# Why is the harmonic oscillator so important?

I've been wondering what makes the harmonic oscillator such an important model. What I came up with:

• It is a (relatively) simple system, making it a perfect example for physics students to learn principles of classical and quantum mechanics.

• The harmonic oscillator potential can be used as a model to approximate many physical phenomena quite well.

The first point is sort of meaningless though, I think the real reason is my second point. I'm looking for some materials to read about the different applications of the HO in different areas of physics.

• The second point is really important. Almost any system near equilibrium is at least approximately harmonic because you can expand the potential energy in a Taylor series and the linear term is zero by construction. This applies to everything from atoms in a crystal to quantum fields. – DanielSank Jan 12 '15 at 17:20
• The first reason is not meaningless. It provides a starting point for modeling more complex dynamic systems. For example the harmonic oscillator assumes linear damping, but Duffing extended the simple linear oscillator to one where the damping is nonlinear. And this extends the coverage of the modeling to other physical systems not so well modeled by the linear oscillator. You need to walk before you can run. – docscience Jan 12 '15 at 17:23
• Remarkably, this question does not appear to have been asked yet (correct me if I'm wrong!). If so, this has the potential to become a really great, canonical question for this site; I'm looking forward to reading some good answers. – Mark Mitchison Jan 12 '15 at 17:41
• @DepeHb for the study of the quantum harmonic oscillator a formalism is developed (you probably will learn it some later), with raising and lowering operators. This formalism will accompany you in whatever theory that uses the 2nd quantization, i.e. in which the number of particles of a certain type is not constant. – Sofia Jan 12 '15 at 22:07
• "The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction." Sidney Coleman. – Davidmh Jan 13 '15 at 19:18

To begin, note that there is more than one incarnation of "the" harmonic oscillator in physics, so before investigating its significance, it's probably beneficial to clarify what it is.

What is the harmonic oscillator?

There are at least two fundamental incarnations of "the" harmonic oscillator in physics: the classical harmonic oscillator and the quantum harmonic oscillator. Each of these is a mathematical thing that can be used to model part or all of certain physical systems in either an exact or approximate sense depending on the context.

The classical version is encapsulated in the following ordinary differential equation (ODE) for an unknown real-valued function $f$ of a real variable: \begin{align} f'' = -\omega^2 f \end{align} where primes here denote derivatives, and $\omega$ is a real number. The quantum version is encapsulated by the following commutation relation between an operator $a$ on a Hilbert space and its adjoint $a^\dagger$: \begin{align} [a, a^\dagger] = I. \end{align} It may not be obvious that these have anything to do with one another at this point, but they do, and instead of spoiling your fun, I invite you to investigate further if you are unfamiliar with the quantum harmonic oscillator. Often, as mentioned in the comments, $a$ and $a^\dagger$ are called ladder operators for reasons which we don't address here.

Every incarnation of harmonic oscillation that I can think of in physics boils down to understanding how one of these two mathematical things is relevant to a particular physical system, whether in an exact or approximate sense.

Why are these mathematical models important?

In short, the significance of both the classical and quantum harmonic oscillator comes from their ubiquity -- they are absolutely everywhere in physics. We could spend an enormous amount of time trying to understand why this is so, but I think it's more productive to just see the pervasiveness of these models with some examples. I'd like to remark that although it's certainly true that the harmonic oscillator is a simple an elegant model, I think that answering your question by saying that it's important because of this fact is kind of begging the question. Simplicity is not a sufficient condition for usefulness, but in this case, we're fortunate that the universe seems to really "like" this system.

Where do we find the classical harmonic oscillator?

(this is by no means an exhaustive list, and suggestions for additions are more than welcome!)

1. Mass on a Hooke's Law spring (the classic!). In this case, the classical harmonic oscillator equation describes the exact equation of motion of the system.
2. Many (but not all) classical situations in which a particle is moving near a local minimum of a potential (as rob writes in his answer). In these cases, the classical harmonic oscillator equation describes the approximate dynamics of the system provided its motion doesn't appreciably deviate from the local minimum of the potential.
3. Classical systems of coupled oscillators. In this case, if the couplings are linear (like when a bunch of masses are connected by Hooke's Law springs) one can use linear algebra magic (eigenvalues and eigenvectors) to determine normal modes of the system, each of which acts like a single classical harmonic oscillator. These normal modes can then be used to solve the general dynamics of the system. If the couplings are non-linear, then the harmonic oscillator becomes an approximation for small deviations from equilibrium.
4. Fourier analysis and PDEs. Recall that Fourier Series, which represent either periodic functions on the entire real line, or functions on a finite interval, and Fourier transforms are constructed using sines and cosines, and the set $\{\sin, \cos\}$ forms a basis for the solution space of the classical harmonic oscillator equation. In this sense, any time you are using Fourier analysis for signal processing or to solve a PDE, you are just using the classical harmonic oscillator on massively powerful steroids.
5. Classical electrodynamics. This actually falls under the last point since electromagnetic waves come from solving Maxwell's equations which in certain cases yields the wave equation which can be solved using Fourier analysis.

Where do we find the quantum harmonic oscillator?

1. Take any of the physical systems above, consider a quantum mechanical version of that system, and the resulting system will be governed by the quantum harmonic oscillator. For example, imagine a small system in which a particle is trapped in a quadratic potential. If the system is sufficiently small, then quantum effects will dominate, and the quantum harmonic oscillator will be needed to accurately describe its dynamics.
2. Lattice vibrations and phonons. (An example of what I assert in point 1 when applied to large systems of coupled oscillators.
3. Quantum fields. This is perhaps the most fundamental and important item on either of these two lists. It turns out that the most fundamental physical model we currently have, namely the Standard Model of particle physics, is ultimately based on quantizing classical fields (like electromagnetic fields) and realizing that particles basically just emerge from excitations of these fields, and these excitations are mathematically modeled as an infinite system of coupled, quantum harmonic oscillators.

The harmonic oscillator is important because it's an approximate solution to nearly every system with a minimum of potential energy.

The reasoning comes from Taylor expansion. Consider a system with potential energy given by $U(x)$. You can approximate $U$ at $x=x_0$ by $$U(x) = U(x_0) + (x-x_0) \left.\frac{dU}{dx}\right|_{x_0} + \frac{(x-x_0)^2}{2!} \left.\frac{d^2U}{dx^2}\right|_{x_0} + \cdots$$ The system will tend to settle into the configuration where $U(x)$ has a minimum --- but, by definition, that's where the first derivative $dU/dx = 0$ vanishes. Also a constant offset to a potential energy usually does not affect the physics. That leaves us with $$U(x) = \frac{(x-x_0)^2}{2!} \left.\frac{d^2U}{dx^2}\right|_{x_0} + \mathcal O(x-x_0)^3 \approx \frac12 k (x-x_0)^2$$ which is the harmonic oscillator potential for small oscillations around $x_0$.

• I wonder whether there is any relevant example of an oscillation around x₀ with d²U/dx²=0 at x₀ – Walter Tross Jan 13 '15 at 23:11
• @WalterTross I remember talking about a quartic oscillator in some class, but I don't remember if there was a physical context or if it was just a made-up potential. That would make a good question. – rob Jan 14 '15 at 5:05
• The quartic oscillator is not the most important counterexample to "everything can be approximated harmonically". More relevantly, if a particle's ground state in some quantum-well is unsharp over a range where the higher-order expansion terms of the potential can't be neglected, then the harmonic oscillator is utterly useless for describing the system. As an extreme case, the hydrogen atom is totally not harmonic, you can't even Taylor-expand the potential at all! – leftaroundabout Jan 14 '15 at 11:12
• @leftaroundabout but can you use other transforms to deal with it, like Fourier? – Ooker Sep 8 '17 at 11:37
• @Ooker sure, but those only converge in an $L^2$ sense, not in a locally-pointwise sense. – leftaroundabout Sep 8 '17 at 12:26

## The harmonic oscillator is common

It appears in many everyday examples: Pendulums, springs, electronics (such as the RLC circuit), standing waves on a string, etc. It's trivial to set up demonstrations of these phenomena, and we see them constantly.

## The harmonic oscillator is intuitive

We can picture the forces on systems such as pendulum or a plucked string. This makes it simple to study in the classroom. In contrast, there are many "everyday" examples that are not intuitive, such as the infamous Bernoulli effect lifting a disk by blowing air downwards. These paradoxes are great puzzles, but they would confuse (most) beginner students.

## The harmonic oscillator is mathematically simple

Math is part of physics. In studying simple harmonic motion, students can immediately use the formulas that describe its motion. These formulas are understandable: for example, the equation for frequency shows the intuitive result that increasing spring stiffness increases frequency. At a more advanced level, students can derive the equations from first principles. The ability to solve a real-life problem so easily is a clear demonstration of how physics uses math.

Engineering benefits greatly as well. Many systems, even very complex ones, are linear. Complicated linear systems act as multiple harmonic oscillators. For example, a pinned string naturally vibrates at frequencies that are multiples of its fundamental. Any motion of the string can be represented as a sum of each component vibration, with each components independent of other components. This superposition allows us to model things like plucking the string. Circular plates, guitar chambers, skyscrapers, radio antennas, and even molecules are more complex. However, superposition and other tools from linear systems theory still allows us to take massive shortcuts on the computation and trust the results. These computation methods also make good teaching tools for topics in linear algebra and differential equations.

Because the harmonic oscillator is a familiar system that is so tightly connected with fundamental topics in math, science, and engineering it is one of the most widely studied and understood systems.

The other answers already cover many of the most important aspects. One interesting application is in finding out how the form of the harmonic oscillator is connected to Gaussian (normal) distribution, another often used mathematical construct. I might've suggested this to joshphysics' list, but as it requires some details to appreciate, decided to make it a standalone answer (but it really is more of a drawn out comment).

Take $N$ independent random variables $X_i$, each with variance $\sigma$ and, for simplicity, mean $0$. Now the characteristic function for an arbitrary probability distribution $P_X$ is $G_X(k) = \langle e^{ikX} \rangle = \int e^{ikx}P_X(x) \mathrm{d}x$. Writing out the exponential in Taylor series (where we cut off all the terms past the quadratic) $e^{ikx} \approx 1 + ixk - \frac{1}{2}x^2k^2$, we have $G_X(k) \approx 1 - \frac{1}{2}\sigma^2k^2$.

Now define a new random variable $Z = \frac{\sum_{i=1}^N X_i}{\sqrt{N}}$, so $G_Z(k) = \left(G_X\left(\frac{k}{\sqrt{N}}\right)\right)^N \approx \left(1 - \frac{\sigma^2k^2}{2N}\right)^N$ and as $N\to\infty$ (all the higher order terms in the sum drop) we have by definition, $G_Z(k) = e^{-\frac{1}{2}\sigma^2k^2}$, which then gives the Gaussian distribution $$P_Z(z) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{z^2}{2\sigma^2}}$$

This is a simplistic derivation of the central limit theorem, which is of huge importance in several areas of science and probably among the most fundamental results of statistics.

Note that in the derivation all the higher order terms dropped out (as $N\to\infty$), and the only remaining one was the quadratic, harmonic, term. This happens regularly in applications across different domains, yet I can't really name a fundamental reason why it should be so.

I think Rob's answer pretty much inclusive and true. I just want add something. If you expand the potential via Taylor series, the second derivative is looking for a tangential vector at point $x_0$ which is the minimum point of the curve, so it is zero. So, we have a potential of the form $\frac{1}{2}k(x-x_0)^2$ which we have shifted the the origin of x to the location of $x_0$. So we would approximate the curve of the potential to a parabola. That makes harmonic oscillator important for physics.

• Sorry I do not agree with you. I edit my answer. approximating any curve like potential energy or etc. makes our life easier for calculating anything around the shifted point. – aQuestion Jan 13 '15 at 15:17

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