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Physical interpretations of the differentiation rule for the Dirac delta function

Denys I. Bondar, Robert R. Lompay, Wing‐Ki Liu

Published Jul 24, 2010• 0 citations
OpenAlex

Abstract

The canonical commutation relation between the quantum mechanical operators of momentum and coordinate and the Ehrenfest theorem in the free particle case are derived solely from the rule of differentiation of the delta function [$x\delta'(x) = - \delta(x)$].

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The canonical commutation relation between the quantum mechanical operators of momentum and coordinate and the Ehrenfest theorem in the free particle case are derived solely from the rule of differentiation of the delta function [xδ′(x)=−δ(x)x\delta'(x) = - \delta(x)xδ′(x)=−δ(x)].
Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value ihihih where hhh is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.
This chapter introduces an elementary form of the fundamental axioms of quantum mechanics, focusing on the structure and interpretation of state functions within a vector space framework. The properties of linear operators acting on state functions are explored, along with the physical meaning of the state function itself. From this interpretation, boundary conditions and normalization requirements are derived. The chapter presents key results such as the normalization theorem, the time derivative of a physical observable, and Ehrenfest's theorem. It also defines probability density and probability current density, deriving the continuity equation for probability. The uncertainty principle is formally established, and energy conservation is addressed. The eigenvalues and eigenstates of the momentum operator are determined, and normalization using the Dirac delta function is introduced. The free particle wavefunction is derived, concluding with a discussion on the classical limit of quantum mechanics. Metadata/Keywords: quantum axioms, state function, linear operators, probability density, probability current, Ehrenfest's theorem, uncertainty principle, Dirac delta function, momentum eigenstates, classical limit