Quantum Theory of Tunneling
DOI: 10.1063/1.1688073
Tunneling is one of the most striking of quantum phenomena. The term comprises a wealth of different effects having in common the possibility that a quantum system will be found in a classically forbidden region of space. Despite the multitude of tunneling processes, textbooks on quantum mechanics usually limit their discussions to the passage of a particle through a square potential barrier. Several more specialized books exist, but surprisingly no comprehensive monograph on quantum tunneling had been published before this past year. That lacuna was filled by Mohsen Razavy’s Quantum Theory of Tunneling . The book provides a good collection of examples of tunneling effects in atomic, molecular, condensed matter, and nuclear physics in the last three chapters; the rest of the book is essentially devoted to a rather impressive sweep of theoretical techniques.
Razavy has considerable experience in the field and has contributed to many of the topics included in his book. He pays a good deal of attention to semiclassical techniques, both in the simple case of a structureless particle in one dimension and in the more complicated multichannel and multidimensional cases. But he also discusses other techniques, such as the variable reflection amplitude method, inverse scattering methods, and techniques involving Feynman path integrals, Wigner functions, Heisenberg’s equations of motion, complex scaling, and optical potentials. Other subjects covered include tunneling times, group and signal velocities, classical descriptions, time-dependent barriers, Gamow’s theory, solvable models, motion in a space bounded by a surface of revolution, deviations from exponential decay, and decay widths calculated with scattering theory. One subject not addressed is dissipative tunneling, which is Razavy’s main area of interest.
Each of the book’s 26 short chapters has its own references. Razavy’s style is quite laconic. He usually gives a very brief introduction immediately followed by formalism. He offers little or no discussion, but clearly explains the mathematics and works out calculations step by step.
The price paid for the broad scope is a certain lack of depth that, in some cases, can be misleading. I would like to have read an account of the limitations of the Wigner trajectory concept discussed in chapter 14. I also regard chapters 17–19 on tunneling times more as a sample of results than as a balanced review. They do not really do justice to the enormous number of publications on the topic. In particular, chapter 17 would benefit from a discussion going beyond the elementary but unsatisfactory treatment of Ken Stevens’s work on evanescent waves. Chapter 19 considers Francis Low and Paul Mende’s objections to the use of scattering theory for a Gaussian state that is initially close to a potential barrier, but does not adequately discuss work that has clarified how to handle those objections. Another example of the danger of brevity is the discussion of the time–energy uncertainty principle and tunneling given in chapter 2. I find the arguments vague and unconvincing, as usually happens when the “principle” is invoked without making precise its exact meaning.
A technical shortcoming of the book is the poor resolution of the three-dimensional plots. Moreover, those plots have no axis labels, so interpreting them may be quite difficult for some readers. Some of the 2D plots are also missing axis labels; absent as well are calculational details such as units or masses that are necessary for reproducing or interpreting the displayed curves. I have detected name misspellings and, in chapter 5, displaced reference numbers.
All in all, though, I enjoyed Quantum Theory of Tunneling and think that it will be useful for students and researchers alike. The main strength of the book is Razavy’s concise summaries of the many techniques he reviews. It would be easy to adapt part or all of the material for various graduate courses—not necessarily about tunneling—because several of the methods discussed in the book are applicable to a wide range of phenomena.
More about the Authors
Juan Gonzalo Muga. University of the Basque Country, Bilbao, Spain .
