The Position-Momentum Commutator as a Generalized Function: Resolution of the Apparent Discrepancy Between Continuous and Discrete Bases

It has been known for many years that the matrix representation of the one-dimensional position-momentum commutator calculated with the position and momentum matrices in a finite basis is not proportional to the diagonal matrix, contrary to what one expects from the continuous-space commutator. This discrepancy has correctly been ascribed to the incompleteness of any finite basis, but without the details of exactly why this happens. Understanding why the discrepancy occurs requires calculating the position, momentum, and commutator matrix elements in the continuous position basis, in which all are generalized functions. The reason for the discrepancy is revealed by replacing the generalized functions with sequences approaching them as their parameter approaches zero. Besides explaining the discrepancy in the discrete and continuous models, this investigation finds an unusual double-peaked sequence for the Dirac delta function.