In Paul Halmos’s celebrated automathography, I want to be a Mathematician, he claims that most articles in the Scientific American are too hard for him. In the preface of this book by Thomas F. Jordan, the author states that most people who read the Scientific American can read this book. Nevertheless I found this book quite readable and a delightful introduction to some of the basic ideas in matrix mechanics, as well as offering a nice on-ramp to quantum mechanics that could be used in undergraduate mathematics or physics courses.
The book divides naturally into two parts, the first of which is decidedly more elementary than the second. The first half of the book uses only elementary facts about matrices, high school algebra and some very simple probability. Mostly this first half is an introduction to complex numbers and relatively simple matrix and probability calculations that would be accessible to bright undergraduates and high school students.
As with any attempt to take a very complex subject and make it accessible to a lay audience, some topics have to be omitted. While the book strives to minimize prerequisites, there is a cost, in that many important ideas are glossed over. Eigenvalues and eigenvectors are not mentioned even though they are used implicitly. The author goes through considerable contortions to describe what it means for a matrix to represent a physical quantity; in effect, he is describing observables and Hermitian matrices, though these terms are nowhere used. State spaces are not mentioned, though they are lurking behind many of the examples. Neither are bras nor kets nor wave functions. Entanglement is only vaguely alluded to in connection with Einstein’s criticisms. The Pauli spin matrices are introduced and used, but no hint of where they come from is given. That said, the book does a good job of illustrating how matrices and probabilities are used with two and four state quantum systems such as spin angular momentum, a quantum analogue of angular momentum and its corresponding magnetic moment, a quantum analogue of a compass needle.
The mathematics of quantum mechanics has two fundamental aims, to describe the spectrum of possible results from an experiment and to finding the probability of each result in this spectrum. In other words, the goal is to predict the outcomes of experiments. Jordan’s book gives some nice examples of this. The connection with quantum physics makes for an exciting introduction to matrices.
Some additional nice features of the book include an enlightening explanation in terms of elementary matrices for why simultaneous observables (again the author does not use this word) must commute and a proof of why Heisenberg’s famous formula \(QP-PQ=i\hbar\) implies the uncertainty principle for position and momentum \(Q\) and \(P\).
The second half of the book is considerably more advanced. Topics discussed include harmonic oscillators and their energy quantization, the Bohr model of the atom, as well as the quantization of orbital angular momentum and the hydrogen atom.
There are some exercises in the book, but really too few. The range of complexity of the matrix algebraic calculations range from the very simple (in the first part of the book) to the quite involved later. Still, despite some of these drawbacks, I would rate this book highly.
There is only one minor thing I might quibble with. On page 149 the author asserts that Dirac invented the word commute. This term was used in the mathematics literature as early as 1814 by the French mathematician François-Joseph Servois and appears in Cayley’s famous 1858 paper on matrices. Dirac certainly used it and may have been the first to use it in the quantum physics literature but the term was not new.
Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.