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DESCRIPTION OF STATES BY WAVE FUNCTIONS

Cyril Norman Hinshelwood

Published Oct 13, 2005• 0 citations
Publisher

Abstract

Abstract This chapter discusses rules for the description of microscopic phenomena. Topics covered include Pauli principle in terms of wave functions, evidence from the helium spectrum, quantum statistics, evidence from rotational specific heats, transitions and the symmetry of states, and electron distributions from wave functions.
DOI
10.1093/acprof:oso/9780198570257.003.0009

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This chapter provides the quantum mechanical definition of the multipole moments, and study those of their properties that follow from symmetry considerations. It offers both the power and limitations of symmetry arguments. The interaction between the electromagnetic moments of atoms and nuclei determines the hyperfine structure of the spectrum, and because of this hyperfine spectroscopy is a powerful technique for studying the electromagnetic properties of nuclei. The chapter describes the quantum theory of systems that have a Correspondence Principle limit in classical point mechanics. It explores the detailed connection between the abstract formalism and the wave functions of Schrodinger's theory by computing the x-p transformation function. The chapter offers the symmetry properties of two-nucleon systems. The statistical mechanics of systems obeying the exclusion principle was worked out by Fermi and Dirac and is called Fermi-Dirac statistics; particles that obey this type of statistics are called fermions.
The approach is to start with some of the elementary ideas of classical mechanics and to translate these concepts to their equivalents in quantum mechanics. This will be done by changing the classical components of the total energy of particle into their quantum mechanical equivalents and arriving at the Schrodinger equation. The conjugate variables, position and momentum, in the Hamiltonian functions can be determined precisely for classical mechanical applications. This chapter deals with the Schrodinger model of the behavior of single electrons as affected by potential wells. The wave mechanic approach is extended to provide an outline of the interactions of many electrons with nuclei. In terms of the Pauli exclusion principle, the four quantum numbers of the electron states must be different. As additional electrons are added, the states available for occupation can be filled only according to the rules governing the quantum numbers.
Abstract For macroscopic systems, we take the continuity of many variables for granted, including the continuity in energy. For example, the heat flux along a rod through conduction, according to the Fourier law, can be continuously varied to any desired value by controlling the temperature difference and the material properties. The microscopic picture of energy, however, is entirely different. According to quantum mechanical principles, the permissible energy levels of matter (electrons, crystals, molecules, and so on) are often discontinuous. Differences in allowable energy levels among materials are major factors that distinguish them from each other. For example, why is glass transparent in the visible light range but not silicon, and why are some materials electrical insulators but others are conductors? In this chapter, we introduce the basic quantum mechanical concepts necessary to appreciate various energy states found in different materials. It should be remembered that these energy states represent the range of possibilities for the matter but do not tell which state the matter will be in. The latter depends on the temperature, a topic we will discuss in chapter 4. Important concepts that should be mastered through this chapter include the wave-particle duality, the Schrodinger equation and the meaning of the wave function, the Pauli exclusion principle, quantum states, and degeneracy. Solutions of the Schr6dinger equation for various simple yet very common potentials will be given. Key concepts and results of this chapter are summarized in the last section of the chapter.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.