The bicomplex quantum Coulomb potential problem

TL;DR

This paper extends quantum mechanics to bicomplex numbers, providing an analytical solution to the Coulomb potential problem, including eigenvalues, eigenfunctions, and an orthonormal system within a bicomplex Hilbert space.

Contribution

It introduces a bicomplex Hamiltonian and extends quantum formalism, solving the Coulomb problem in this new mathematical framework with explicit eigenvalues and eigenfunctions.

Findings

Eigenvalues of the bicomplex Hamiltonian are derived.

Explicit solutions for the bicomplex Schrödinger equation are provided.

An orthonormal basis in bicomplex Hilbert space is constructed.

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 [X_i,P_k] = i_1 hbar xi delta_{ik}, where xi 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 Schrodinger's time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.