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arXiv:0806.3832 · Europhysics Letters (EPL)

Simultaneous measurement of coordinate and momentum on a von Neumann lattice

A. Mann, M. Revzen, J. Žák

Published Jun 18, 2008• 4 citations
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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.

DOI
10.1209/0295-5075/83/10007

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arXiv:quant-ph/0202092•Journal of Physics A Mathematical and General•Mar 4, 2002•0 citations
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.
An orthonormal basis of modified coherent states is constructed. Each member of the basis is an infinite sum of coherent states on a von Neumann lattice. A single state is assigned to each unit cell of area h (Planck constant) in the phase plane. The uncertainties of the coordinate x and the square of the momentum p 2 for these states are shown to be similar to those for the usual coherent states. Expansions in the newly established set are discussed and it is shown that any function in the kq -representation can be written as a sum of two fixed kq -functions. Approximate commuting operators for x and p 2 are defined on a lattice in phase plane according to von Neumann's prescription.
An averaging procedure in phase plane is developed leading to orthonormality of coherent states on a von Neumann lattice. These states correspond to entire unit cells of area h (Planck constant) in the phase plane and they can be specified by ( mb , n 2πℏ/ b ), where m and n are integers and b is a constant related to the spread of the harmonic oscillator ground state. The product of uncertainties for the co-ordinate x and momentum p in these states is very close to ℏ. This is by a factor of 2π smaller than the unit cell area, and makes it, in principle, possible to measure simultaneously x and p for the location of a unit cell in the phase plane.
The prime number decomposition of a finite-dimensional Hilbert space reflects itself in the representations that the space accommodates. The representations appear in conjugate pairs for factorization to two relative prime factors which can be viewed as two distinct degrees freedom. These, Schwinger's quantum degrees of freedom, are uniquely related to a von Neumann lattices in the phase space that characterizes the Hilbert space and specifies the simultaneous definitions of both (modular) positions and (modular) momenta. The area in phase space for each quantum state in each of these quantum degrees of freedom, is shown to be exactly hhh, Planck's constant.
Abstract Starting from a complete but not overcomplete set of coherent states defined on a lattice in the phase space, we construct the orthonormal basis using the procedure due to Lödwin, in which all states to be normalized enter the procedure on equal footing. We show that our normalized states may be interpreted as eigenstates of the operator of simultaneous unsharp measurement of coordinate and momentum. In a sense we followed the classical idea of von Neumann but without drawbacks which in his construction of such an operator were due to the use of the Gram‐Schmidt orthonormalization procedure which is somewhat inappropriate for this purpose. We discuss the obtained results.
On the basis of the recently introduced averaging procedure in phase space, a new type of entropy is defined on the von Neumann lattice. This quantity can be interpreted as a measure of the uncertainty associated with simultaneous measurement of the position and momentum observables in a discrete subset of the phase space. Evaluating for a class of coherent states, it is shown that this entropy takes a stationary value for the ground state, modulo a unit cell of the lattice in such a class. This value for the ground state depends on the ratio of the position lattice spacing and the momentum lattice spacing. It is found that its minimum is realized for the perfect square lattice, i.e., absence of squeezing. Numerical evaluation of this minimum gives 1.386....