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arXiv:1207.0766 · Canadian Journal of Physics

The bicomplex quantum Coulomb potential problem

J. Mathieu, L. Marchildon, Dominic Rochon

Published Jul 27, 2013• 12 citations
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Abstract

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.

DOI
10.1139/cjp-2013-0261

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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.
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.
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.
The research for quantum features of molecular Coulomb interactions subjected to a time-dependent non-central potential has many applications. The wave functions of this system can be obtained by using an invariant operator, which is necessary for investigating a time-dependent Hamiltonian system. Regarding time dependence of the system, one can confirm that the formula of this operator is in general somewhat complicated. Hence, in order to solve its eigenvalue equation, special mathematical techniques beyond separation of variables method, such as the unitary transformation method, the Nikiforov–Uvarov method, and the asymptotic iteration method, should be employed. The double ring-shaped generalized non-central potential of which evolution explicitly depends on time is introduced as a particular case. The complete quantum solutions of the system can be identified from the eigenstates of the invariant operator. These solutions are useful for analyzing dynamical properties of the system.
Quantum features of a dynamical system subjected to time-dependent non-central potentials are investigated. The entire potential of the system is composed of the inverse quadratic potential and the Coulomb potential. An invariant operator that enables us to treat the time-dependent Hamiltonian system in view of quantum mechanics is introduced in order to derive Schrödinger solutions (wave functions) of the system. To simplify the problem, the invariant operator is transformed to a simple form by unitary transformation. Quantum solutions in the transformed system are easily obtained because the transformed invariant operator is a time-independent simple one. The Nikiforov-Uvarov method is used for solving eigenvalue equation of the transformed invariant operator. The double ring-shaped generalized non-central time-dependent potential is considered as a particular case for further study. From inverse transformation of quantum solutions obtained in the transformed system, the complete quantum solutions in the original system are identified. The quantum properties of the system are addressed on the basis of the wave functions.