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Journal of Mathematical Physics

Amplitudes on von Neumann lattices

M. H. Boon, J. Žák

Published May 1, 1981• 33 citations

Abstract

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.

DOI: 10.1063/1.524992

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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

\mathrm{kq}

representation is used in an elementary proof of completeness and orthogonality of states on a discrete phase plane lattice.

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.

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.

The concepts of von Neumann lattices and tight frames are used for defining discrete quantum mechanical transforms in the phase plane. These transforms are obtained by finite shifts la and mb in the coordinate x and momentum p, respectively, of the Weyl–Heisenberg group, and they are called the discrete Weyl–Heisenberg transforms ψ(la,mb). Here ab=h/N with h the Planck constant, l and m integers, and N a positive integer. A construction is carried out of ψ(la,mb) for a general Weyl–Heisenberg set by using the kq-representation, in which a useful formula is established for the frame operator. The construction is illustrated on an example of the ground state of a harmonic oscillator. It is shown that any physical quantity can be described by the discrete Weyl–Heisenberg transform. Connections are established between ψ(la,mb), the Bargmann representation, and the Husimi distribution function.

By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials.We also derive the new reciprocal relation of Laguerre polynomials n=0(1) n l n L n(x) = x l l!,and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated.Some new expansion identities about the operator Laguerre polynomial are also derived.This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique.