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American Journal of Physics

On the Radial Momentum Operator

Richard L. Liboff, Isaiah Nebenzahl, Hans H. Fleischmann

Published Aug 1, 1973• 22 citations
Publisher

Abstract

The validity of the radial momentum operator as an observable equivalent is investigated. In addition to the fundamental mathematical prescription, purely physical arguments are presented which demonstrate its inadmissibility. This remains in spite of its correct commutator relationship with the radial coordinate. In the course of presenting these arguments, the condition r1/2Ψ(r)←0, r←0, is derived to be a necessary condition that Ψ be an acceptable wave function.

DOI
10.1119/1.1987445

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The validity of the radial momentum operator as an observable equivalent is investigated. In addition to the fundamental mathematical prescription, purely physical arguments are presented which demonstrate its inadmissibility. This remains in spite of its correct commutator relationship with the radial coordinate. In the course of presenting these arguments, the condition r1/2Ψ(r)←0, r←0, is derived to be a necessary condition that Ψ be an acceptable wave function.
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.
There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.
With acceptance of the Dirac's observation that the canonical quantization entails using Cartesian coordinates, we examine the operator erPr rather than Pr itself and demonstrate that there is a decomposition of erPr into a difference of two self-adjoint but noncommutative operators, in which one is the total momentum and another is the transverse one. This study renders the operator Pr indirectly measurable and physically meaningful, offering an explanation of why the mean value of Pr over a quantum mechanical state makes sense and supporting Dirac's claim that Pr "is real and is the true momentum conjugate to r".
We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.