Journal of Mathematical Physics

Integral Forms for Quantum-Mechanical Momentum Operators

Peter D. Robinson

Published Nov 1, 1966• 8 citations

Abstract

The usual differential form P0 for the quantum-mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P0 = − i(g−½) ∂/∂q (g½), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P0 is not always self-adjoint on the domain 𝒟 of physically acceptable bound-state wavefunctions, as a proper quantum-mechanical operator should be. An integral form is proposed for P, defined by Pf(q)=(2π)−12g−12(q) ∫ −∞∞exp (ikq)kF(k) dk, α≤q≤β,where F(k)=(2π)−12 ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ, f∈D. The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P0 only at the end-points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta-function terms to P0, but it suffers from none of the defects, since delta-functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain 𝒟 is presented as an appendix.

DOI: 10.1063/1.1704889

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