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Journal of Physics Conference Series

Designing energy spectra in finite phase plane

J. Žák

Published Mar 1, 2008• 1 citation
Publisher

Abstract

In recent years the finite phase plane has been used to accommodate spin 1/2 particles which play a decisive role as a tool in quantum computing [1]. For this, one often uses conjugate operators in the framework of the Wigner function approach. Conjugate variables are important both in classical and quantum mechanics. The best known example of such operators are the coordinate x and the momentu p [2]. In elementary quantum mechanics there are very few such pairs of conjugate operators. One of the striking results in a finite phase plane of dimension M is that there exist more than one pair of conjugate operators in the xp-plane. In this talk we show how to build 2 conjugate kq-representations (k-quasimomentum, q- quasicoordinte) [3]. These conjugate representations are then applied to Harper-like Hamiltonians, and it's shown how to design physical systems with energy spectra with any desired number of discrete energy levels, and with prescribed degeneracy. A general theory of quantum mechanics in finite phase plane was developed by Schwinger [4].

DOI
10.1088/1742-6596/104/1/012013

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In recent years the finite phase plane has been used to accommodate spin 1/2 particles which play a decisive role as a tool in quantum computing [1]. For this, one often uses conjugate operators in the framework of the Wigner function approach. Conjugate variables are important both in classical and quantum mechanics. The best known example of such operators are the coordinate x and the momentu p [2]. In elementary quantum mechanics there are very few such pairs of conjugate operators. One of the striking results in a finite phase plane of dimension M is that there exist more than one pair of conjugate operators in the xp-plane. In this talk we show how to build 2 conjugate kq-representations (k-quasimomentum, q- quasicoordinte) [3]. These conjugate representations are then applied to Harper-like Hamiltonians, and it's shown how to design physical systems with energy spectra with any desired number of discrete energy levels, and with prescribed degeneracy. A general theory of quantum mechanics in finite phase plane was developed by Schwinger [4].
The known symmetries in one-dimensional systems are inversion and translations. These symmetries persist in finite phase plane, but a novel symmetry arises in view of the discrete nature of the coordinate x i and the momentum p i : x i and p i can undergo permutations. Thus, if x i assumes M discrete values, i = 0, 1,2,..., M – 1, a permutation will change the order of the set x 0 , x 1 ,..., x M −1 into a new ordered set. Such a symmetry element does not exist for a continuous x -coordinate in an infinite phase plane. Thus, in a finite phase plane, translations can be replaced by permutations. This is also true for the inversion operator. The new permutation symmetry has been used for the construction of conjugate representations and for the splitting of the M -dimensional vector space into independent subspaces. This splitting is exhaustive in the sense that if M = i M i with M i being prime numbers, the M -dimensional space splits into M 1 , M 2 ,... M n -dimensional independent subspaces. It is shown that following this splitting one can design new potentials with appropriate constants of motion. A related problem is the Weyl-Heisenberg group in the M-dimensional space which turns into a direct product of its subgroups in the M i -dimensional subspaces. As an example we consider the case of M = 8.
This paper deals with certain aspects of the phase-space formulation of quantum mechanics. Noting the ambiguities involved in quantizing the classical phase-space dynamics, a generalized quantum-mechanical phase space is introduced in terms of eigenvalues of two mutually incompatible complete sets of operators. General definitions are given for the phase-space distribution function corresponding to a given abstract state of the system and the phase-space function for an arbitrary operator. Our formulation goes over to the Wigner formulation when the complete sets correspond to position and momentum operators in Cartesian coordinates. Explicit examples are discussed with use of action-angle and polar coordinates for simple one-, two-, and three-dimensional systems. Symmetry changes in phase space are discussed by applying Gel'fand-Levitan transformations and it is shown that, in general, the degeneracies of motion in phase space are not reflected in the energy eigenvalues.
In this work, the version of Wigner transforms and Wigner functions on discrete systems are formulated. Using frame quantization, the star-product of phase space functions are also computed for a particular finite group. Specifically, the Wigner transform, Wigner function and star-product are computed for the dicyclic group Dien. Frames are the discrete version of coherent states. In the literature, quasiprobability distributions are essential in the study of the phase space representations of quantum mechanics, and are naturally associated to frame quantizations. For this work, the unitary irreducible representations of a finite group play an essential role in the construction.
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.
In 1960 Schwinger (J Schwinger 1960 Proc. Natl Acad. Sci. 46 570–9) proposed the algorithm for factorization of unitary operators in the finite M -dimensional Hilbert space according to a coprime decomposition of M . Using a special permutation operator A we generalize the Schwinger factorization to every decomposition of M . We obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as Schwinger's. We apply the new factorization to two problems. First, we show how to generate two kq -like mutually unbiased bases for any composite dimension. Then, using a Harper-like Hamiltonian model in the finite dimension M = M 1 M 2 , we show how to design a physical system with M 1 energy levels, each having degeneracy M 2 .