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Introduction to Quantum Theory

Wai‐Kee Li, Gong-Du Zhou, Thomas C. W. Mak

Published Mar 27, 2008• 42 citations
Publisher

Abstract

Abstract This chapter covers the rudiments of quantum theory, including the dual nature of light and matter, the Uncertainty Principle and probability concept, the electronic wavefunction, and the probability density function. Numerical examples are given to show that given the electronic wavefunction of a system, the probability of finding an electron in a volume element around a certain point in space can be readily calculated. Finally, the electronic wave equation, the Schrödinger equation, is introduced. This discussion is followed by the solution of a few particle-in-a-box problems, with the ‘box’ having the shape of a wire (one-dimensional), a cube, a ring, or a triangle. Where possible, the solutions of these problems are then applied to a chemical system.

DOI
10.1093/acprof:oso/9780199216949.003.0001

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Abstract This chapter covers the rudiments of quantum theory, including the dual nature of light and matter, the Uncertainty Principle and probability concept, the electronic wavefunction, and the probability density function. Numerical examples are given to show that given the electronic wavefunction of a system, the probability of finding an electron in a volume element around a certain point in space can be readily calculated. Finally, the electronic wave equation, the Schrödinger equation, is introduced. This discussion is followed by the solution of a few particle-in-a-box problems, with the ‘box’ having the shape of a wire (one-dimensional), a cube, a ring, or a triangle. Where possible, the solutions of these problems are then applied to a chemical system.
Introduction to Quantum Theory and Atomic Structure envelops the basic concepts needed as background for this topic, and discusses atomic structure but not molecular applications. The first two chapters are concerned with the basic ideas and problems of wave-particle duality, the nature of wavefunction, and its statistical interpretation. Chapter 3 discusses some important applications of Schrödinger's equation to chemically relevant situations. Chapters 4 and 5 deal respectively with the hydrogen atom, and with the structure of many-electron atoms and the periodic table of elements. The emphasis throughout is on the physical concepts and their concrete application.
Abstract This chapter contains a review of quantum mechanics of one-electron systems, suitable for undergraduate students with a limited background of modern physics. It is first explained that matter consists of electrons and nuclei, and the concept of wave functions is introduced. The Schrödinger equation is then presented, and it is shown how the static Schrödinger equation emerges from the stationary limit of the time-dependent Schrödinger equation. Two examples are discussed: the particle in a box and plane waves. The chapter finishes by listing important concepts and properties of quantum mechanics, such as probability density, normalization, observables, operators, energy spectra, and orthonormal sets.
Assuming a background in basic classical physics, multivariable calculus, and differential equations, A Concise Introduction to Quantum Mechanics provides a self-contained presentation of the mathematics and physics of quantum mechanics. The relevant aspects of classical mechanics and electrodynamics are reviewed, and the basic concepts of wave-particle duality are developed as a logical outgrowth of experiments involving blackbody radiation, the photoelectric effect, and electron diffraction. The Copenhagen interpretation of the wave function and its relation to the particle probability density is presented in conjunction with Fourier analysis and its generalization to function spaces. These concepts are combined to analyze the system consisting of a particle confined to a box, developing the probabilistic interpretation of observations and their associated expectation values. The Schrödinger equation is then derived by using these results and demanding both Galilean invariance of the probability density and Newtonian energy-momentum relations. The general properties of the Schrödinger equation and its solutions are analyzed, and the theory of observables is developed along with the associated Heisenberg uncertainty principle. Basic applications of wave mechanics are made to free wave packet spreading, barrier penetration, the simple harmonic oscillator, the Hydrogen atom, and an electric charge in a uniform magnetic field. In addition, Dirac notation, elements of Hilbert space theory, operator techniques, and matrix algebra are presented and used to analyze coherent states, the linear potential, two state oscillations, and electron diffraction. Applications are made to photon and electron spin and the addition of angular momentum, and direct product multiparticle states are used to formulate both the Pauli exclusion principle and quantum decoherence. The book concludes with an introduction to the rotation group and the general properties of angular momentum.
Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)
We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the ε j 's and g j 's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics. We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.
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.