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Journal of Physics A Mathematical and General

Wilson-like functions on von Neumann lattices

J. Žák

Published Jun 21, 2002• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/35/26/104

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•Jan 1, 2002•0 citations
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.
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.
A rational von Neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers. Zeros of harmonic oscillator functions in the kq representation on such lattices are found. It is shown that the number of zeros of the kq function determines the number of states by which a set on a von Neumann lattice is overcomplete. Interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation.
A formula is derived connecting any wave function 〈k,q‖f〉 in the kq-representation with the corresponding amplitudes of the state ‖f〉 on von Neumann lattices of states. The formula is used for establishing a possible interpretation for these amplitudes, for obtaining linear relationships between them, and for finding sum rules for the squares of their absolute values, and other related sum rules. It can also be used for establishing completeness criteria for the lattices of states and for defining a modified Hilbert space in which they become strictly complete. Particular attention is given to the coherent state lattice, but the discussion is extended to von Neumann lattices generated from an arbitrary state. Lattices generated from harmonic oscillator states are studied explicitly, and shown incidentally to lead to a wealth of summation expressions for Laguerre polynomials.
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