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Physical review. B, Condensed matter

Discrete coherent states on the von Neumann lattice

M. H. Boon, J. Žák

Published Dec 15, 1978• 51 citations
Publisher

Abstract

The discrete subset of coherent states on a lattice in phase space is used as an expansion basis for states and operators in quantum mechanics. The nature of the expansion as well as the uniqueness of the expansion coefficients is investigated. An explicit example is given for the harmonic-oscillator states.
DOI
10.1103/physrevb.18.6744

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Physical Review A•Oct 1, 1983•3 citations
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.
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 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.
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....
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.