Exact diagonalization of the D-dimensional spatially confined quantum harmonic oscillator

Adegoke Kunle,Afees Tiamiyu,Akintujoye Funmi,Adenike Olatinwo,Henry Otobrise

doi:10.5937/kgjsci1638029k

Abstract

In the existing literature various numerical techniques have been developed to quantize the confined harmonic oscillator in higher dimensions. In obtaining the energy eigenvalues, such methods often involve indirect approaches such as searching for the roots of hypergeometric functions or numerically solving a differential equation. In this paper, however, we derive an explicit matrix representation for the Hamiltonian of a confined quantum harmonic oscillator in higher dimensions, thus facilitating direct diagonalization.

Highlights

The d − dimensional confined harmonic oscillator of mass m and frequency ω is described by the Hamiltonian
Where we have introduced a d − dimensional vector c whose components are N d × N d symmetric binary matrices, ci with elements given by d
We introduce yet another d − dimensional vector, α, whose components, αi, are N d × N d symmetric binary matrices, in terms of which the ci matrices may be expressed

Summary

INTRODUCTION

The d − dimensional confined harmonic oscillator (cho) of mass m and frequency ω is described by the Hamiltonian. Various techniques have been employed by researchers to numerically diagonalize the Hamiltonian of a confined oscillator. These methods usually involve searching for roots of hypergeometric functions, as can be seen for example in references (AL-JABER, 2008) and (MONTGOMERY et al 2010). In (CAMPOY et al 2002) a method based on the expansion of the wavefunction as well as numerical integration of an ordinary differential equation were used to obtain the energy eigenvalues and wavefunctions of a one-dimensional confined oscillator. In this paper we will derive an explicit matrix representation for the Hamiltonian of the confined d − dimensional harmonic oscillator.

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SUMMARY