Illustrated Physics
physics, drawn
Concept

Harmonic approximation — where it appears

The replacement of a potential near its minimum by the parabola matching the curvature there, available because the linear term vanishes. It is excellent for frequencies and returns exactly zero for thermal expansion and for the coupling between modes, so it is silent about them rather than approximate.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

One parabola, several wells. Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 1% of the parabola's own value there: a pendulum at 0.35, a chemical bond at 0.01, a pair of atoms at 0.0015. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

Every minimum is a parabola

A pendulum, a bond between two atoms and a ship rolling in a swell obey the same equation, and the reason is not that they are alike. It is that a function with a minimum has no linear term there, so the first thing every potential well looks like is the same well.

mechanics · Harmonic approximation The two normal modes of a coupled pair at kc/k = 0.1. Two equal masses, each held to a wall by a spring of stiffness k and to each other by a coupling spring of 0.1k. Above: the in-phase mode, in which both masses move the same way by the same distance, the coupling spring never changes length, and the frequency is therefore 1.0000√(k/m) — the coupling does not appear in it at all. Below: the out-of-phase mode, in which the coupling spring changes length by twice the displacement, so each mass feels k + 2kc and the frequency rises to 1.0954√(k/m), a ratio of 1.0954. Both displacement patterns are the eigenvectors of the pair's stiffness matrix, obtained from its trace and determinant and checked against those two square roots. The red arrows are the force each mass is pulled back by, computed as −Kx: 1.00kA in the first mode against 1.20kA in the second, a factor of 1.20, which is the square of the frequency ratio because ω² is a stiffness over a mass. Any motion of the pair whatsoever is a sum of these two and nothing else.

The two pendulums that will not stop swapping

Coupled oscillators joined by a weak spring appear to hand energy back and forth. Nothing is handed anywhere: the system has only a pair of motions that keep their shape, at √(k/m) and √((k+2kc)/m), and the apparent traffic is the beat between them — 51 swings from one handover to the next at a coupling of one part in fifty. Extend the same arithmetic to N masses and it produces a dispersion relation with a hard ceiling, near 7 THz in copper.

mechanics · Harmonic approximation The average position of something that is only shaking. The mean displacement of an oscillator against temperature, taken as a Boltzmann average over each well rather than from any expansion of it, with the temperature measured against each well's own depth so that unlike bonds can share an axis. A symmetric well gives exactly zero at every temperature — heating a harmonic solid makes it vibrate harder and does not make it larger. The others drift outward, because the outward side is the shallower one, and the measured slopes are a pendulum 0.000, a chemical bond 0.766, a pair of atoms 0.150. Thermal expansion is not a property a spring has; it is one a spring lacks.

Why heating a perfect spring changes nothing

A harmonic solid vibrates harder when heated and does not get any longer. Thermal expansion lives entirely in the term that the harmonic approximation throws away — and so does the fact that a solid conducts heat at a finite rate, which is the same discarded term doing a second job nobody would have connected to the first.

thermodynamics · Harmonic approximation A drive at one frequency, and what comes back at three times it. The Fourier components of the steady motion of an oscillator driven at a single frequency ω = 1.35, for drive strengths of 0.02, 0.05, 0.1. The equation is a harmonic oscillator with a cubic term added, and the components are projected out of the integrated motion rather than assumed. A linear oscillator answers only in the first column. This one answers in the third as well, because x³ of a cosine contains a cosine of three times the angle. The third harmonic grows as the drive to the power 3.00 where the fundamental grows as the power 1.00, so it is negligible at small drive and not at large — which is why nonlinearity in an instrument is a specification rather than a yes or no.

The oscillator that answers at three times the question

Push a spring hard enough that the parabola stops being the whole story, and three things happen that a linear oscillator cannot do: it emits frequencies nobody supplied, its resonance leans over, and its amplitude at one drive frequency depends on where the drive has been.

mechanics · Harmonic approximation The area under the spacings is the bond. The gap between each vibrational level and the next, against level number, for a Morse well and a Lennard-Jones well with the same depth and the same curvature at the bottom, Λ = 100, in units of the well depth. The sum of the gaps from the lowest level up is the energy needed to break the bond from there. For the Morse well the gaps fall in a straight line to zero, and a straight line through the first 5 of them, extended to zero, encloses 0.9473 against the true 0.9473. For the Lennard-Jones well the gaps start almost the same and fall slightly faster at first — which is what the straight line sees — then trail off in a long tail of small gaps near the top; the same extrapolation from the first 5 encloses 0.7988 against the true 0.9474 — 15.7 per cent short, because the straight line cannot see the crowded levels its data never reached.

The spectrum that weighs a bond

A vibrational spectrum measured near the bottom of a molecule's well gives its curvature, and the curvature says nothing about how deep the well is. Measured further up, it says everything. The levels of a real bond crowd together towards the top, the gap between neighbours falls towards zero, and the sum of all the gaps is the energy it takes to break the bond. A straight line drawn through the gaps that have been measured, and extended to zero, was for decades how bond energies were found — and for a bond with the tail of two neutral atoms it comes out a sixth short.

mechanics · Harmonic approximation A bound state that lives outside its well. The radial probability density of the one bound state of a square well of radius R, against distance in units of R, for four depths just past the one that first binds a state. In units of ε = ħ²/2μR², a state bound by 0.00589 ε has 93 per cent of its probability outside the well and a mean radius of 7.0 R; a state bound by 0.0397 ε has 82 per cent of its probability outside the well and a mean radius of 3.0 R; a state bound by 0.245 ε has 62 per cent of its probability outside the well and a mean radius of 1.5 R; a state bound by 1.4 ε has 35 per cent of its probability outside the well and a mean radius of 0.9 R. The more weakly the state is bound, the further its exponential tail reaches, and the less of it is where the force is.

The molecule that lives outside its own well

The harmonic picture describes the bottom of a well, where a molecule vibrates near the length of its bond. The last bound level lives at the top, and when it is bound by almost nothing it is not near the bond at all. Two helium atoms whose potential well bottoms out at three ångströms form a molecule whose atoms sit on average about fifty ångströms apart, most of the time where the force between them is negligible. Such a state forgets the shape of the well that made it. Its size, its binding and the way it scatters are all set by one number, and nothing inside the well can be read from them.

mechanics · Harmonic approximation

Named alongside it

The objects these essays reach for when they reach for this one.

AnharmonicityPotential wellSimple harmonic motionNormal modesTaylor expansionZero-point energyAmplitude dependenceApproximationBeatsBinding energyThe Boltzmann factorBound state

All concepts