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 3 essays across one field — 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

Named alongside it

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

Classical limitCoherent stateUncertainty principleZero-point energyAnharmonicityCommutatorCorrespondenceCorrespondence principleDe broglie wavelengthDecoherenceEhrenfest theoremExpectation value

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