What’s Wrong With These Equations?

The 4^th edition of Intermediate Physics for Medicine and Biology is full of equations: thousands of them. Each one must fit into the text in a way to make the book easy to read. How?

N. David Mermin wrote a fascinating essay that appeared in the October 1989 issue of Physics Today titled “What’s Wrong With These Equations?” You can find it online at www.cvpr.org/doc/mermin.pdf. It begins

A major impediment to writing physics gracefully comes from the need to embed in the prose many large pieces of raw mathematics. Nothing in freshman composition courses prepares us for the literary problems raised by the use of displayed equations.

Mermin then presents three rules “that ought to govern the marriage of equations to readable prose”

• Rule 1 (Fisher’s rule): Number all displayed equations.


• Rule 2 (Good Samaritan rule): When referring to an equation identify it by a phrase as well as a number.


• Rule 3 (Math is Prose rule): End a displayed equation with a punctuation mark.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I violate Fisher’s rule: some of our displayed equations are not numbered. All I can say is, there are lots of equations in our book, and revising it to obey Fisher’s rule would require more effort than we are willing to expend.

I know you are wondering how an essay about punctuating and numbering equations could possibly be interesting, but Mermin makes the subject entertaining. And if you ever find yourself writing an article that contains equations, obeying his three rules will make the article easier to read.

Boojums All the Way Through,
by N. David Mermin.

Many physicists know Mermin for his renowned textbook Solid State Physics with Neil Ashcroft. His series of “Reference Frame” essays in Physics Today are all delightful, particularly the ones with Professor Mozart. Several Reference Frame essays are reprinted in his book Boojums All the Way Through: Communicating Science in a Prosaic Age. The title essay describes Mermin’s quest to establish the whimsical word “Boojum” as a scientific term for a phenomenon in superfluidity. If you want to learn to write physics well, read Mermin.

The Feynman Lectures

The Feynman Lectures on Physics,
by Richard Feynman.

On page 318 of the 4^th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite The Feynman Lectures on Physics. Reading The Feynman Lectures, written by Nobel Prize winner Richard Feynman, is a rite of passage for future physicists. Feynman describes how he came to present the lectures in his preface:

These are the lectures in physics that I gave last year and the year before to the freshman and sophomore classes at Caltech. The lectures are, of course, not verbatim—they have been edited, sometimes extensively and sometimes less so. The lectures form only part of the complete course. The whole group of 180 students gathered in a big lecture room twice a week to hear these lectures and then they broke up into small groups of 15 to 20 students in recitation sections under the guidance of a teaching assistant. In addition, there was a laboratory section once a week.

Although written for freshman and sophomores, most physics students read The Feynman Lectures a bit later in their education. I recall reading them the summer between graduation from the University of Kansas and starting graduate school at Vanderbilt University. There is some biological and medical physics in the lectures. For instance, Chapters 35 and 36 of Volume 1 are about vision and the eye. In Chapter 3 (The Relation of Physics to Other Subjects), Feynman describes his reductionist point of view about biology:

Certainly no subject or field is making more progress on so many fronts at the present moment, than biology, and if we were to name the most powerful assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms, and that everything that living things do can be understood in terms of jigglings and wigglings of atoms.

And in Volume 2 (Chapter 1), Feynman had this to say about the impact of electricity and magnetism on life:

Now we realize that the phenomena of chemical interaction and, ultimately, of life itself are to be understood in terms of electromagnetism.

He closed that chapter with a favorite quote of mine:

From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.

Anyone wondering what to get an aspiring physicist for a holiday gift might want to consider The Feynman Lectures. If you are looking for lighter reading, I suggest two autobiographical books by Feynman: Surely You’re Joking Mr. Feynman, and What Do You Care What Other People Think? They’re delightful and hilarious.

Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles

Quantum Physics of Atoms,
Molecules, Solids, Nuclei, and Particles,
by Eisberg and Resnick.

One of the sources that Russ Hobbie and I cite most often in the 4th edition of Intermediate Physics for Medicine and Biology is Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, by Robert Eisberg and Robert Resnick. I used the first (1974) edition of this textbook when I was an undergraduate studying physics at the University of Kansas. Quantum Physics was the book where I was first introduced to the ideas of quantum mechanics, to the Schrodinger equation, and to nuclear physics. A second edition was published in 1985, but I can find nothing about a third edition in the last 25 years. Despite it being somewhat out-of-date, I still consider this book to be one of the best sources of information about modern physics. Below is the first paragraph of the preface:

The basic purpose of this book is to present clear and valid treatments of the properties of almost all the important quantum systems from the point of view of elementary quantum mechanics. Only as much quantum mechanics is developed as is required to accomplish the purpose. Thus we have chosen to emphasize the applications of the theory more than the theory itself. In so doing we hope that the book will be well adapted to the attitudes of contemporary students in a terminal course on the phenomena of quantum physics. As students obtain an insight into the tremendous explanatory power of quantum mechanics, they should be motivated to learn more about the theory. Hence, we hope that the book will be equally well adapted to a course that is to be followed by a more advanced course in formal quantum mechanics.

I have never taught the modern physics class here at Oakland University, but if I did I would certainly consider using Eisberg and Resnick’s book. When I have taught the undergraduate quantum mechanics class (taken after modern physics) I used another wonderful book, Introduction to Quantum Mechanics by David Griffiths. There are several good quantum mechanics books at the graduate level, but I—a biomedical physicist—have never been asked to teach graduate quantum mechanics. (Are they telling me something?)

Intermediate Physics for Medicine and Biology doesn’t make much use of quantum ideas, except at a very qualitative level. Schrodinger’s equation is only mentioned once (on page 49), and is never written out. The idea of discrete quantum energy levels is introduced in Chapter 3 when we discuss statistical mechanics, and again in Chapter 14 when explaining atomic spectra. However, concepts related to quantization of light are important. For instance, thermal (blackbody) radiation is discussed in Section 14.7 (and is covered elegantly in the first chapter of Eisberg and Resnick) and Compton scattering is analyzed in Sec. 15.4. Quantum Physics should provide all the background you will need to understand these and other modern physics topics.

Clark and Plonsey

Problem 30 in Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology is based on a paper by John Clark and Robert Plonsey (“The Extracellular Potential of a Single Active Nerve Fiber in a Volume Conductor,” Biophysical Journal, Volume 8, Pages 842–864, 1968). This paper shows how to calculate the extracellular potential from the transmembrane potential, with results shown in our Fig. 7.13.

The calculation involves some mathematical concepts that are slightly advanced for Intermediate Physics for Medicine and Biology. First, the potentials are written in terms for their Fourier transforms. Russ Hobbie and I don’t cover Fourier analysis until Chapter 11, so the problem just assumes a sinusoidal spatial dependence. We also introduce Bessel functions for the first time in the book (to be precise, modified Bessel functions of the first and second kind). Bessel functions arise naturally when solving Laplace’s equation in cylindrical coordinates.

I have admired Clark and Plonsey’s paper for years, and was glad to see this problem introduced into the 4th edition of our book. Robert Plonsey was a professor at Case Western Reserve University from 1968-1983. He then moved to Duke University, where he was when I came to know his work while I was a graduate student. I am most familiar with his research on the bidomain model of cardiac tissue, often in collaboration with Roger Barr (e.g., “Current Flow Patterns in Two-Dimensional Anisotropic Bisyncytia with Normal and Extreme Conductivities,” Biophysical Journal, Volume 45, Pages 557–571 and “Propagation of Excitation in Idealized Anisotropic Two-Dimensional Tissue,” Biophysical Journal, Volume 45, Pages 1191–1202). Plonsey was elected as a member of the National Academy of Engineering in 1986 for “the application of electromagnetic field theory to biology, and for distinguished leadership in the emerging profession of biomedical engineering.” He retired from Duke in 1996 as the Pfizer Inc./Edmund T. Pratt Jr. University Professor Emeritus of Biomedical Engineering. He has won many awards, such as the 2000 Millennium Medal from the IEEE Engineering in Medicine and Biology Society and the 2004 Ragnar Granit Prize from the Ragnar Granit Foundation. John Clark is currently a Professor of Electrical and Computer Engineering at Rice University. He is a Life Fellow in the Institute of Electrical and Electronics Engineers (IEEE) “for contributions to modeling in electrophysiology and cardiopulmonary systems.”

One of my earliest papers was an extension of Clark and Plonsey’s model to a strand of cardiac tissue, using the bidomain model (“A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue,” IEEE Transactions of Biomedical Engineering, Volume 33, Pages 467–469, 1986.) The mathematics is almost the same as in their paper—Fourier transforms and Bessel functions—but the difference is that I modeled a multicellular strand of tissue, like a papillary muscle in the heart, that contains of both intracellular and interstitial spaces (the two domains of the “bidomain” model). A comparison of my paper to Clark and Plonsey’s earlier work indicates how influential their research was on my early development as a scientist. They were cited in the first sentence of my paper.