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American Journal of Physics

A Simplified Treatment of the Quantum-Mechanical Scattering Problem Using Wave Packets

Russell K. Hobbie

Published Dec 1, 1962• 3 citations
Publisher

Abstract

This paper presents a treatment of quantum-mechanical scattering suitable for a senior level course in modern physics. The entire treatment is based on the use of wave packets instead of unrealistic plane waves. The asymptotic behavior of the scattered wave packet is derived from physical considerations, and the Born approximation is then obtained for a wave packet in a manner which displays the details of the scattering process.

DOI
10.1119/1.1941838

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This paper presents a treatment of quantum-mechanical scattering suitable for a senior level course in modern physics. The entire treatment is based on the use of wave packets instead of unrealistic plane waves. The asymptotic behavior of the scattered wave packet is derived from physical considerations, and the Born approximation is then obtained for a wave packet in a manner which displays the details of the scattering process.
We discuss "the plane wave approximation" to quantum mechanical scattering using simple one-dimensional examples. The central points of the paper are that (a) plane waves should be thought of as infinitely wide wave packets, and (b) the calculations of reflection and transmission probabilities RRR and TTT in standard textbook presentations involve an approximation which is almost never discussed. We argue that it should be discussed explicitly, and that doing so provides a simple and intuitively revealing alternative way to derive and understand certain formulas. Using an under-appreciated exact expression for wave packet scattering probabilities, we calculate, for two standard examples, expressions for RRR and TTT for an incident Gaussian wave packet. Comparing these results to the corresponding ones calculated using the plane wave approximation helps illuminate the domain of applicability of that approximation.
A wave-packet formalism is presented for obtaining the differential cross section for the elastic scattering of a single particle by a potential. The method avoids difficulties inherent in the more common formulations such as interference between incident and scattered waves, box normalization, and density of final states. The time sequence of events in the scattering process is pictured in an intuitively appealing manner. The initial form of the wave packet is specified explicitly, and the time development of the wave packet is determined from the time-translation operator. The asymptotic form of the wave packet is investigated after a sufficiently long time lapse, and the cross section is obtained from a calculation of particle flux densities.
Beginning with the conventional asymptotic description of quantum scattering, we propose a simple wave function that is valid at all distances and compute the corresponding Bohmian trajectories. This 'simulated experiment' allows a concrete visualization of the process throughout the scattering interaction. Not only is this of pedagogic value at the undergraduate level, it also illustrates several important themes in time-dependent scattering theory, such as the importance of wave packets, the role of cross sections in describing measurement and the wave and particle pictures of quantum mechanics.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
Over the course of the past two to three decades, new tools of presentation and mathematical treatment have emerged and the subject matter of quantum mechanics has gone through significant changes. A Textbook on Modern Quantum Mechanics presents the selected elementary, intermediate, and advance topics with rejuvenated approach to the subject matter. Newly merged topics from contemporary physics and chemistry are included in the text as well as solved examples. The book covers: (i) fundamental discoveries that are the foundation of modern quantum mechanics; (ii) solution of Schrödinger’s wave equation for 1D problems and their importance; (iii) matrix and vector formulation of quantum mechanics; (iv) transformations, symmetries, and conservation laws; (v) angular and spin momenta; (vi) solution of Schrödinger equation for central potentials; (vii) time-independent perturbation theory, variational method and WKB approximation; (viii) quantum theory of scattering; (xi) many-particle systems and their quantum mechanical treatments; (x) time-dependent perturbations and the interaction of fields with matter; (xi) relativistic quantum mechanics; and (xii) quantization of fields and the second quantization. Key Features: It provides everything a student needs to know for succeeding at all levels of the undergraduate and graduate studies. It covers most of the topics that are taught under (a) elementary, (b) intermediate, and (c) advance courses of quantum mechanics at universities and colleges. It has detailed and elegant mathematical treatment with contemporary style of interpretation and presentation in simple English. Solved examples and unsolved exercises that are part of each chapter to consolidate the readers' understanding of fundamental concepts. The subject matter of the book is well tested on the students taught by the author over a period of 30 years. This is a valuable textbook for students pursuing Bachelor of Science, Master of Science, and Doctor of Philosophy (PhD) degrees in the subjects of Physics, Chemistry, and materials science in India, South Asian countries, the United States, and Europe.