Abstract
The Schrödinger equation for an isotropic three-dimensional harmonic oscillator is solved using ladder operators. The starting point is the shape invariance condition, obtained from supersymmetric quantum mechanics. Generalized ladder operators can be constructed for the three spherical spatial coordinates. Special emphasis is given to the adaptation made to each of these coordinates. The approach used is general and is indicated as an alternative method to solve the Schrödinger equation.
Highlights
The harmonic oscillator is one of the most fundamental systems studied in quantum mechanics [1,2,3]
The Schrodinger equation for the three-dimensional (3D) harmonic oscillator is solved by using the spherical coordinates
The development of Supersymmetric Quantum Mechanics (SQM) [7, 8] has contributed to increase algebraic methods [9] applied to quantum mechanical problems and the shape invariance property has an underlying algebraic structure associated with Lie algebras [5, 10,11,12,13]
Summary
The harmonic oscillator is one of the most fundamental systems studied in quantum mechanics [1,2,3]. An algebraic approach is used to factorize the differential equations and ladder operators are built for each coordinate [4, 5] This approach is general and it can be used to solve the Schrodinger equation exactly, fitting into the context of Supersymmetric Quantum Mechanics (SQM) [6]. The isotropic spherical harmonic oscillator can be used as an introductory problem for formalism and for the construction of generalized ladder operators. The Schrodinger equation for the φ coordinate has been little explored in the literature using generalized ladder operators This equation is similar to the particle in a box problem and it has a hidden shape invariance [27].
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