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THE RADIAL EQUATIONS OF A PARTICLE OF ANY SPIN IN INTERACTION WITH A CENTRAL POTENTIAL

André Jolivet

Published Feb 1, 1964• 0 citations
OpenAlex

Abstract

A simple expression is given for the irreducible forms of the matrix system used to describe a particle with any spin. The radial equations for a particle with any spin were determined for the case of an interaction in spherical symmetry. (tr-auth)

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A straightforward derivation is given for the Hermitian operator pr ≡ 1r ħi δδrr corresponding to the radial component of linear momentum in the central force problem. The purpose is purely pedagogical; we believe that the radial term of the Schrödinger equation in spherical coordinates is more comprehensible when it appears in the form (pr2/2m)ψ = [(1r ħi δδrr)2/2m]ψ than when it appears, as it does in most textbooks, in the form −ħ22mr2 δδr[r2δδr]ψ. The latter expression is usually arrived at by a tedious transformation of variable done “in the appendix”; instead one may derive pr with a minimum of algebra and with the focus where it belongs, on the concepts of quantum mechanics.
One of the most important problems in quantum mechanics concerns the motion of a particle in a potential of the form V ( r ) = V ( r ), where r ≡ | r | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203741481/4ab22e1a-c1c5-4151-9df7-a6d7ea2b05e0/content/eq763.tif"/> . This system has the simplifying feature of spherical symmetry, which allows the angular dependence of the motion to be reduced to purely geometrical considerations and hence solved once and for all, while the physics enters only through the radial motion. Treating this problem in detail also provides a good opportunity to illustrate the quantum mechanical formalism.