# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent Schro¨dinger equations are introduced along with a discussion of wave functions for particles in a potential field. Some instructors may wish to omit the first or both of these chapters or to present abbreviated versions. Chapter 3 is the heart of the book. It presents the postulates of quantum mechanics and the mathematics required for understanding and applying the postulates. This chapter stands on its own and does not require the student to have read Chapters 1 and 2, although some previous knowledge of quantum mechanics from an undergraduate course is highly desirable. Chapters 4, 5, and 6 discuss basic applications of importance to chemists. In all cases the eigenfunctions and eigenvalues are obtained by means of raising and lowering operators. There are several advantages to using this ladder operator technique over the older procedure of solving a second-order differ- viii ential equation by the series solution method. Ladder operators provide practice for the student in operations that are used in more advanced quantum theory and in advanced statistical mechanics. Moreover, they yield the eigenvalues and eigenfunctions more simply and more directly without the need to introduce generating functions and recursion relations and to consider asymp- totic behavior and convergence. Although there is no need to invoke Hermite, Legendre, and Laguerre polynomials when using ladder operators, these func- tions are nevertheless introduced in the body of the chapters and their proper- ties are discussed in the appendices. For traditionalists, the series-solution method is presented in an appendix. Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the free-electron gas and Bose\u2013 Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. The first-year graduate course in quantum mechanics is used in many chemistry graduate programs as a vehicle for teaching mathematical analysis. For this reason, this book treats mathematical topics in considerable detail, both in the main text and especially in the appendices. The appendices on Fourier series and the Fourier integral, the Dirac delta function, and matrices discuss these topics independently of their application to quantum mechanics. Moreover, the discussions of Hermite, Legendre, associated Legendre, La- guerre, and associated Laguerre polynomials in Appendices D, E, and F are more comprehensive than the minimum needed for understanding the main text. The intent is to make the book useful as a reference as well as a text. I should like to thank Corpus Christi College, Cambridge for a Visiting Fellowship, during which part of this book was written. I also thank Simon Capelin, Jo Clegg, Miranda Fyfe, and Peter Waterhouse of the Cambridge University Press for their efforts in producing this book. Donald D. Fitts Preface ix 1 The wave function Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon\u2013the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and tech- nology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schro¨dinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schro¨dinger\u2019s wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schro¨dinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 1 1.1 Wave motion Plane wave A simple stationary harmonic wave can be represented by the equation \u142(x) cos 2ðx º and is illustrated by the solid curve in Figure 1.1. The distance º between peaks (or between troughs) is called the wavelength of the harmonic wave. The value of \u142(x) for any given value of x is called the amplitude of the wave at that point. In this case the amplitude ranges from 1 to ÿ1. If the harmonic wave is A cos(2ðx=º), where A is a constant, then the amplitude ranges from A to ÿA. The values of x where the wave crosses the x-axis, i.e., where \u142(x) equals zero, are the nodes of \u142(x). If the wave moves without distortion in the positive x-direction by an amount x0, it becomes the dashed curve in Figure 1.1. Since the value of \u142(x) at any point x on the new (dashed) curve corresponds to the value of \u142(x) at point xÿ x0 on the original (solid) curve, the equation for the new curve is \u142(x) cos 2ð º (xÿ x0) If the harmonic wave moves in time at a constant velocity v, then we have the relation x0 vt, where t is the elapsed time (in seconds), and \u142(x) becomes \u142(x, t) cos 2ð º (xÿ vt) Suppose that in one second, í cycles of the harmonic wave pass a fixed point on the x-axis. The quantity í is called the frequency of the wave. The velocity \u142(x) x0 º ºº/2 3º/2 º2 x Figure 1.1 A stationary harmonic wave. The dashed curve shows the displacement of the harmonic wave by x0. 2 The wave function v of the wave is then the product of í cycles per second and º, the length of each cycle v íº and \u142(x, t) may be written as \u142(x, t) cos 2ð x º ÿ ít \ufffd \ufffd It is convenient to introduce the wave number k, defined as k \ufffd 2ð º (1:1) and the angular frequency ø, defined as ø \ufffd 2ðí (1:2) Thus, the velocity v becomes v ø=k and the wave \u142(x, t) takes the form \u142(x, t) cos(kxÿ øt) The harmonic wave may also be described by the sine function \u142(x, t) sin(kxÿ øt) The representation of \u142(x, t) by the sine function is completely equivalent to the cosine-function representation; the only difference is a shift by º=4 in the value of x when t 0. Moreover, any linear combination