A Fundamental Approach to Modern Physics

Harvey Mudd College

As an undergraduate, my first course in quantum mechanics was in the spring semester of my senior year. This was, frankly, too little, too late. As a sophomore, I had taken a fairly standard course in modern physics. Despite the fact that I thought the professor teaching the course did a good job, I was not pleased with the content. The course seemed to be a summary of phenomenology, without giving me any understanding of the underlying physics. To a budding physicist, this was not a satisfying experience.

Today, I am confident we do better by our students, at least
in the upper-division physics curriculum. This confidence is inspired by the
quality and nature of the quantum mechanics textbooks that we use there, one of
which (*A Modern Approach to Quantum Mechanics*) I am proud to have
authored. At the introductory level the changes have been, in my judgment, less
clear cut. I believe that students deserve a serious introduction to quantum
mechanics, comparable to the introduction they receive to the subjects of
mechanics and electromagnetism. Moreover, with the appropriate grounding in
quantum mechanics, it is possible to give students real understanding and
insight into an array of topics that often fall under the rubric of modern
physics. Students can see in quantum mechanics a common thread that ties these
topics together into a coherent picture of how the world works, a picture that
gives students confidence that quantum mechanics itself really works, too.

While I have used the term "modern physics" to describe the material typically taught in an introductory course, I believe this term has reached the end of its useful life, at least in the way it is commonly used. Most if not all modern physics textbooks follow an historical ordering of the material, with, in order of appearance, Planck, Einstein, Rutherford, Bohr, and Schrödinger among the key characters in the story. Now I enjoy the history as much as anyone, and I try to weave it into the text in a natural way. But I don't think following the historical ordering so closely makes a lot of sense. After all, a story that starts in the early 1900s does not sound modern to students learning physics in the 21st century. Moreover, times have changed. We now have the advantage of truly modern experiments, such as single-photon and single-atom interferometry experiments, that have replaced the thought experiments that characterized much of the early discussions of quantum mechanics. So why not start with real experiments, which is what physics is really based on, after all.

Chapter 1 focuses on the quantum nature of light. While this
chapter does include discussion of the photoelectric effect (the key to
understanding the operation of a photodetector) and Compton scattering, the
single-photon anticoincidence and interference experiments carried out by Alain
Aspect and coworkers in the late 1980s are at the center of this chapter.
Understanding the results of these experiments leads us to the concept of a
probability amplitude and to the rules for multiplying and, in particular,
adding these probability amplitudes when there are multiple paths that a photon
can take between the source and the detector. This is really the sum-over-paths
approach to quantum mechanics pioneered by Richard Feynman. One of the
advantages of this approach is that students can see right away how quantum
mechanics can explain everyday phenomena such as the the law of reflection,
Snell's law, and a diffraction grating (in, say, the reflection of light from a
CD) as straightforward extensions of the sum-over-paths approach from a few
paths to many paths (leading naturally to Fermat's principle of least time).
Although the approach that I follow in Chapter 1 is not the same as that given
by Feynman in his short series of lectures titled *QED*, it is inspired by
these lectures.

Chapter 2 starts with the double-slit experiment, a topic that was discussed in Chapter 1 as an illustration of the sum-over-paths approach to quantum mechanics. But in Chapter 2 the key experiment is a double-slit experiment with helium atoms carried out by Jürgen Mlynek's group in the 1990s. This beautiful experiment really brings home to students the strangeness of quantum mechanics. Since the sum-over-paths approach is not as useful for determining the behavior of particles such as electrons when they travel microscopic distances, Chapter 2 moves naturally toward the Schrödinger equation. This wave equation plays a similar role for nonrelativistic particles to that played by the wave equation for light in Chapter 1. Other topics in this chapter include wave packets, phase and group velocities, expectation values and uncertainty, and Ehrenfest's equations.

Chapter 3 and Chapter 4 are all about solving the Schrodinger equation for a variety of one-dimensional potentials. The centerpiece of Chapter 3 is the particle in a box, a great laboratory for seeing many quantum effects. Chapter 4 includes discussion of the finite square well, the harmonic oscillator, and the Dirac delta function potential, both as a simple model for an atom and, more interestingly, as a double well that can be used to capture the key features of molecular binding. Chapter 4 also includes a discussion of scattering (and tunneling) in one-dimensional quantum mechanics. One relatively novel feature of Chapter 3 at this level is the use of the particle in a box to illustrate the key features of the energy eigenvalue equation, including the principles of superposition and completeness and the way these principles are utilized to calculate the probability of events for a wave function that is not an energy eigenfunction. These ideas are generalized in Chapter 5 to the more general class of Hermitian operators corresponding to observables. Here the role that commuting and, in particular, noncommuting operators and uncertainty relations play in quantum mechanics is emphasized.

Chapter 6 extends the discussion of quantum mechanics to
three-dimensional systems. Because the particle in a three-dimensional box, the
orbital angular momentum eigenvalue problem, and the hydrogen atom can be
attacked by the technique of separation of variables, these systems have much in
common with the treatment of one-dimensional potentials from the earlier
chapters. I make an effort to keep the mathematical level accessible to
students. Some of the details, such as solving the hydrogen atom by a
power-series solution, are left to an appendix for the interested reader. But
the simple, direct way in which the eigenvalue problem for the *z* component of the orbital angular momentum, for example, quantizes the
eigenvalues to integral multiples of hbar is a
very important and natural extension of the techniques introduced in the earlier
chapters. And the fact that the *z *component of the intrinsic spin angular
momentum of an electron takes on only half-integral multiples of hbar
tells
us that intrinsic spin, real angular momentum that it is, is not connected to
wave functions or to anything physically rotating, as is the case for orbital
angular momentum.

Given the profound impact that the intrinsic spin of identical particles plays in multiparticle systems in quantum mechanics, one can argue that intrinsic spin is perhaps the single most important attribute of a particle. After introducing the exchange operator and seeing how the Pauli Principle arises from basic quantum mechanics, Chapter 7 goes on to examine systems with identical fermions (multielectron atoms, electrons in a solid, and white dwarf and neutron stars) and identical bosons (cavity radiation, Bose-Einstein condensation, and lasers). Again, the focus is on showing how the behavior of these systems follows directly from quantum mechanics.

The remaining three chapters of the book are devoted to applications as well. But here too the focus is restricted so as to avoid the peril of too much phenomenology and too little explanation of the underlying physics. Chapter 8 is devoted to crystalline solids, namely solids for which the periodic nature of the potential energy leads to an energy band structure, allowing us to understand the electrical properties of metals, insulators, and semiconductors. The role that this band structure of semiconductors plays in the electronics/computer revolution in which we are all participating is emphasized. Chapter 9 is devoted to nuclear physics, focusing in one form or another on the all-important "curve of binding energy," including its impact on radioactivity, nuclear fission, and nuclear fusion. This chapter concludes with a discussion of the history and physics of nuclear weapons, a subject with which humanity is still struggling to deal. Finally, Chapter 10 is devoted to particle physics. One of the benefits of starting this book with a sum-over-paths approach to quantum mechanics is the natural way it leads into a description of relativistic quantum mechanics. Although it is not possible to explain particle physics at the same level of completeness as the earlier topics, it is important for students to see the role that quantum mechanics plays in the interactions of the particles in the Standard Model, where the probability amplitudes can be represented graphically by Feynman diagrams. This chapter concludes with a discussion of the close connection between symmetries and conservation laws and the fundamental role symmetry plays in determining the detailed nature of the interactions in quantum electrodynamics and quantum chromodynamics.

Although most of this book focuses on nonrelativistic quantum mechanics, the first chapter with its discussion of the quantum mechanics of light, the ninth chapter with its discussion of the curve of binding energy, and the last chapter with its discussion of Feynman diagrams presume some knowledge of special relativity. Consequently, a discussion of the basics of special relativity is included in an appendix for the benefit of students who have not had a previous exposure to this subject.

A comprehensive solutions manual for the instructor is available from the publisher, upon request of the instructor.

It is a pleasure to acknowledge the people who have helped me during the writing of this book. I have benefited greatly from the assistance of my colleagues at Harvey Mudd College. Tom Helliwell, Theresa Lynn, Dan Petersen, and Patti Sparks read portions of the manuscript and gave me helpful feedback. Tom Donnelly and Peter Saeta read the entire manuscript and gave me many valuable comments. In addition to giving me feedback on Chapter 10, Vatche Sahakian has been unfailingly patient and helpful with my LaTeX questions. For the better part of the past decade, the introductory physics course that is taken by all students at HMC in their first semester has combined an introduction to special relativity (see Appendix A) and an introduction to quantum mechanics, as detailed in Chapter 1. At Swarthmore College, where I completed a preliminary version of the text while on sabbatical, Frank Moscatelli provided thoughtful comments on the whole manuscript and Eric Jensen helped me fine-tune the astrophysics sections. My Swarthmore College Physics 14 students were a great audience for field testing this book. I very much appreciate the feedback and encouragement that I received from them. Jeff Dunham of Middlebury College was kind enough to give me detailed feedback on the entire manuscript. In addition to Harvey Mudd College and Swarthmore College, a number of institutions, including Bryn Mawr College, California State University (San Bernardino and San Marcos), Grinnell College, Haverford College, Pomona College, Rice University, and Ursinus College, have used a prepublication version of the text. I want to thank Michael Schulz, Michael Burin, Mark Schneider, Dwight Whitaker, Paul Padley, and Tom Carroll for their input based on their experiences teaching with the book. I also want to thank the Mellon Foundation for its support, Bob Wolf for assistance with the book's subtitle, Lee Young for helpful copyediting, Christine Taylor and her staff at Wilsted & Taylor Publishing Services for producing the book, Jane Ellis of University Science Books for all her efforts in overseeing the publishing process, and my publisher, Bruce Armbruster, for his support of this project.

Please do not hesitate to contact me if you find errors or have suggestions that might improve the book.

John S. Townsend

Harvey Mudd College