8.9: Harmonic Oscillator

Introduction

The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment of such diverse phenomena as elasticity, acoustics, AC circuits, molecular and crystal vibrations, electromagnetic fields and optical properties of matter.

Classical Oscillator

A simple realization of the harmonic oscillator in classical mechanics is a particle which is acted upon by a restoring force proportional to its displacement from its equilibrium position. Considering motion in one dimension, this means

\[ F = −kx \label{1}\]

Such a force might originate from a spring which obeys Hooke’s law, as shown in Figure \(\PageIndex{1}\). According to Hooke’s law, which applies to real springs for sufficiently small displacements, the restoring force is proportional to the displacement—either stretching or compression—from the equilibrium position.

Figure \(\PageIndex{1}\): Spring obeying Hooke’s law.

The force constant \(k\) is a measure of the stiffness of the spring. The variable \(x\) is chosen equal to zero at the equilibrium position, positive for stretching, negative for compression. The negative sign in Equation \(\ref{1}\) reflects the fact that \(F\) is a restoring force, always in the opposite sense to the displacement \(x\).

Applying Newton’s second law to the force from Equation \(\ref{1}\), we find \(x\)

\[ F = m \dfrac{d^2 x}{dx^2} = -kx \label{2} \]

where \(m\) is the mass of the body attached to the spring, which is itself assumed massless. This leads to a differential equation of familiar form, although with different variables:

\[ \ddot{x}(t)+ \omega^2x(t)= 0 \label{3}\]

with

\[\omega^2 \equiv \dfrac{k}{m}\]

The dot notation (introduced by Newton himself) is used in place of primes when the independent variable is time. The general solution to Equation \(\ref{3}\) is

\[ x(t) = A\sin ωt + B\cos ωt \label{4}\]

which represents periodic motion with a sinusoidal time dependence. This is known as simple harmonic motion and the corresponding system is known as a harmonic oscillator. The oscillation occurs with a constant angular frequency

\[ \omega = \sqrt{\dfrac{k}{m}}\; \text{radians per second} \label{5} \]

This is called the natural frequency of the oscillator. The corresponding circular (or angular) frequency in Hertz (cycles per second) is

\[ \nu = \dfrac{\omega}{2\pi } = \dfrac{1}{2\pi} \sqrt{\dfrac{k}{m}}\; \text{Hz} \label{6}\]

The general relation between force and potential energy in a conservative system in one dimension is

\[ F =\dfrac{−dV}{dx} \label{7}\]

Thus the potential energy of a harmonic oscillator is given by

\[ V(x) = \dfrac{1}{2}kx^2 \label{8}\]

which has the shape of a parabola, as drawn in Figure \(\PageIndex{2}\). A simple computation shows that the oscillator moves between positive and negative turning points \(\pm x_{max}\) where the total energy \(E\) equals the potential energy \(\dfrac{1}{2} k x_{max}^{2}\) while the kinetic energy is momentarily zero. In contrast, when the oscillator moves past \(x = 0\), the kinetic energy reaches its maximum value while the potential energy equals zero.

Figure \(\PageIndex{2}\): Potential energy function and first few energy levels for harmonic oscillator.

Harmonic Oscillator in Quantum Mechanics

Given the potential energy in Equation \(\ref{8}\), we can write down the Schrödinger equation for the one-dimensional harmonic oscillator:​

\[ -\frac{\hbar^{2}}{2m} \frac{d^2\psi(x)}{dx^2} + \frac{1}{2}kx^2 \psi(x) = E \psi(x) \label{9}\]

This equation is challenging to solve for \(\psi\) because the potential varies with position and is less than infinity in all of space. This means the particle can be found at any x between \(\pm\infty\). Solutions for it have been found by others by what are essentially sophisticated guess and check methods. The results are:

\[\psi_{n}(x)=\left(\frac{\sqrt{\alpha}}{2^{n}n!\sqrt{\pi}}\right)^{1/2} H_{n}(\sqrt{\alpha}x) e^{-\alpha x^{2}/2} \label{26}\]

where 

\[\alpha = \sqrt{\frac{mk}{\hbar^2}}\]

and \(H_{n}(\xi)\), with \(\xi = \sqrt{\alpha}x\) represents the Hermite polynomial of degree \(n\). The first few Hermite polynomials are

\[H_{0}(\xi)=1\]

\[H_{1}(\xi)=2\xi\]

\[H_{2}(\xi)=4\xi^{2}-2\]

\[H_{3}(\xi)=8\xi^{3}-12\xi \label{27}\]

The four lowest harmonic-oscillator eigenfunctions are plotted in Figure \(\PageIndex{3}\). Note the topological resemblance to the corresponding particle-in-a-box eigenfunctions.

Figure \(\PageIndex{3}\): Harmonic oscillator eigenfunctions for n=0, 1, 2, 3 and probability densities. Two things to note: these wavefunctions do not reach zero until x = \(\pm\infty\); a particle represented by any of these wavefunctions have a probability of being in the classically forbidden regions where the energy is less than the potential energy (the particle exhibits tunneling).

The eigenvalues are given by the simple formula

\[E_{n}=\left(n+\dfrac{1}{2}\right)\hbar\omega \label{28}\]

where \(\omega\) is as defined in equation \(\ref{5}\). These energy levels are drawn in Figure \(\PageIndex{2}\), on the same scale as the potential energy. The ground-state energy \(E_{0}=\dfrac{1}{2}\hbar\omega\) is greater than the classical value of zero, again a consequence of the uncertainty principle. This means that the oscillator is always oscillating.

Note that the difference between successive energy eigenvalues has a constant value

\[\Delta E=E_{n+1}-E_{n}=\hbar\omega=h\nu \label{29}\]

This is reminiscent of Planck’s formula for the energy of a photon. It comes as no surprise then that the quantum theory of radiation has the structure of an assembly of oscillators, with each oscillator representing a mode of electromagnetic waves of a specified frequency.

 

Contributors and Attributions

• Seymour Blinder (Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor)

• Jonathan Gutow (UW Oshkosh).