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CRC Press eBooks

Simple Harmonic Oscillator (SHO): V = ½ kx2

Steven M. Pascal

Published Jul 12, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781003356172-12

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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.
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.
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.
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.
In this chapter, the authors introduce the Hamilton-Jacobi equation showing that classical mechanics can be described with a wave equation for the classical action. In order to find the equation showing that classical mechanics already contains a wave behavior in matter particles, the people need to go back to the Lagrange/Hamilton formalism. Davisson-Germer experiment has shown without any doubt that electrons behave as massive particles whose dynamics is governed by some sort of wave. A free particle is apparently the conceptually simplest system to study because the TISE will reduce to its simplest form. The simple (linear) harmonic oscillator is of paramount importance in classical as well as quantum mechanics. A proper solution to the Harmonic oscillator problem can be found by solving Schrodinger equation analytically. Charles Hermite was a French mathematician who gave important contributions in many fields of mathematics used in physics like, for example, Hermitian operators and Hermite polynomials.
The Wave-particle duality and the Schrodinger cat paradox pose important notions on quantum mechanical systems, in which a quantum system can exist in two physically opposite or distinct states. This motivates scientists from the days of Louis de Broglie to confidently look at matter as having a dual nature, which is a physical particle that can also be visualized as a wave entity. This paper seeks to provide an elegant method of solving the time-independent Schrodinger Equation (TISE) for a quantum simple harmonic oscillator (QSHO). A fully algebraic method is presented, the paper explains in detail how simple algebra, coupled with basic quantum mechanical operator commutation algebra can be employed to solve the TISE without resorting to complicated mathematical methods of solving second-order differential equations. The construction of the Hamiltonian is explained and an analogy between the classical simple harmonic oscillator (CSHO) and the QSHO is drawn to develop the solution method. The action of creation and annihilation operators on wave functions is exploited to generate states with higher and lower energies respectively and to generate a Hamiltonian that can act on a system and generate required quantum states. The aim is to generate the ground state of the QSHO and its associated energy value. From this state, other higher energy states can be generated by the action of the creation operator. The use of algebraic methods has been seen to be versatile as it does not explicitly rely on any particular basis or coordinate system. The method allows operators to be manipulated in different bases for example the momentum operator can be given as P ̂ x=-iℏ (∂/∂x) in a spatial coordinate basis or as a pure number p ̂ =p in the momentum basis.