During much of the last century, British publishers, such as Methuen and Oliver and Boyd, put out a series of inexpensive little books on mathematics and physics, meant as introductory guides to one or another undergraduate topic. One of the earliest (1910) and best known was Silvanus P. Thompson's Calculus Made Easy, which begins with “What one fool can do, another can.” (It's still in print; Feynman was familiar with this book and quoted its epigraph once or twice.) These little books did not pretend to be full-blown texts. But like a well-written guide to a mountain trail, they made the beginner's ascent a great deal easier. Such a book cannot be exhaustive, but it need not be superficial. Instead, the basics can be treated carefully, as the topics are deliberately few. Some years ago, Cambridge University Press introduced a fine series of short introductions under the general title “A Student's Guide.” The series thus far includes Maxwell's equations, analytical mechanics, and Fourier transforms, among other topics. The author of several popular titles in the series, Daniel A. Fleisch, has now tackled the Schrödinger equation. This seems an apt topic for a little book. I would have been very glad to have it when I was learning quantum theory.
A student beginning quantum mechanics has to take a great deal on faith, not merely the Schrödinger equation itself, which appears out of nowhere, but also eigenfunctions and operators and commutators, the delta function, and a flurry of special functions: Hermite polynomials, spherical harmonics, and associated Laguerre polynomials. Ideally, a beginner would have had good courses previously in linear algebra, multivariable calculus, and partial differential equations. Judging from my own experience, most of us made first contact with many species in the quantum menagerie just as we were trying to learn the strangest physics we'd encountered. A little help would have been very welcome. (Heisenberg himself did not know what a matrix was when he arrived at non-commutative operators. Fortunately, Max Born, one of Heisenberg's teachers, did.)
All this is to say that there are two separate mountains to be scaled while learning quantum mechanics, one physical and the other mathematical. The student must attain a worldview in terms of probability waves and put aside for a while Newton's deterministic mechanics. At the same time, she must acquire a large set of new tools and the requisite skills to use them. An intrepid author hoping to provide a traveler's guide to the Schrödinger equation, and hence to quantum mechanics, has a difficult task, particularly if the guide is to fit within two hundred pages. For the most part, this book succeeds admirably.
The book is divided into five short chapters, treating sequentially vectors and functions; operators and eigenfunctions; the form and meaning of the Schrödinger equation (in one and three dimensions); solutions to the equation with an introduction to Fourier methods; and worked examples of solving the equation for some simple potentials: infinite and finite square wells and the one-dimensional harmonic oscillator potential, both analytically and algebraically. Important terms are printed in bold type. At the end of each section in the first two chapters, there is a useful summary of the section's results; readers would benefit from summaries in the other chapters as well. Each chapter ends with ten fairly easy problems (exactly the right level for a study guide). These topics provide a good introduction to a first semester of quantum mechanics, and on the whole, the author does an excellent job. He is particularly good at the fine details of the mathematical notation (explaining, as he did in his Student's Guide to Maxwell's Equations, what each factor in a given equation means and what it does) and also in graphical analysis of the meaning of a solution or an expression. The book has an accompanying website with additional references, appendices showing various arguments and techniques ancillary to the main development, a glossary of important terms, and videos of the author working through each section of the book. Additionally, there are worked solutions for every problem in the book, and both Octave and Matlab code for simulations. (I lack Matlab and could not make any of the three simulations I sampled run in Octave, which may simply indicate my lack of experience with the program.) What's in the book is excellent although I have some concerns about where the content appears and how some of it is presented. The author displays a peculiar reticence concerning some important conclusions.
He succeeds in making a lot of new mathematics less intimidating although advanced concepts are occasionally introduced with inadequate scaffolding, and sometimes, their subsequent discussion is spotty. In his first description of Dirac bra and ket notation, he differentiates bra vectors from ket vectors in function space as “linear functionals,” also describing them as “covectors” and vectors “in a dual space,” like “one-forms.” While all of this is true, none of it is likely to be helpful to students whose mathematical background is such that they need help understanding the Laplacian, to which he sensibly devotes several pages in Chapter 4. He defines a linear functional abstractly as “a mathematical device that maps a vector to a scalar.” A simple example is readily at hand, for the particular case of a square integrable function (a ket vector in a Hilbert space of functions): a definite integral (the meaning of both a one-form and a function space bra vector), as is found in some other books on quantum theory. Eventually, the author provides this example, but it has to wait nine pages. The extensive parallelism between two dimensional and three-dimensional vectors and the more abstract Hilbert space, particularly function space, is hinted repeatedly; functions are described as “like vectors,” but not called “vectors in (a) Hilbert space”. The author provides a strong visual representation of a function as a kind of vector, with an infinite number of components, and its argument as a correspondingly infinite component index. It seems to me, though, that this parallelism is not developed to the point where a student would be likely to make the appropriate connections between linear combinations and eigenfunction expansions. (The terms “function space” and “linear combination” do not appear, or at least I failed to find them in a careful reading; “functions as abstract vectors” is given in the index on p. 14, but a better reference would be to the bottom of p. 17.)
Some other concepts are introduced early on, but not quite fleshed out; and when they appear later, there is no reference back to what was discussed earlier. In Chapter 3, a plausibility argument is given for the form of the Schrödinger equation, with the expression for the total energy given not in terms of E and p but in terms of E and k. Later on, we are millimeters away from the usual identification of energy and momentum with derivatives acting on wave functions, and yet, we don't get there. The momentum's identification with a derivative is made eventually but not the energy's. The Schrödinger equation's connection via the derivative replacements of E and p to the usual expression for total energy is not made as clearly as it could be; instead, operator H is described as “an operator associated with total energy E.” This seems to me a missed opportunity. The representation of momentum as a derivative is rederived in Chapter 4, but there is no indication that this was found in the previous chapter. There is a nice discussion of uncertainty and variance in Chapter 2, but the author does not state plainly that the uncertainty equals the square root of the variance. Before (2.61), he writes “Calling the uncertainty in position Δx, the square of the uncertainty is given by” (an expression equal to the variance). He writes down the appropriate equations and comes close to saying it, but the bald statement is not there. Beginners need simple, clear, and unambiguous statements. There is an excellent, explicit treatment of uncertainty with respect to Gaussian wave functions, and the minimum Heisenberg uncertainty bound is derived, but there is no mention of the earlier discussion of uncertainty. Also, while the non-commutation of the position and momentum is described as “the essence of quantum indeterminacy,” the author leaves the student to infer that, perhaps, when two or more variables do commute, as in x and y, or px and py, then no uncertainty arises, another missed opportunity. In the second discussion of the Dirac bra and ket notation (Chapter 4), it is mentioned that the eigenstate of a particle localized to a position x1 is a delta function, δ(x − x1), and a brief argument is given that if xf(x1) = x1 f(x1), then f(x1) must equal this delta function. Why not then state as a rule the formula f(x)δ(x − a) = f(a)δ(x − a)? A relation for ⟨x | k⟩ as a plane wave is stated (4.19), but there is no corresponding relation for ⟨x | p⟩ (although it is given in the online solution to problem 3 in Chapter 4). Explicit statements that ⟨x | x′⟩ = δ(x − x′) and ⟨p | p′⟩ = δ(p − p′) are also absent. Sometimes, important results are simply not stressed; they're present, but they're not given the emphasis they should receive. For example, in Chapter 2, a student will learn the critical results that a Hermitian operator must have only real eigenvalues and that if two eigenfunctions of a Hermitian operator have different eigenvalues, they must be orthogonal. But she would have to read the page pretty carefully for those lessons to make an impression. As a last example, in Chapter 4, the author states that “Quantum wave functions must be square integrable.” Two chapters earlier he has told the reader that for functions to be in Hilbert space, they have to be square integrable. That's the mathematical reason. But there is also a physical reason: if the Copenhagen interpretation (previously discussed carefully) is to hold, then as a measure of probability, the wave functions must be normalizable to unity—impossible if the norm is not finite.
It's a trivial gripe, but the notation is not ideal. This may be beyond the author's power to change. Dot products are denoted with a composition symbol, a small open circle, as opposed to a dark dot, large enough not to be overlooked. Vectors are written with overhead arrows, which might be fine if, at the same time, carets were not used to indicate operators (the carets seem to me a good thing). A better choice would be to use bold types for vectors, to avoid vector operators wearing two hats. Finally, two overlines for matrices? Why not san-serif capital letters? We have a wonderful (and free!) mathematical typesetting standard in LaTeX; there is no longer any excuse for poor mathematical notation in print.
In summary, this is a very good book and likely to be very helpful to a student beginning the study of quantum mechanics. If I were teaching quantum mechanics, I would recommend it to my students. But it could be a great book, and I hope that it will achieve that in a second edition.
David Derbes recently retired from the University of Chicago Laboratory Schools, where he taught high school physics for 33 years. Trained as a particle physicist, his post-career occupation is manuscript salvage, shepherding lecture notes and neglected texts into new books. He can be reached at [email protected].
