Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

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) ˆ 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) ˆ cos 2ð
º
(xÿ x0)
If the harmonic wave moves in time at a constant velocity v, then we have the
relation x0 ˆ vt, where t is the elapsed time (in seconds), and \u142(x) becomes
\u142(x, t) ˆ 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 ˆ íº
and \u142(x, t) may be written as
\u142(x, t) ˆ 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 ˆ ø=k and the wave \u142(x, t) takes the form
\u142(x, t) ˆ cos(kxÿ øt)
The harmonic wave may also be described by the sine function
\u142(x, t) ˆ 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 ˆ 0. Moreover, any linear combination