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Oxford University Press eBooks

Rotational motion and the hydrogen atom

Peter Joshua Atkins, Ronald Friedman

Published Nov 11, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1093/hesc/9780199541423.003.0003

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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.
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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 focuses on the motion first on a circle and then on the surface of a sphere. It discusses the imposition of certain constraints on the wavefunction which shows how the energy and angular momentum of a rotating particle are quantized. It also examines the energy and the angular momentum of a rotating object which are quantized. The chapter reviews quantum mechanics and shows what they reveal about rotational motion. The chapter also considers a particle of mass m moving in a circular path of radius r in the xy-plane. It highlights translational motion which focuses on linear momentum, p and analogous angular momentum, J if rotational motion is considered.
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 pays particular attention to the quantum mechanics of motion. It is not possible to explain the structures and properties of atoms and molecules, where the role of electrons is central, without using quantum mechanics. Many of the processes of biochemistry involve the transfer of electrons and protons from a donor to an acceptor, and this migration can be understood only in terms of quantum theory. Here, the chapter introduces three fundamental types of motion — translation, rotation, and vibration — along with their characteristic quantum mechanical behaviours. It shows how various unexpected properties emerge, such as the ability of particles to penetrate into classically forbidden regions, the quantization of energy and angular momentum, and the existence of energy that cannot be removed from the system.
This chapter deals with some solutions of Schrödinger's equation in situations that are relatively simple, but which nevertheless have important applications in chemistry. Most of these deal with the quantization of energy. However, there is another remarkable prediction of quantum mechanics, known as tunnelling. In classical physics, kinetic energy cannot be negative; in other words, the total energy cannot be less than the potential energy. Schrödinger's equation can, however, be solved when this is the case. In some situations, these solutions are physically significant. The chapter then looks at the harmonic oscillator, which is one of the most important elementary models in mechanics, and is especially relevant in chemistry in connection with the vibration of molecules. It also considers rotations and angular momentum.