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AIP conference proceedings

Hilbert Space of the Bicomplex Quantum Harmonic Oscillator

Raphaël Gervais Lavoie, ‎Louis Marchildon, Dominic Rochon, Andrei Yu. Khrennikov, Gregg S. Jaeger, M. Schlosshauer, G. Weihs

Published Jan 1, 2011• 2 citations
Publisher

Abstract

Bicomplex numbers are pairs of complex numbers with a multiplication law that makes them a commutative ring. The problem of the quantum harmonic oscillator is investigated in the framework of bicomplex numbers. Starting with the commutator of the bicomplex position and momentum operators, we find eigenvalues and eigenkets of the bicomplex harmonic oscillator Hamiltonian. Coordinate‐basis eigenfunctions of the Hamiltonian are then obtained in terms of hyperbolic Hermite polynomials, and some of them are graphically illustrated. These eigenfunctions form a basis of an infinite‐dimensional module over bicomplex numbers, and this module can be given the structure of a bicomplex Hilbert space.

DOI
10.1063/1.3578696

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Bicomplex numbers are pairs of complex numbers with a multiplication law that makes them a commutative ring. The problem of the quantum harmonic oscillator is investigated in the framework of bicomplex numbers. Starting with the commutator of the bicomplex position and momentum operators, we find eigenvalues and eigenkets of the bicomplex harmonic oscillator Hamiltonian. Coordinate‐basis eigenfunctions of the Hamiltonian are then obtained in terms of hyperbolic Hermite polynomials, and some of them are graphically illustrated. These eigenfunctions form a basis of an infinite‐dimensional module over bicomplex numbers, and this module can be given the structure of a bicomplex Hilbert space.
Bicomplex numbers are pairs of complex numbers with a multiplication law that makes them a commutative ring. The problem of the quantum harmonic oscillator is investigated in the framework of bicomplex numbers. Starting with the commutator of the bicomplex position and momentum operators, we find eigenvalues and eigenkets of the bicomplex harmonic oscillator Hamiltonian. Coordinate‐basis eigenfunctions of the Hamiltonian are then obtained in terms of hyperbolic Hermite polynomials, and some of them are graphically illustrated. These eigenfunctions form a basis of an infinite‐dimensional module over bicomplex numbers, and this module can be given the structure of a bicomplex Hilbert space.
Generalizations of the complex number system underlying the mathematical formulation of quantum mechanics have been known for some time, but the use of the commutative ring of bicomplex numbers for that purpose is relatively new. This paper provides an analytical solution of the quantum Coulomb potential problem formulated in terms of bicomplex numbers. We define the problem by introducing a bicomplex hamiltonian operator and extending the canonical commutation relations to the form [Formula: see text], where ξ is a bicomplex number. Following Pauli’s algebraic method, we find the eigenvalues of the bicomplex hamiltonian. These eigenvalues are also obtained, along with appropriate eigenfunctions, by solving the extension of Schrödinger’s time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.
We investigate bicomplex Hamiltonian systems in the framework of an analogous version of the Schrödinger equation. Since in such a setting three different types of conjugates of bicomplex numbers appear, each is found to define, in a natural way, a separate class of time reversal operator. However, the induced parity ( )-time ( )-symmetric models turn out to be mutually incompatible, except for two of them which could be chosen uniquely. The latter models are then explored by working within an extended phase space. Applications to the problems of harmonic oscillator, inverted oscillator and isotonic oscillator are considered and many new interesting properties are uncovered for the new types of symmetries.
Quantum mechanics of a linear harmonic oscillator in a finite-dimensional Hilbert space satisfying the correct equations of motion is studied. The connections to Weyl's formulation of the algebra of bounded unitary operators in finite space as well as to a truncated quantized linear harmonic oscillator are discussed. It is pointed out that the discrete Fourier transformation (DFT) plays a central role in determining the actual form of the position, the momentum, the number and the Hamiltonian operators. The explicit form of these operators in different bases is exhibited for some low values of the dimension of the Hilbert space. In this formulation, it is shown that the Hamiltonian is indeed the logarithm of the DFT and that by modifying Weyl's framework to include position and momentum operators with non-uniformly spaced spectra the equations of motion are satisfied.
The harmonic oscillator is considered within generalized frameworks using complex and quaternionic numbers. The classical oscillator is considered in terms of a complex position function, and quantum oscillators are examined in terms of complex wave functions, and in terms of quaternionic wave functions as well. Both of the quantum solutions are obtained within the real Hilbert space formalism. The results reveal the complex and quaternionic descriptions as suitable frameworks for non-stationary processes, including damped oscillations, forced oscillations, and additionally self-interacting processes that cannot be appropriately described otherwise.
In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.