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Canadian Journal of Physics

SCATTERING OF AN ELECTRON BY AN ATOM AND REARRANGEMENT COLLISIONS

Ta‐You Wu

Published Feb 1, 1956• 5 citations
Publisher

Abstract

For the problem of the scattering of an electron by a hydrogen atom, the complete equivalence is shown of the treatment of Trainor and Wu based on symmetry considerations and that of Altshuler giving a direct proof of the method of Mott and Massey. A generalization of the treatment of Altshuler to scattering processes involving non-identical particles or other rearrangement collisions is made. Also, the usual treatment involving matrix elements for transitions between states belonging to two different Hamiltonians is discussed and justified. Finally, the recent treatment by Dirac of rearrangement collisions is discussed, and it is shown that a correct formulation of the method and interpretation of the result give exactly the same result as in the usual treatment.

DOI
10.1139/p56-020

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For the problem of the scattering of an electron by a hydrogen atom, the complete equivalence is shown of the treatment of Trainor and Wu based on symmetry considerations and that of Altshuler giving a direct proof of the method of Mott and Massey. A generalization of the treatment of Altshuler to scattering processes involving non-identical particles or other rearrangement collisions is made. Also, the usual treatment involving matrix elements for transitions between states belonging to two different Hamiltonians is discussed and justified. Finally, the recent treatment by Dirac of rearrangement collisions is discussed, and it is shown that a correct formulation of the method and interpretation of the result give exactly the same result as in the usual treatment.
Abstract The problem of the collision of two particles which act upon each other with forces varying as the inverse square of the distance between them has been solved exactly on the basis of the new quantum mechanics, and the solution is the same as that given by classical mechanics. In a recent paper, however, Mott has pointed out that this agreement between the predictions of classical mechanics and wave mechanics depends upon the dissimilarity of the colliding particles; if the particles are identical the scattering laws given by the wave mechanics will be very different from those of classical theory. Mott has considered the two types of collision between similar particles, (1) in which the particles possess spin, such as the collisions of electrons with electrons or protons with hydrogen nuclei, and (2) in which the particles have no spin, such as the collisions of α-particles with helium nuclei.
Abstract It is well-known that the problem of the collision between two particles interacting according to the inverse square law is exactly soluble on the wave mechanics, and that the solution yields the same scattering laws as the classical theory. If, however, the two particles are identical, e.g., two electrons or two α-particles, this is not necessarily the case; for the wave functions used must be antisymmetrical or symmetrical in the co-ordinates of the two particles; and this may affect the scattering laws. In this paper we shall discuss the collision between two particles possessing spin, such as electrons, and also between two particles without spin, such as α-particles. Assuming an inverse square law force between the particles, and neglecting the actual spin forces, we shall deduce from the symmetry properties of the wave functions a scatter­ing law differing considerably from the classical. We shall also mention the various methods by which the effect could be observed, and give some experi­mental evidence in its favour. The application of the exclusion principle to collision problems has been discussed by the author in a previous paper. Suppose we wish to describe the motion of two particles interacting in any field of force. We obtain a solution w(r1 r2) of the wave equation, where r1 refers to the position of the first particle, and r2 to that of the second. If we did not use antisymmetrical wave functions, we should argue that the probability that the first particle should be at r1 and the second at r 2 would be |w (r1 r2)|2, and therefore the probability that one particle should be at r1 and the other at r2 would be |w (r1 r2)|2 + |w(r2 r1)|
The scattering process is equivalent to the collision process and relates the physics of gaseous states, solid state physics (electrons colliding with phonons in the crystal lattice), plasma physics and so on. When we cannot directly visualize particles, scattering can be a valuable probe for their study. An example is the hydrogen atom that scattering experiments confirm it to have a nucleus with an electron orbiting around it. Scattering experiments also confirm protons to have some constituents called quarks. So the scattering theory is pivotal in understanding of the quantum nature of particles. In principle, we can formulate all problems of quantum mechanics in the language of scattering theory.
We present an exactly solvable quantum field theory which allows rearrangement collisions. We solve the model in the relevant sectors and demonstrate the orthonormality and completeness of the solutions, and construct the S-matrix. In light of the exact solutions constructed, we discuss various issues and assumptions in quantum scattering theory, including the isometry of the Möller wave matrix, the normalization and completeness of asymptotic states, and the nonorthogonality of basis states. We show that these common assertions are not obtained in this model. We suggest a general formalism for scattering theory which overcomes these and other shortcomings and limitations of the existing formalisms in the literature.