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Physica Scripta

Fractional Dimensional Momentum Operator for a System of One Degree of Freedom

Alex Matos-Abiague

Published Aug 1, 2000• 10 citations
Publisher

Abstract

By introducing a Cartesian-like pseudocoordinate, and starting with the general Wigner commutation relations for the canonical variables of a Bose-like oscillator of a single degree of freedom, an expression for the momentum operator is obtained. The wave function that describes the motion of a free particle in a fractional dimensional space is also found.

DOI
10.1238/physica.regular.062a00106

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By introducing a Cartesian-like pseudocoordinate, and starting with the general Wigner commutation relations for the canonical variables of a Bose-like oscillator of a single degree of freedom, an expression for the momentum operator is obtained. The wave function that describes the motion of a free particle in a fractional dimensional space is also found.
Abstract. After introducing Wigner Quantum Systems, we give a short review of the one-dimensional Wigner Quantum Oscillator. Then we define the three-dimensional N-particle Wigner Quantum Oscillator, and its relation to the Lie su-peralgebra sl(1|3N). In this framework (and first for N = 1), energy, coordinates, momentum and the angular momentum of the particles are investigated. Wigner Quantum Systems (WQSs) are quantum systems in which the canonical com-mutation relations (CCRs) are replaced by a compatibility condition between the Heisenberg equations and Hamilton’s equations. By dropping the CCRs, WQSs offer a natural framework for quantum mechanics with non-commutative coordinates. Here we consider in particular the three-dimensional N-particle Wigner Quantum Oscillator (WQO). As an introductory ex-ample, some properties of the one-dimensional WQO are reviewed, with an emphasis on the different particle probability distributions (as compared to the ordinary one-dimensional os-cillator). For the three-dimensional N-particle WQO, a solution related to the Lie superalge-bra sl(1|3N) is considered. For N = 1 we investigate the operators corresponding to coordi-nates, momentum and angular momentum of the particle in a class of representation spaces.
A detailed study is made of the wave-mechanical representation of a one-dimensional Bose-like oscillator whose canonical variables satisfy the general commutation relations first proposed by Wigner. The eigenvalue problems of the momentum and Hamiltonian operators are completely solved, and this is made possible only when wavefunctions in general are allowed to be hyperfunctions. The equivalence between the wave- and matrix-mechanical representations is thereby established for any value of c (a characteristic parameter of the theory), contrary to the conclusion reached previously by Yang. It is also found that for the case −1/2<c<0 or 0<c<1/2 there exist two classes of eigenfunctions that are mutually separated by a superselection rule.
In noncommutative spaces to maintain Bose–Einstein statistics for identical particles at the non-perturbation level described by deformed annihilation-creation operators when the state vector space of identical bosons is constructed by generalizing one-particle quantum mechanics it is explored that the consistent ansatz of commutation relations of phase space variables should simultaneously include space–space non-commutativity and momentum–momentum non-commutativity, and a new type of boson commutation relations at the deformed level is obtained. Consistent perturbation expansions of deformed annihilation-creation operators are obtained. The influence of the new boson commutation relations on dynamics is discussed. The non-perturbation and perturbation property of the orbital angular momentum of two-dimensional system are investigated. Its spectrum possesses fractional eigen values and fractional intervals.
A time fractional quantum framework has been introduced into quantum mechanics.A new version of the space-time fractional Schrödinger equation has been launched.The introduced space-time fractional Schrödinger equation has a new scale parameter, which is a fractional generalization of Planck's constant in quantum physics.It has been shown that the presence of a time fractional time derivative in the space-time fractional Schrödinger equation significantly impacts quantum mechanical fundamentals.Time fractional quantum mechanical operators of coordinate, momentum and angular momentum were defined and their commutation relationships were established.The pseudo-Hamilton operator was introduced and its Hermicity has been proven.Two new functions related to the Mittag-Leffler function have been introduced to solve the space-time fractional Schrödinger equation.Energy of a time fractional quantum system has been defined and calculated in terms of the newly introduced functions.It has been found that in the framework of time fractional quantum mechanics there are no stationary states, and the eigenvalues of the pseudo-Hamilton operator are not the energy levels of the time fractional quantum system.A free particle solution to the space-time fractional Schrödinger equation was found.A free particle space-time fractional quantum mechanical kernel has been found and expressed in terms of the H-function.Renormalization properties of a free particle solution and the space-time fractional quantum kernel were established.Some particular cases of time fractional quantum mechanics have been analyzed and discussed.
The eigenstates of a free quantum particle propagating in a plane are derived in the context of phase space quantum mechanics. Two possibilities are analysed. First, the case of a particle with fixed energy and angular momentum is considered. A special choice of coordinates on the four-dimensional phase space, suitable for representing the eigenstates of the particle under consideration, is presented. A phase-space counterpart of the product of operators, known as the Moyal product, is derived in these coordinates. The eigenvalue equations are solved, and their physically acceptable solutions, called Wigner functions, are identified. Second, a particle with fixed components of the Cartesian momentum is considered. A relationship is found between the Wigner function of the particle with fixed Cartesian momentum components and some functions characterising the particle with fixed energy and angular momentum.
After a general introduction to Wigner Quantum Systems, we define the three-dimensional n-particle Wigner Quantum Oscillator, and its relation to the Lie superalgebra sl(1 vertical bar 3n). In this framework, coordinates, momentum and the angular momentum of the particles are defined and investigated. This investigation is done in state spaces, using representation theory of sl(1 vertical bar 3n). For the case n = 1, the main properties are listed, with an emphasis on the non-commutative coordinates of the particle and its unconventional consequences. For general n, we study energy spectra, angular momentum, and in particular the particle configuration and its interpretation. Throughout, a comparison with the canonical oscillator solution is given.