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arXiv:1912.08355 · American Journal of Physics

A completely algebraic solution of the simple harmonic oscillator

M. Rushka, J. K. Freericks

Published Oct 21, 2020• 22 citations
arXivHTMLPDF

Abstract

We present a full algebraic derivation of the wavefunctions of a simple harmonic oscillator. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the coordinate-space or momentum-space wavefunctions from the energy eigenvectors. It is simple to incorporate into the undergraduate and graduate curricula. We provide a summary of the history of operator-based methods as they are applied to the simple harmonic oscillator. We present the derivation of the energy eigenvectors along the lines of the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schrödinger factorization method). We supplement it by employing the appropriate translation operator to determine the coordinate-space and momentum-space wavefunctions algebraically, without any derivatives.

DOI
10.1119/10.0001702

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We present a full algebraic derivation of the wavefunctions of a simple harmonic oscillator. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the coordinate-space or momentum-space wavefunctions from the energy eigenvectors. It is simple to incorporate into the undergraduate and graduate curricula. We provide a summary of the history of operator-based methods as they are applied to the simple harmonic oscillator. We present the derivation of the energy eigenvectors along the lines of the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schrödinger factorization method). We supplement it by employing the appropriate translation operator to determine the coordinate-space and momentum-space wavefunctions algebraically, without any derivatives.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
The harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics.
The factorization method of Schrödinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
Abstract The special and recurring role that the simple harmonic oscillator (SHO) plays in quantum mechanics can be attributed both to its physical relevance and its simple solutions. The fact that the ground state solution is the minimum uncertainty wave packet (Section 12.4) and the highly constrained connection between the position-space and momentum-space wavefunctions (P4.23 and Section 9.2.2) also indicates that this problem occupies a special niche. The symmetry between x and p present in the solutions is obviously a reflection of the fact that only for the SHO is the potential energy function quadratic in x. It is often the case in physics where systems with a high degree of symmetry are amenable to solution in a variety of ways, sometimes quite unexpected. In this chapter, we discuss a powerful method of solving the harmonic oscillator problem involving the factorization of the differential equation using differential operators; we then (briefly) discuss extensions and applications of operator methodsÉ to other physical problems.
Abstract Momentum and position operators are essential in many areas of quantum mechanics. This chapter introduces these operators and discusses their mathematical and conceptual foundations. These operators are then applied to the widely used simple harmonic oscillator model. Simple methods lead to the formal description of both the coordinate-space and momentum-space wave functions for the oscillator model. The Heisenberg uncertainty relation for the oscillator model embodies the important concept of wave–particle duality in quantum mechanics.