N.1 Why this book?

With the current emphasis on nanotechnology, quantum mechanics is becoming increasingly essential to engineering students. Yet, the typical quantum mechanics texts for physics students are not written in a style that most engineering students would likely feel comfortable with. Furthermore, an engineering education provides very little real exposure to modern physics, and introductory quantum mechanics books do little to fill in the gaps. The emphasis tends to be on the computation of specific examples, rather than on discussion of the broad picture. Undergraduate physics students may have the luxury of years of further courses to pick up a wide physics background, engineering graduate students not really. In addition, the coverage of typical introductory quantum mechanics books does not emphasize understanding of the larger-scale quantum system that a density functional computation, say, would be used for.

Hence this book, written by an engineer for engineers. As an engineering professor with an engineering background, this is the book I wish I would have had when I started learning real quantum mechanics a few years ago. The reason I like this book is not because I wrote it; the reason I wrote this book is because I like it.

This book is not a popular exposition: quantum mechanics can only be described properly in the terms of mathematics; suggesting anything else is crazy. But the assumed background in this book is just basic undergraduate calculus and physics as taken by all engineering undergraduates. There is no intention to teach students proficiency in the clever manipulation of the mathematical machinery of quantum mechanics. For those engineering graduate students who may have forgotten some of their undergraduate calculus by now, there are some quick and dirty reminders in the notations. For those students who may have forgotten some of the details of their undergraduate physics, frankly, I am not sure whether it makes much of a difference. The ideas of quantum mechanics are that different from conventional physics. But the general ideas of classical physics are assumed to be known. I see no reason why a bright undergraduate student, having finished calculus and physics, should not be able to understand this book. A certain maturity might help, though. There are a lot of ideas to absorb.

My initial goal was to write something that would “read like a mystery novel.” Something a reader would not be able to put down until she had finished it. Obviously, this goal was unrealistic. I am far from a professional writer, and this is quantum mechanics, after all, not a murder mystery. But I have been told that this book is very well written, so maybe there is something to be said for aiming high.

To prevent the reader from getting bogged down in mathematical details, I mostly avoid nontrivial derivations in the text. Instead I have put the outlines of these derivations in notes at the end of this document: personally, I enjoy checking the correctness of the mathematical exposition, and I would not want to rob my students of the opportunity to do so too. In fact, the chosen approach allows a lot of detailed derivations to be given that are skipped in other texts to reduce distractions. Some examples are the harmonic oscillator, orbital angular momentum, and radial hydrogen wave functions, Hund’s first rule, and rotation of angular momentum. And then there are extensive derivations of material not even included in other introductory quantum texts.

While typical physics texts jump back and forward from issue to issue, I thought that would just be distracting for my audience. Instead, I try to follow a consistent approach, with as central theme the method of separation-of-variables, a method that most mechanical graduate students have seen before already. It is explained in detail anyway. To cut down on the issues to be mentally absorbed at any given time, I purposely avoid bringing up new issues until I really need them. Such a just-in-time learning approach also immediately answers the question why the new issue is relevant, and how it fits into the grand scheme of things.

The desire to keep it straightforward is the main reason that topics such as Clebsch-Gordan coefficients (except for the unavoidable introduction of singlet and triplet states) and Pauli spin matrices have been shoved out of the way to a final chapter. My feeling is, if I can give my students a solid understanding of the basics of quantum mechanics, they should be in a good position to learn more about individual issues by themselves when they need them. On the other hand, if they feel completely lost in all the different details, they are not likely to learn the basics either.

That does not mean the coverage is incomplete. All topics that are conventionally covered in basic quantum mechanics courses are present in some form. Some are covered in much greater depth. And there is a lot of material that is not usually covered. I include significant qualitative discussion of atomic and chemical properties, Pauli repulsion, the properties of solids, Bragg reflection, and electromagnetism, since many engineers do not have much background on them and not much time to pick it up. The discussion of thermal physics is much more elaborate than you will find in other books on quantum mechanics. It includes all the essentials of a basic course on classical thermodynamics, in addition to the quantum statistics. I feel one cannot be separated from the other, especially with respect to the second law. While mechanical engineering students will surely have had a course in basic thermodynamics before, a refresher cannot hurt. Unlike other books, this book also contains a chapter on numerical procedures, currently including detailed discussions of the Born-Oppenheimer approximation, the variational method, and the Hartree-Fock method. Hopefully, this chapter will eventually be completed with a section on density-functional theory. (The Lennard-Jones model is covered earlier in the section on molecular solids.) The motivation for including numerical methods in a basic exposition is the feeling that after a century of work, much of what can be done analytically in quantum mechanics has been done. That the greatest scope for future advances is in the development of improved numerical methods.

Knowledgeable readers may note that I try to stay clear of abstract mathematics when it is not needed. For example, I try to go slow on the more abstract vector notation permeating quantum mechanics, usually phrasing such issues in terms of a specific basis. Abstract notation may seem to be completely general and beautiful to a mathematician, but I do not think it is going to be intuitive to a typical engineer. The discussion of systems with multiple particles is centered around the physical example of the hydrogen molecule, rather than particles in boxes. The discussion of solids in chapter 10 avoids the highly abstract Dirac comb (delta functions) mathematical model in favor of a physical discussion of more realistic one-di­men­sion­al crystals. The Lennard-Jones potential is derived for two atoms instead of harmonic oscillators.

The book tries to be as consistent as possible. Electrons are grey tones at the initial introduction of particles, and so they stay through the rest of the book. Nuclei are red dots. Occupied quantum states are red, empty ones grey. That of course required all figures to be custom made. They are not intended to be fancy but consistent and clear. I also try to stay consistent in notations throughout the book, as much as is possible without deviating too much from established usage.

When I derive the first quantum eigenfunctions, for a pipe and for the harmonic oscillator, I make sure to emphasize that they are not supposed to look like anything that we told them before. It is only natural for students to want to relate what we told them before about the motion to the completely different story we are telling them now. So it should be clarified that (1) no, they are not going crazy, and (2) yes, we will eventually explain how what they learned before fits into the grand scheme of things.

Another difference of approach in this book is the way it treats classical physics concepts that the students are likely unaware about, such as canonical momentum, magnetic dipole moments, Larmor precession, and Maxwell’s equations. They are largely “derived“ in quantum terms, with no appeal to classical physics. I see no need to rub in the student's lack of knowledge of specialized areas of classical physics if a satisfactory quantum derivation is readily given.

This book is not intended to be an exercise in mathematical skills. Review questions are targeted towards understanding the ideas, with the mathematics as simple as possible. I also try to keep the mathematics in successive questions uniform, to reduce the algebraic effort required. There is an absolute epidemic out there of quantum texts that claim that “the only way to learn quantum mechanics is to do the exercises,” and then those exercises turn out to be, by and large, elaborate exercises in integration and linear algebra that take excessive time and have nothing to do with quantum mechanics. Or worse, they are often basic theory. (Lazy authors that claim that basic theory is an exercise avoid having to cover that material themselves and also avoid having to come up with a real exercise.) Yes, I too did waste a lot of time with these. And then, when you are done, the answer teaches you nothing because you are unsure whether there might not be an algebraic error in your endless mass of algebra, and even if there is no mistake, there is no hint that it means what you think it means. All that your work has earned you is a 75/25 chance or worse that you now know something that is not true. Not in this book.

Finally, this document faces the very real conceptual problems of quantum mechanics head-on, including the collapse of the wave function, the indeterminacy, the nonlocality, and the symmetrization requirements. The usual approach, and the way I was taught quantum mechanics, is to shove all these problems under the table in favor of a good sounding, but upon examination self-contradictory and superficial story. Such superficiality put me off solidly when they taught me quantum mechanics, culminating in the unforgettable moment when the professor told us, seriously, that the wave function had to be symmetric with respect to exchange of bosons because they are all truly the same, and then, when I was popping my eyes back in, continued to tell us that the wave function is not symmetric when fermions are exchanged, which are all truly the same. I would not do the same to my own students. And I really do not see this professor as an exception. Other introductions to the ideas of quantum mechanics that I have seen left me similarly unhappy on this point. One thing that really bugs me, none had a solid discussion of the many worlds interpretation. This is obviously not because the results would be incorrect, (they have not been contradicted for half a century,) but simply because the teachers just do not like these results. I do not like the results myself, but basing teaching on what the teacher would like to be true rather on what the evidence indicates is true remains absolutely unacceptable in my book.