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Elementary Quantum Mechanics in One Dimension
Article in American Journal of Physics · May 2005
DOI: 10.1119/1.1862639
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These personal anecdotes, and the orientation of the articles toward connecting with, and reflecting on, work by
Hawking provide a thread that binds together these reviews
of highly specialized subjects. Overall, this volume provides
a rewarding experience for researchers interested in gravity,
cosmology, and fundamental physics.
Sean M. Carroll is Assistant Professor in the Department of
Physics, Enrico Fermi Institute, and Kavli Institute for Cosmological Physics at the University of Chicago. His research
concerns theoretical aspects of gravitation, field theory, and
cosmology.
Elementary Quantum Mechanics in One Dimension. Robert Gilmore, 229 pp. Johns Hopkins University Press,
Baltimore, 2004. Price: $69.95 共cloth兲 ISBN: 0-80188014-9; $24.95 共paper兲 ISBN: 0-8018-8015-7. 共Alan J.
DeWeerd, Reviewer.兲
A text focusing exclusively on one-dimensional quantum
mechanics may sound too limited in scope or a bit dull, but
Gilmore’s book is neither. He makes a persuasive case that
much can be learned from detailed study of one-dimensional,
time-independent problems. This is accomplished by using a
computational method that can be applied to a wide variety
of interesting problems.
Elementary Quantum Mechanics is intended to supplement standard quantum mechanics textbooks, so it assumes
familiarity with Schrödinger’s equation and concepts like
transmission coefficients and bound states. Gilmore starts by
introducing transfer matrices and explaining how to factorize
them so that computations involve real 2⫻2 matrices. This
simplification is possible for piecewise-constant potentials,
which can be used to approximate many realistic potentials.
The transfer matrix method is a natural extension to the application of boundary conditions usually introduced in the
context of step and barrier potentials, so there would not be
too much overhead involved with teaching it.
There is a brief, but very good, comparison of the transfer
matrix with the scattering matrix. Gilmore acknowledges
that scattering matrices are more suitable for some advanced
problems. In particular, transfer matrices do not generalize
easily to three dimensions. However, they provide a unified
approach to one-dimensional systems that students can use to
explore some relatively sophisticated problems.
After covering these foundations in a mere 34 pages, the
book’s remaining three sections show how transfer matrices
can be used in calculations for scattering, bound states, and
periodic potentials. Each of these cases requires a slightly
different type of boundary condition. Gilmore provides detailed algorithms for performing computations, rather than
providing computer code in a specific language. In each section, the results for analytically solvable examples are given
to validate that a program is working correctly. The transfer
matrix approach is simple enough that it can be implemented
in a spreadsheet program.
The wide variety of applications covered makes this an
intriguing book. It includes some topics that are not usually
part of an undergraduate course, such as how bound and
scattering states are related and avoided level crossings.
Gilmore manages to present these in a way that is accessible
to undergraduates. The computational methods allow students to explore these topics without the more advanced
mathematical tools usually used to study them.
The explanation of the transfer matrix method is very
clear, with a couple of exceptions. The calculations of wave
functions and probability densities are not described completely, especially for the asymptotic regions. Filling in the
details would be a good homework assignment for students.
In the section on periodic potentials, Gilmore switches the
form of the wave functions without mentioning it for a few
chapters. This is somewhat confusing because the transfer
matrices change without explanation. Readers are advised to
look ahead at Chap. 42 to see the wave functions that are
used in the earlier chapters of Part IV.
The text is lacking in two relatively minor ways. First, it
contains almost no references, except in the concluding
chapter on solar cells. Suggestions for further reading would
be useful since the explanations of some topics are a bit
terse. Second, there are only few end-of-chapter problems. In
fact, so many results are presented that students using this
text might not feel the need to perform computations themselves. If students do not have a copy of the book, there will
be more for them to do.
This book is a rich source of ideas for activities or projects
suitable for students in an intermediate quantum mechanics
or a computational physics course. It is highly recommended
to those teaching undergraduate quantum mechanics.
Alan J. DeWeerd is an Associate Professor of Physics at the
University of Redlands. He and his students do research in
optics. He has also published several articles on optics in
physics education journals.
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Am. J. Phys., Vol. 73, No. 5, May 2005
Book Reviews
480