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Global Journal of Pure and Applied Sciences

Exact solutions of the spherically symmetric multidimensional isotropic harmonic oscillator

K. J. Oyewumi, Amos Wale Ogunsula

Published Jun 2, 2004• 2 citations
Publisher

Abstract

The complete orthonormalised energy eigenfunctions and the energy eigenvalues of the spherically symmetric isotropic harmonic oscillator in N dimensions, are obtained through the methods of separation of variables. Also, the degeneracy of the energy levels are examined. KEY WORDS: - Schrödinger Equation, Isotropic harmonic oscillator, Hyperspherical harmonics, Multidimensional space. Global Jnl Pure & Applied Sciences Vol.10(2) 2004: 343-349

DOI
10.4314/gjpas.v10i2.16404

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The complete orthonormalised energy eigenfunctions and the energy eigenvalues of the spherically symmetric isotropic harmonic oscillator in N dimensions, are obtained through the methods of separation of variables. Also, the degeneracy of the energy levels are examined. KEY WORDS: - Schrödinger Equation, Isotropic harmonic oscillator, Hyperspherical harmonics, Multidimensional space. Global Jnl Pure & Applied Sciences Vol.10(2) 2004: 343-349
We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius rcr_crc​ with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.
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.
Bound state solutions of the Schrödinger equation have been investigated for n -dimensional ( n ⩾ 2) harmonic oscillator potential decorated with any finite number ( P ) of Dirac delta functions. The potential is radially symmetric and given as , where σ i s are arbitrary real numbers, r 1 < r 2 < ⋅⋅⋅ < r P and r i ∊ (0, + ∞). We have demonstrated that addition of Dirac delta functions lifts the accidental degeneracies of n -dimensional harmonic oscillator energy levels and leaves only the degeneracy due to the radial symmetry. Explicit forms of bound state eigenfunctions and the eigenvalue equation are given for n , l values, where n is the space dimension and l is the degree of n -dimensional spherical harmonics. We have shown that, for given n and l , there are a countably infinite number of bound state energy levels which are continuous functions of ω, σ i s and at most P of them can be negative.
The solution of non-central PT -symmetric potentials is discussed by the separation of the variables in polar and angular coordinates. Conditions are formulated to guarantee the separation of the variables and the PT symmetry of the potential. The original eigenvalue equation is separated into one-dimensional Schrödinger-type differential equations. The importance of the boundary conditions, especially that of the periodic boundary condition of the azimuthal equation is pointed out. Further conditions leading to exact solutions of the whole problem are also formulated. An example combining the harmonic oscillator and the Scarf I potential in the radial and polar equation is discussed in detail, and the bound-state wave functions and the energy eigenvalues are derived. The spectrum exhibits partial degeneracies similar to those observed in the spectrum of the isotropic harmonic oscillator.