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Oxford University Press eBooks

Simple harmonic oscillators

Tom Lancaster, Stephen J. Blundell

Published Apr 17, 2014• 1 citation
Publisher

Abstract

The simple harmonic oscillator is well known from basic quantum physics as an elementary model of an oscillating system. The chapter solves this simple model using creation and annihilation operators and shows that the solutions have the characteristics of particles. Linking masses by springs into a chain allows for a generalization of this problem and the solutions are phonons.

DOI
10.1093/acprof:oso/9780199699322.003.0003

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The simple harmonic oscillator is well known from basic quantum physics as an elementary model of an oscillating system. The chapter solves this simple model using creation and annihilation operators and shows that the solutions have the characteristics of particles. Linking masses by springs into a chain allows for a generalization of this problem and the solutions are phonons.
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.
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.
Another "simple" potential is one experienced by a spring, the simple harmonic oscillator potential (SHO). To understand how a quantum particle behaves in such a potential, first we study how a classical particle, such as a mass on a spring, behaves. Then we change to using the TISE with the above potential. Solving this TISE is not a simple task. One way of solving a differential equation is to see if someone else has already solved it. Fortunately, a French mathematician named Hermite solved a related equation in the 19 th century. His solutions can be multiplied by a simple exponential function, and these modified Hermite polynomials can be shown to solve the TISE for the SHO. Interestingly, the resulting trapped particles have quantized energies that are evenly spaced. It is for this reason that bond vibrations, which can be modeled as springs on a mass, are found to be quantized in nearly evenly spaced energy levels, at least for relatively small vibrations. The SHO TISE also provides an opportunity to introduce two new operators: the creation and annihilation operators.
Chapter 9 is devoted to a detailed study of the harmonic oscillator in one dimension. The time-dependent Schrödinger equation is used to construct the coherent oscillator packet with minimal dispersion, which provides an opportunity to appreciate the similarities and differences between the dynamics of the classical and quantum oscillators. In a second step, we study the stationary solutions of the oscillator, which represent one of the most important orthogonal sets of quantum eigenfunctions because of its many applications, properties, and relative simplicity. Further, the use of raising and lowering (or creation and annihilation) operators is shown to be a powerful way of dealing with both material and field oscillators in general. Finally, the problem of two coupled oscillators is used to generalize the Schrödinger equation to the case of more than one particle.
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.