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American Journal of Physics

Through scattering theory with gun and camera: Coping with conventions in collision theory

Michael A. Morrison, Andrew N. Feldt

Published Dec 13, 2006• 11 citations
Publisher

Abstract

Four ambiguities are inherent in time-independent quantum-mechanical scattering theory: two in the normalization of continuum functions, and two in the relation between various scattering matrices. These ambiguities allow the use of different conventions in scattering theory. This freedom has led to inconsistencies among textbooks, monographs, review articles, and research papers. We identify these ambiguities, indicate their context and origin in scattering theory, give the key equations of scattering theory in a form that can by adapted to any convention, and show interrelations between these equations that can lead to confusion and error. We also give several diagnostic approaches for determining the conventions used in a particular source, and reference conventions adopted in widely used books on scattering theory.

DOI
10.1119/1.2358156

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Four ambiguities are inherent in time-independent quantum-mechanical scattering theory: two in the normalization of continuum functions, and two in the relation between various scattering matrices. These ambiguities allow the use of different conventions in scattering theory. This freedom has led to inconsistencies among textbooks, monographs, review articles, and research papers. We identify these ambiguities, indicate their context and origin in scattering theory, give the key equations of scattering theory in a form that can by adapted to any convention, and show interrelations between these equations that can lead to confusion and error. We also give several diagnostic approaches for determining the conventions used in a particular source, and reference conventions adopted in widely used books on scattering theory.
The major themes of quantum mechanics are the physical interpretations of interactions in bound systems and particle scatterings. Stability of atoms and molecules is necessary in order that materials in nature could have any given determined physical and chemical properties, according to which these materials are distinguished and identified. Standard scattering theory has originally been introduced in nuclear physics, where the interactions are of short range, so that in the asymptotic region, the aggregates can be considered as being fully free. The required behaviours of the total scattering state are determined by the very nature of the interactions between the widely separated aggregates in the asymptotic region of scattering. In the old literature a terminological difference has sometimes been made between the notions of scattering and collision.
We begin by considering the following scattering experiment. A projectile, e.g., an atom A in a given electronic state, collides with a target which we will take as a diatomic molecule BC in its ground electronic, vibrational and rotational state. For a short period of time an activated complex [ ABC ] # is generated. As time passes the activated complex can break into different products. For instance, in our example these products will be A + BC , B + AC, C + AB and A + B + C . Each one of the possible products can be in different electronic, vibrational and rotational quantum states. The total energy which is originally the sum of the electronic and translational energies of the projectile A and the electronic, vibrational, rotational and translational energies of target BC is conserved during the scattering process. Time-independent scattering theory enables one to calculate the probability of obtaining the specific products in given quantum states and the kinetic energy distribution of the products as a function of the total energy of the system without the need to solve the time-dependent Schrödinger equation. The time-independent formulation of scattering theory is based on the ability to propagate analytically an initial given wavepacket, Φ(0), to infinite times. That is, we need to get a closed form expression for lim t →±∞ e −iĤ t /ħ |Φ( t = 0)〉. To quote from the introduction of the excellent book on scattering theory written by Taylor: “The most important experimental technique in quantum physics is the scattering experiment. That this is so is clear from even the briefest review of modern physics”.
In their article, Chandralekha Singh, Mario Belloni, and Wolfgang Christian focus exclusively on “functional understanding of quantum mechanics,” which they claim “is quite distinct from the foundational issues alluded to by Feynman.”But are the foundational and the functional really so distinct? The work of other physics education researchers suggests not. For example, in a classic article, Alan Van Heuvelen discusses students’ prevalent and frustrating use of “primitive formula-centered problem-solving strategies” 1 1. A. Van Heuvelen, Am. J. Phys. 59, 891 (1991). https://doi.org/10.1119/1.16667 and suggests that physical, intuitive understanding developed through qualitative diagrams and models “must come before students start using math in problem solving. The equations become crutches that short-circuit attempts at understanding.” Van Heuvelen also urges that “instead of thinking of [problems] as an effort to determine some unknown quantity, [teachers] might … encourage students to think of the problem statement as describing a physical process—a movie of a region of space during a short time interval or of an event at one instant of time.” I suspect Singh, Belloni, and Christian would agree with this advice. They comment that such “qualitative understanding of quantum mechanics is much more challenging than facility with the technical aspects.”But isn't the main barrier to such intuitive, qualitative understanding the nature of quantum mechanics itself—at least, the version of the theory advocated by Niels Bohr, Werner Heisenberg, and virtually every textbook writer since? Why should we expect students to invest the time and energy necessary to, say, visualize the time-dependence of |ψ|2 when we also preach the ambiguous and contradictory Copenhagen dogma that ψ does not represent anything physically real, yet still provides a complete description of physical reality? Why are we surprised that students are confused about, and don't take seriously, something that we assure them is, at best, some kind of algorithmic fantasy? Is there really any difference between “shut up and calculate” and “plug and chug”?Why not teach them Bohmian mechanics—an alternative (deterministic) version of quantum theory in which particles are particles (and really exist, all the time) and the same dynamical laws apply whether anyone is looking or not? 2 2. S. Goldstein, in Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/qm-bohm. About this alternative theory John S. Bell asked, “Why is [it] ignored in text books? Should it not be taught … as an antidote to the prevailing complacency? To show that vagueness, subjectivity, and indeterminism are not forced upon us by experimental facts, but by deliberate theoretical choice?” 3 3. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge U. Press, New York (2004), p. 173. https://doi.org/10.1017/CBO9780511815676.021 If we really want to help students understand quantum mechanics, the first step is to reject the confusion-spawning foundational vagueness, ambiguity, and philosophical absurdity of Copenhagen quantum theory, and adopt a clearer, more scientific, less fuzzy version. (See Sheldon Goldstein's two-part article “Quantum Theory Without Observers,” Physics Today, March 1998, page 42, and Physics Today, April 1998, page 38) The first step, in short, is to present them with a theory that can be understood.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Van Heuvelen, Am. J. Phys. 59, 891 (1991). https://doi.org/10.1119/1.16667 , Google ScholarCrossref2. S. Goldstein, in Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/qm-bohm. Google Scholar3. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge U. Press, New York (2004), p. 173. https://doi.org/10.1017/CBO9780511815676.021 , Google ScholarCrossref© 2007 American Institute of Physics.
A general expression for spherical harmonics as an explicit function of the angle coordinates is derived using standard methods of group-representation theory. This expression is more useful to physicists than a corresponding expression available in the mathematical literature for two reasons: (1) Its normalization is the one most commonly used by physicists, rather than one of several other normalization conventions used mainly by mathematicians. (2) Its form is especially convenient for use in the operator equations which occur in quantum mechanics. Furthermore, the derivation serves the pedagogical purpose of providing a simple illustration of the methods of group-representation theory. A set of relations useful in solving the Dirac equation are listed, both because their verification provides an example of the utility of the expression derived, and because the form in which these relations are stated in the literature is apt to result in confusion stemming from the various different normalization conventions in use.