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Journal of Physics A Mathematical and General

Quantum mechanical models in fractional dimensions

Max A. Lohe, Alagu Thilagam

Published May 25, 2004• 38 citations
Publisher

Abstract

We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P , Q satisfy the R -deformed Heisenberg relations, which depend on an operator ν. We find representations of P , Q in which the dimension d and angular momentum ℓ appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity ℓ is quantized. The representations of P , Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d , both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.

DOI
10.1088/0305-4470/37/23/015

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We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P , Q satisfy the R -deformed Heisenberg relations, which depend on an operator ν. We find representations of P , Q in which the dimension d and angular momentum ℓ appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity ℓ is quantized. The representations of P , Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d , both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.
We construct representations of the Lie algebra using representations of the momentum and position operators satisfying the R -deformed Heisenberg relations, in which the fractional dimension d and angular momentum ℓ appear as parameters. The Bargmann index κ, which characterizes representations of the positive discrete series of , can take any positive value. We construct coherent states in fractional dimensions, in particular we extend the two well-known analytic representations of coherent states for , Perelomov and Barut–Girardello states, from dimension one to any dimension d . We generalize this construction to time-dependent coherent states by means of the symmetries of the quantum time-dependent harmonic oscillator in fractional dimensions. We investigate the uncertainty relations of the momentum and position operators with respect to these coherent states, and their dependence on the dimension.
We study the properties of the Berry phase in quantum systems where momentum and position operators satisfy the R -deformed Heisenberg relations and the noninteger dimension d and angular momentum ℓ appear as parameters. We focus on the dynamics of the generalized time-dependent harmonic oscillator which satisfies the Schrödinger wave mechanics formulated in arbitrary dimensions. We also examine quantum systems which can be expressed as linear combination of generators with time-dependent coefficients in arbitrary dimensions. We show the explicit dependence of the Berry phase on d and the parity ℓ of quantum systems which satisfy the R -deformed Heisenberg relations. Using an explicit relation of the Fubini–Study metric in fractional dimensions, we highlight a geometrical interpretation of d and ℓ of systems which are quantized in arbitrary dimensions.
Position deformation of a Heisenberg algebra and Hilbert space representation of both maximal length and minimal momentum uncertainties may lead to loss of Hermiticity of some operators that generate this algebra. Consequently, the Hamiltonian operator constructed from these operators is also not Hermitian. In this paper, with an appropriate positive-definite Dyson map, we establish the Hermiticity of these operators by means of a similarity transformation. We then construct Hilbert space representations associated with these Hermitian operators that generate a Hermitian Heisenberg algebra. With the help of these representations, we establish the path integral formulation of any systems in this Hermitian algebra. Finally, using the path integral of a free particle as an example, we demonstrate that the Euclidean propagator, action, and kinetic energy of this system are constrained by the standard classical mechanics limits.
Abstract Wavefunctions with rotational symmetry (i.e., zero angular momentum) in D dimensions, are called s‐waves. In quantum quadratic systems (free particle, harmonic and repulsive oscillators), their radial parts obey Schrödinger equations with a fictitious centrifugal (for integer D ≥ 4) or centripetal (for D = 2) potential. These Hamiltonians close into the three‐dimensional Lorentz algebra so(2,1), whose exceptional interval corresponds to the critical range of continuous dimensions 0 < D < 4, where they exhibit a one‐parameter family of self‐adjoint extensions in ℒ︁2(ℜ︁+). We study the characterization of these extensions in the harmonic oscillator through their spectra which – except for the Friedrichs extension – are not equally spaced, and we build their time evolution Green function. The oscillator is then contracted to the free particle in continuous‐D dimensions, where the extension structure is mantained in the limit of continuous spectra. Finally, we compute the free time evolution of the expectation values of the Hamiltonian, dilatation generator, and square radius between three distinct sets of ‘heat’‐diffused localized eigenstates. This provides a simple group‐theoretic description of the purported contraction/expansion of Gaussian‐ring s‐waves in D > 0 dimensions.
The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.
In this research, we investigate the extended Dunkl quantum mechanics with two parameters. We apply this formalism to study general quantum phenomena. Using cartesian coordinate, we calculate the solution to problem of a spinless particle confined to a box in three dimensions. Then, we derive the creation and annihilation operators to analyze harmonic oscillator in three dimensions via algebraic approach. Subsequently, utilizing the same operators and the generators of [Formula: see text] Lie algebra, we examine the Dirac harmonic oscillator in [Formula: see text] dimension. Finally, we construct the hydrodynamical formulation of the extended Dunkl quantum mechanics by deriving continuity and quantum Hamilton–Jacobi equations.