Equation of the Day #14: The Harmonic Oscillator

The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.”
Sidney Coleman

One of the first physical systems students get to play with is the harmonic oscillator. An oscillator is a system with some property that varies repeatedly around a central value. This variation can be in displacement (such as a mass on a spring), voltage (like the electricity that comes out of the wall), or field strength (like the oscillations that make up light). When this variation has a fixed period, we call the motion harmonic oscillation, and the simplest harmonic oscillation is known as — you may have guessed — simple harmonic oscillation.

A simple harmonic oscillator is a system under the influence of a force proportional to the displacement of the system from equilibrium. Think of a mass on a spring; the further I pull the mass from the spring’s natural length, the more the spring pulls back, and if I push the mass so that the spring compresses, the spring will push against me. Mathematically, this is known as Hooke’s Law and can be written as

F=-k\Delta x

where F is the net force applied to the system, Δx is the displacement of the system from equilibrium, and k is some proportionality constant (often called the “spring constant”) which tells us how strongly the spring pushes or pulls given some displacement – a larger k indicates a stronger spring. If I let go of the mass when it is displaced from equilibrium, the mass will undergo oscillatory motion. What makes this “simple” is that we’re ignoring the effects of damping or driving the oscillation with an outside force.

How do we know that such a restoring force causes oscillatory motion? Utilizing Newton’s second law,

\begin{aligned} F &= ma\\ &= -kx\\ \Rightarrow \dfrac{d^2x}{dt^2} &= -\dfrac{k}{m} x \end{aligned}

The solution to this equation is sinusoidal,

x(t) = A\cos(\omega t +\phi),

where A is the amplitude of oscillation (the farthest the mass gets from equilibrium),

\omega \equiv \sqrt{\dfrac{k}{m}}

is the angular frequency of oscillation, and ϕ is the phase, which captures the initial position and velocity of the mass at time t = 0. The period is related to the angular frequency by

T=\dfrac{\tau}{\omega}

For this reason, harmonic oscillators are useful timekeepers, since they oscillate at regular, predictable intervals. This is why pendulums, coiled springs, and currents going through quartz crystals have been used as clocks. What other physical systems does this situation apply to? Well, if you like music, simple harmonic oscillation is what air undergoes when you play a wind, string, or membrane instrument. What you’re doing when you play an instrument (or sing) is forcing air, string(s), or electric charge (for electronic instruments) out of equilibrium. This causes the air, string(s), and voltage/current to oscillate, which creates a tone. Patch a bunch of these tones together in the form of chords, melodies, and harmonies, and you’ve created music. A simpler situation is blowing over a soda/pop bottle. When you blow air over the mouth of the bottle, you create an equilibrium pressure for the air above the mouth of the bottle. Air that is slightly off of this equilibrium will oscillate in and out of the bottle, producing a pure tone.


Image: Wikipedia

Now for the fun part: what happens when we enter the quantum realm? Quantum mechanics says that the energy of a bound system is quantized, and an ideal harmonic oscillator is always bound. The total energy of a harmonic oscillator is given by

\begin{aligned} E &= \dfrac{1}{2} mv^2 + \dfrac{1}{2} kx^2\\  &= \dfrac{1}{2m} \left(p^2 + (m\omega x)^2\right), \end{aligned}

where the first term is the kinetic energy, or energy of motion, and the second term is the potential energy, or energy due to location. I used the facts that p = mv and k = 2 to go from the first line to the second line. The quantum prescription says that p and x become mathematical operators, and the energy takes a role in the Schrödinger equation. For the harmonic oscillator, solving the Schrödinger equation yields the differential equation

\begin{aligned} \dfrac{\hbar}{m\omega}\dfrac{d^2\psi}{dx^2} + \left(\dfrac{2E}{\hbar\omega} - \dfrac{m\omega}{\hbar}\, x^2 \right) \psi(x) = 0 \end{aligned}

where ħ is the (reduced) Planck constant, and ψ is the quantum mechanical wave function. After solving this differential equation, the allowed energies turn out to be

E_n = \hbar\omega \left(n+\dfrac{1}{2}\right)

where n = 0, 1, 2, . . . is a nonnegative integer. Unlike the classical picture, the quantum states of the harmonic oscillator with definite energy are stationary and spread out over space, with higher energy states spread out more than lower energy states. There is a way, though, to produce an oscillating state of the quantum harmonic oscillator by preparing a superposition of pure energy states, forming what’s known as a coherent state, which actually does behave like the classical mass on a spring. It’s a weird instance of classical behavior in the quantum realm!


Classical simple harmonic oscillators compared to quantum wave functions of the simple harmonic oscillator.
Image: Wikipedia

An example of a quantum harmonic oscillator is a molecule formed by a pair of atoms. The bond between the two atoms gives rise to a roughly harmonic potential, which results in molecular vibrational states, like two quantum balls attached by a spring. Depending on the mass of the atoms and the strength of the bond, the molecule will vibrate at a specific frequency, and this frequency tells physicists and chemists about the bond-lengths of molecules and what those bonds are made up of. In fact, the quantum mechanical harmonic oscillator is a major topic of interest because the potential energy between quantum objects can often be approximated as a Hooke’s Law potential near equilibrium, even if the actual forces at play are more complex at larger separations.

Additionally, the energy structure of the harmonic oscillator predicts that energies are equally spaced by the amount ħω. This is a remarkable feature of the quantum harmonic oscillator, and it allows us to make a toy model for quantum object creation and annihilation. If we take the energy unit ħω as equal to the rest energy of a quantum object by Einstein’s E = mc2, we can think of the quantum number n as being the number of quantum objects in a system. This idea is one of the basic results of quantum field theory, which treats quantum objects as excitations of quantum fields that stretch over all of space and time. This is what the opening quote is referring to; physicists start off learning the simple harmonic oscillator as classical masses on springs or pendulums oscillating at small angles, then they upgrade to the quantum treatment and learn about its regular energy structure, and then to upgrade to the quantum field treatment where the energies are treated as a number of quantum objects arising from a omnipresent quantum field. I find it to be one of the most beautiful aspects of Nature that such a simple system recurs at multiple levels of our physical understanding of reality.

-A

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