Illustrated Physics
physics, drawn
Concept

Harmonic oscillator — where it appears

A system whose restoring force is proportional to displacement, so its potential is a parabola and its period does not depend on amplitude. Every minimum looks like one at small enough displacement, which is why it is the most-used model in physics and why its exactness is misleading.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Two costs, and the width that balances them. The energy of a particle in a harmonic well against how tightly its wavefunction is squeezed, in units of ħω and of the width that minimises the total. Two terms compete. Squeezing the particle into a smaller region raises its kinetic energy, because the uncertainty relation makes a narrow position spread a wide momentum spread and momentum is squared in the energy; that term rises as the inverse square of the width and goes to infinity as the particle is localised. Letting it spread out raises its potential energy, since the well gets steeper away from the bottom; that term rises as the square of the width. The sum has a minimum at a width of 1.0000 in these units, where the total is 0.5000 ħω and the two terms are equal at a quarter each. That number is exactly the true ground-state energy of a quantum harmonic oscillator, obtained here with nothing but the uncertainty relation and a minimisation. What the figure shows and the formula does not is why there is a floor at all: it is not that the particle happens to keep moving, but that every way of stopping it costs more than it saves.

The motion that cannot be stopped

A particle in a well cannot sit at the bottom of it. Squeezing it into a smaller region costs kinetic energy faster than it saves potential energy, so there is a width that minimises the total — and the minimum is not zero. Helium never freezes because of it.

quantum · Uncertainty An average that follows Newton's law exactly. The centre of a wavepacket in a harmonic well, and the classical orbit started from the same place, drawn on top of each other. They agree to 5.5e-6 over 2.2 periods, which is the split-step integrator's own error and not a physical gap. This is exact and it is exact for every state of a harmonic oscillator, however wide, however lumpy, however far from classical: the theorem needs ⟨−V′(x)⟩ = −V′(⟨x⟩), which for a linear restoring force is true term by term. It is worth being suspicious of how strong that looks. The harmonic oscillator is the one potential where the average force over a state and the force at the state's centre cannot differ, so it is the worst possible example from which to conclude that quantum averages follow classical paths.

The average that obeys Newton

Ehrenfest's theorem says the centre of a wavepacket moves according to the average force over the state. That is exact, it is often quoted as the reason classical mechanics survives, and the two statements are not the same — because the average of a force is the force at the average only when the force is linear, which is to say almost never.

quantum · Correspondence A bundle that swings and never spreads. A Gaussian of the ground state's own width, released at rest from x = 3 in a harmonic well and propagated on a grid by split-step Fourier, drawn at 0 of a period, 0.25 of a period, 0.5 of a period. The packet slides from side to side and its shape does not change: over 2.2 full periods the width moves by 6.6e-5 per cent, and its centre tracks x₀cos t to 8.6e-6. Every other initial width breathes. This one is the displaced ground state, and it is the closest a quantum state comes to being a classical oscillator — a definite thing at a definite place, moving on the classical trajectory, staying the size it was.

The state that swings like a pendulum

Most quantum states of an oscillator look nothing like a swinging weight. One family does: it follows the classical trajectory exactly, never spreads, and sits at the uncertainty minimum for ever — and it is the state a laser and a driven circuit actually produce.

quantum · Correspondence A relativistic mass on a spring, at four amplitudes, scaled to one size. The displacement of a mass on an ideal spring against time, over two periods, with the mass's inertia growing with its energy as relativity requires, at amplitudes of 0.2, 1, 3, 10 times c/ω₀ — the distance light travels in one radian of the slow oscillation. Each curve is divided by its own amplitude and its own period so that shape alone is compared. The small swing is the familiar sine. The large one is a triangle: the mass spends almost the whole cycle moving at nearly the speed of light, turning round abruptly at each end. Periods: 1.007, 1.175, 2.132, 6.431 times the slow period.

The spring that becomes a light-clock

A mass on an ideal spring keeps the same period however far it swings — that is what makes it a clock. Give the mass the inertia relativity demands, pull it back far enough, and the clock breaks. The speed can never pass c, so a large swing spends almost all its time moving at nearly the speed of light, turning round abruptly at each end: the sine becomes a triangle, the period grows until it is simply the time light takes to cross the swing four times, and a clock riding on the mass ages a small fraction of what one beside it does.

relativity · Relativistic dynamics Where a speck lands decides how much it weighs. The fractional drop in frequency of a cantilever's first and second bending modes when a small mass lands on it, per unit of that mass as a fraction of the beam's, against where along the beam it lands, from the clamp to the free tip. The response is the square of the mode's displacement at the landing point divided by twice the mode's effective mass: a quarter of the beam for both modes (0.250 and 0.250), since each is measured against its own tip. At the tip both modes respond fully — the first by 2.0 times the landed fraction, the second by 2.0. Near the clamp neither responds, so a speck landing there is invisible. Halfway along, the first mode drops by 0.23 and the second by 1.02; at 0.783 of the length the second mode has a node and does not notice the speck at all. A single frequency cannot tell a light speck at the tip from a heavier one further in.

The weight a frequency can find

A spring's frequency is the square root of its stiffness over the mass it carries, so adding a little mass lowers the note by half the fraction added. Nothing in that rule cares about gravity, and nothing in it cares about size. It weighs astronauts in orbit by the swing of a chair, films a fraction of an atom thick on a quartz plate, and single molecules on a beam a few micrometres long — and it improves without limit as the thing that vibrates gets lighter. What it cannot do on its own is say where the added mass sat, and that turns out to matter as much as how much there was.

mechanics · Harmonic approximation Three orbits that are one orbit, rescaled. Three orbits round the same mass, each an ellipse of eccentricity 0.6, with sizes in the ratio 1 : 2 : 3, integrated from the inverse-square force. The dots mark twelfths of each orbit's period, at the same fractions of the way round. The shapes are exact enlargements of one another and so are the motions, with time stretched as size to the power 3/2: the measured periods are 1.000, 2.828, 5.196 times the smallest, against 1.000, 2.828, 5.196 from the rule. That is Kepler's third law, obtained without solving for any orbit.

The third law that falls out of a change of scale

Kepler spent a decade of tables to find that the squares of the planets' periods go as the cubes of their distances. The law can be had in three lines, without solving for a single orbit. Gravity's potential energy halves when distances double; stretch every length of an orbit by some factor and every time by that factor to the power three-halves, and the action changes only by a constant, so the stretched orbit is an orbit too. The same move gives the swing of a spring that keeps time at any amplitude, the virial theorem that makes a star heat up as it loses energy, and the spacing of quantum levels.

mechanics · Least action Two ways for a swing to die. A mass on a spring released 10.6 dead-band widths from rest, against time in units of 1/ω. Red: sliding on a dry surface, whose friction is a constant force; each swing is exactly a cosine, and the turning points fall in a straight line by two dead-band widths per half-cycle until the mass stops, 2.5 cycles after release, at −0.60. Blue: the same mass with viscous damping (damping ratio 0.05), whose force is proportional to speed; the swings shrink by the same fraction each cycle, along an exponential, and never stop. The period is the same in both and the same as with no damping at all to within the viscous case's 0.13 per cent.

The swing that dies in a straight line

A pendulum in air loses the same fraction of its swing every cycle, so its amplitude falls along an exponential and, in principle, never quite reaches zero. A mass on a spring sliding over a dry surface does something else. Its friction is a fixed force, so it loses a fixed amount of amplitude every half-swing — the swings shrink in a straight line — and it stops completely, after a countable number of swings, somewhere inside a band around its natural position whose width is the friction divided by the stiffness. Where in that band it stops depends on where it started, which is why a barometer with friction in it is tapped before it is read.

mechanics · Friction

Named alongside it

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

AnharmonicityClassical limitCoherent stateUncertainty principleZero-point energyBohr sommerfeld quantisationCantileverCommutatorCorrespondenceCorrespondence principleCoulomb frictionDamping

All concepts