Simultaneous measurement of coordinate and momentum on a von Neumann lattice

TL;DR

This paper demonstrates that on a finite phase plane, coordinate and momentum measurements on a von Neumann lattice are conjugate, introducing a new wave function for simultaneous measurement of both quantities.

Contribution

It establishes a conjugate relationship between $kq$-coordinates and lattice sites on a finite phase plane and defines a new wave function for joint measurement.

Findings

Coordinate and momentum are conjugate on finite phase planes.

A new wave function for simultaneous measurement is introduced.

The results depend on the phase plane size being factorizable into coprime integers.

Abstract

It is shown that on a finite phase plane the $kq$-coordinates and the sites of a von Neumann lattice are conjugate to one another. This elementary result holds when the number $M$ defining the size of the phase plane can be expressed as a product, $M=M_{1}M_{2}$, with $M_{1}$ and $M_{2}$ being relatively prime. As a consequence of this result a hitherto unknown wave function is defined giving the probability of simultaneously measuring the momentum and coordinate on the von Neumann lattice.