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European Journal of Physics

Quantum harmonic oscillator: an elementary derivation of the energy spectrum

Riccardo Borghi

Published Feb 3, 2017• 7 citations
Publisher

Abstract

Abstract An elementary treatment of the quantum harmonic oscillator is proposed. No previous knowledge of linear differential equation theory or Fourier analysis are required, but rather only a few basics of elementary calculus. The pivotal role in our analysis is played by the sole particle localization constraint, which implies square integrability of stationary-state wavefunctions. The oscillator ground-state characterization is then achieved in a way that could be grasped, in principle, even by first-year undergraduates. A very elementary approach to build up and to characterize all higher-level energy eigenstates completes our analysis.

DOI
10.1088/1361-6404/aa57cf

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Abstract An elementary treatment of the quantum harmonic oscillator is proposed. No previous knowledge of linear differential equation theory or Fourier analysis are required, but rather only a few basics of elementary calculus. The pivotal role in our analysis is played by the sole particle localization constraint, which implies square integrability of stationary-state wavefunctions. The oscillator ground-state characterization is then achieved in a way that could be grasped, in principle, even by first-year undergraduates. A very elementary approach to build up and to characterize all higher-level energy eigenstates completes our analysis.
Abstract The ground state of the linear harmonic oscillator is the solution to a minimization problem, and this opens up two new ways to completely solve Schrödinger’s time-independent equation. The first exploits systematic patterns that emerge from a formal relationship between two different integral representations of the energy. A remarkable invariant form of the ground-state probability-density equation under specific conditions is the leading principle of the second procedure, where a uniform notation for all mathematical object involved facilitates calculations and generates useful schemes to quickly deduce the main functional relationships of the problem. We believe these procedures may represent useful educational alternatives to standard treatments that address this fundamental problem.
In this work we present a heuristic derivation for the energy levels of a linear harmonic oscillator that involves only algebra. This derivation offers the instructor a way to rationalize the energy level spectrum of a harmonic oscillator without having to solve the complicated differential equation prescribed by the Schrödinger equation for the harmonic oscillator. This derivation treats the harmonic oscillator as a quasiparticle in a box and yields qualitative results, these being the correct scaling of the energy with quantum number and vibrational frequency, but does not yield the exact result. By comparing the exact result with the result of the heuristic derivation students are introduced to the idea of tunneling in quantum mechanics and how tunneling can effect the energy levels of quantum mechanical systems.
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.
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. No previous knowledge of calculus of variations is required. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: 'completion of square' and integration by parts. This makes our approach particularly suitable for undergraduates. Moreover, the key role played by particle localization is emphasized through the entire analysis. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. Such an unexpected connection is outlined in the final part of the paper.
ABSTRACT In standard quantum mechanics, the spectrum of the harmonic oscillator: is derived by solving the Schrödinger equation for a parabolic potential. This is a mathematical formalism that gives correct numerical results but does not explain why the levels are as they are – why there is a ground state , why the levels are equally spaced, why the difference between them is . In the TVM framework, the spectrum of the harmonic oscillator is recognized through two independent physical processes that are already embedded in the mechanism of the Universe's evolution: Evolution of the first level – how the energy of the original, primary level decreases through successive transformations. Hierarchy of new levels – how, at each transformation, part of the energy separates from the previous level and forms a new, independent level. These two processes are simultaneous and correlated: the first level cannot evolve without creating new levels, and new levels cannot arise without the evolution of the first level. They are two sides of the same process – the impossibility of perfect division upon reaching the transformation limit . In TVM, the ground state of the harmonic oscillator is not a "zero-point energy of the vacuum" – it is a consequence of the first transformation. The formula: comes directly from the TVM mechanism and fractal recursion, where each new level has its own frequency , and the product of amplitude and momentum grows as . Thus, the harmonic oscillator ceases to be a "special" quantum system – it becomes a general pattern that appears wherever a system undergoes a series of transformations, from the initial state of the Universe to local quantum phenomena. Keywords: TVM, harmonic oscillator, spectrum, fractal recursion, transformation, stationarity, energy defect. NOTE: This is the third in a series of supplementary scientific papers that elaborate the application of the TVM formalism to specific physical phenomena. Starting from the foundations laid in the zeroth paper [1], this work derives the harmonic oscillator spectrum from the transformation mechanism and the hierarchy of levels. It is recommended that the reader first study the First supplementary work (de Broglie relation) [2] and the Second supplementary work (Electron shell filling) [3], as well as Chapters 3 and 4 of the main work [1].