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Journal of Physics A Mathematical and General

von Neumann's hypothesis concerning coherent states

J. Žák

Published Oct 8, 2003• 10 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/36/42/l02

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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.
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.
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.
A general set of states on a lattice in the phase plane is considered. The discrete set of von Neumann coherent states for a harmonic oscillator is a particular case of the above set. The kq\mathrm{kq}kq representation is used in an elementary proof of completeness and orthogonality of states on a discrete phase plane lattice.
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 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.