one-dimensional (1D) quantum mechanical tunneling of particles through potential barriers of arbitrary shape is treated. We show that tunneling is a mixed boundary value problem, and obtain the action integral that leads to Schrodinger’s equation and at the same time gives the equations defining the boundary conditions (BCs).