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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Rotating co-ordinates in scattering theory

A. M. Arthurs, J. T. Lewis

Published Oct 9, 1962• 7 citations
Publisher

Abstract

Abstract The use of rotating co-ordinates in scattering problems is justified under certain conditions by considering a particular class of unitary equivalent Hamiltonians.
DOI
10.1098/rspa.1962.0198

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Abstract Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q(0)r, then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q(0)r.
This paper is a contribution to the discussion of the question: to what extent does the scattering matrix determine the Hamiltonian? The Hamiltonians considered are nonrelativistic, but in extension of previous studies, "nonlocal" potentials and many-body potentials are allowed. A large class of unitary transformations is found which produce Hamiltonians leading to the same SSS matrix. In the last section, it is shown that this equivalence is only a special consequence of the general axiomatic formulation of scattering in field theory.
Abstract This paper is concerned chiefly with the introduction of a co-ordinate operator in the quantum mechanics of the electron. It differs from other work on the same subject in the derivation, of the form of the operator by an analogy with the quantum expression for the angular momentum of the electron. Expressions for velocity and momentum operators are derived, and the relation between them is formally that of the classical theory in the absence of a field of force. By the introduction of reciprocal relations for the co-ordinate and momentum operators a condition is obtained which is applicable when the finite extension of the particle cannot be ignored. This is similar to the condition introduced by Yukawa in the theory of non-local fields. In the course of the development of the work, an explanation is offered of the significance of the fifth co-ordinate introduced in Kaluza’s theory of the gravitational and electromagnetic field and in Klein’s adaptation of it. Further, a new aspect of the question of the quantization of space and time is revealed by the use of co-ordinate operators (Xk), for it appears that the interest lies, not in a property of space-time, but in the quantization of localization of a particle. It should, however, be emphasized that the co-ordinates (xk) are retained with their usual meaning, but it is suggested that they are insufficient as a basis of a mechanical description of the electron. It is by operations upon (Xk) and not upon (xk) that mechanical quantities are obtained. The momentum (uk), like (xk), keeps its usual meaning, but it also is part of an operator (Mk) of wider significance. This is analogous to the case of the angular momentum of the electron where Dirac’s introduction of ‘spin’ shows that the orbital angular momentum is a part of a complete angular momentum operator. The application has been limited to the electron, because the derivation of the form of the operator (Xk) rests upon this analogy. The intention has not been to develop a calculus in which ordinary geometrical co-ordinates are replaced by operators. Finally, in proceeding to a force operator an additional term arises proportional to the derivative of the acceleration with respect to time. This term, which is usually ascribed to the reaction of the field of the electron, occurs here without reference to the self field, and suggests that, in addition to charge and current, a non-Maxwellian quantity must be considered as interacting with the external field.