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Journal of Physics A Mathematical and General

Orthonormal coherent states on von Neumann lattices

J. Žák

Published Jan 25, 2001• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/34/5/310

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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.
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 orthonormal set of functions is defined on a von Neumann lattice in phase space. There is one function assigned to each unit cell of area h . The functions are of the Wilson type in that in the x -direction they are obtained from one another by uniform translations, while in the p -direction they are double-peaked, and cannot be obtained by translations. A similar construction is also carried out with the x - and p -axes interchanged. The results apply to any dimension. An explicit example is worked out for the ground state of a harmonic oscillator, and a relation to coherent states is pointed out.
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....
Abstract. Based on 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 uncertainty associated with simultaneous measurement of the position and momentum observables in the discrete subset of the phase space. Evaluating for a class of the coherent states, it is shown that this entropy takes a stationary value for the ground states modulo of 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.... 1 Ever since the introduction of the Wigner distribution function [1], theory of phase-space representations [2-4] has been playing vital roles in quantum statistical mechanics, quantum chaos and signal processing. The Husimi function, which is the coherent-state representation of a wave function (or a density matrix) and is often called the Q-function, is known to be special in the sense that it is nonnegative and has a good analytical behavior in contrast to the Wigner function and the Sudarshan-Glauber P-function. Since the coherent state of the harmonic oscillator is the most classical pure state with the minimum Heisenberg uncertainty product, the Husimi Q-function is supposed to describe a semiclassical aspect of the system under consideration. However, a measure of classical nature of a normalized state, ψ, was lacking until Wehrl [5] introduced a classical entropy, which is defined by 2 W [ ψ] = −∫ ∫ d α α ψ π 2 2 α ψ ln. (1) π α in this equation is the coherent state given by α = ˆD ( α) 0 ( α ∈C), where
A new general subset of coherent states, which is complete and overcomplete by just one element, is introduced. It is the union of NNN arbitrary nonintersecting von Neumann lattices whose unit-cell areas have a harmonic mean of Nh\mathrm{Nh}Nh. First applications are given in the problems of crystal electrons in rational magnetic fields, incommensurate structures in a magnetic field, and quantum dynamics.
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.