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A Brief Introduction to the Quantum Harmonic Oscillator

Salvish Goomanee

Published Jul 1, 2013• 0 citations
OpenAlex

Abstract

In this short paper, a very brief introduction of the quantum harmonic oscillator has been provided. The understanding of the transition from a classically damped oscillator and the quantum mechanical model of the latter is included via a clear mathematics involving simple calculus. Only the one dimensional time dependent Schrodinger’s equation is being studied. This paper is intended to simply provide an analytical description of the subject, there are no advance calculations implying further suggestions in quantum theory.

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The Journal of Chemical Physics•May 15, 1983•7 citations
In this short paper, a very brief introduction of the quantum harmonic oscillator has been provided. The understanding of the transition from a classically damped oscillator and the quantum mechanical model of the latter is included via a clear mathematics involving simple calculus. Only the one dimensional time dependent Schrodinger’s equation is being studied. This paper is intended to simply provide an analytical description of the subject, there are no advance calculations implying further suggestions in quantum theory.
Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator X^ \hat X X^ in x representation. The evolution of ⟨x^⟩ \left\langle {\hat x} \right\rangle ⟨x^⟩ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.
As the title aptly states, this book deals with harmonic oscillators of various kinds, from classical mechanical and electrical oscillations up to quantum oscillations. It is written in a lively language, and occasional interspersed anecdotes make the reading of an otherwise mathematically oriented text quite a pleasure. Although the author claims to have written an `elementary introduction', it is certainly necessary to have a good deal of previous knowledge in physics (mechanics, electrodynamics, quantum theory), electrical engineering and, of course, mathematics in order to follow the general line of his arguments. The book begins with a thorough treatment of classical oscillators (free, damped, forced) that is followed by an elaboration on Fourier analysis. Lagrange and Hamilton formalisms are then introduced before the problem of coupled oscillations is attacked. A chapter on statistical perspectives leads over to the final discussion of quantum oscillations. With the book comes a diskette containing a number of worksheets (Microsoft Excel) that can be used by the reader for instant visualization to get a better qualitative and quantitative understanding of the material. To the reviewer it seems difficult to pinpoint exactly the range of prospective readership of the book. It can certainly not be intended as a textbook for students, but rather as a reference book for teachers of physics or researchers, who want to look up one or other aspect of harmonic oscillations, for which purpose the diskette represents a very valuable tool.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
Abstract A discussion of the physical meaning of the Schrödinger wave equation can not only constitute an exciting introduction to some of the more abstract ideas of quantum mechanics, but serves more generally as a useful demonstration of the application of mathematics to modern physics. This frontline uses physical concepts and mathematical techniques that would be accessible to a sufficiently interested secondary school student, in order to derive the simplest, one-dimensional case of the time-independent Schrödinger equation: the derivation relies only upon aspects of Newtonian/wave mechanics, quantum theory, complex numbers and calculus that would be covered in an advanced secondary school syllabus.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.