Illustrated Physics
physics, drawn
Concept

Correspondence principle — where it appears

The requirement that a new theory reproduce the predictions of the one it replaces wherever that one was tested. In quantum mechanics it is satisfied by particular states and limits rather than automatically, and identifying which is a substantial part of the subject.

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

Where the particle is likely to be found. A particle confined between two walls one unit apart. States 1, 4, 16 are drawn, each riding on a line at its own energy — 1E₁, 16E₁, 256E₁ — because the energies go as n². The curves are |ψ|², the probability of finding the particle at each position. The dashed line on each is the classical answer: a ball bouncing between the walls at constant speed is equally likely to be anywhere, and the quantum density oscillates about it and converges onto it as n rises.

Where the quantum picture hands back the old one

A confined particle's probability density oscillates violently at every quantum number, and never stops. What makes the classical answer come back is not that the oscillations die away — it is that nothing can resolve them.

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 swing that dies away and comes back. The envelope of the mean position of a coherent state with 9 quanta on average, in an oscillator whose levels carry a small quadratic term, Eₙ = n + n²/240, over one revival time of 240 oscillator periods. The swing collapses within about 9.1 periods, when the packet has spread round its orbit, and it stays at zero for most of the run. It returns whole at half the revival time, on the opposite side, and whole again at the full revival time; the envelope is summed from the state's energy components and checked against α·exp(−2n̄ sin²χt) to a part in a hundred million. The dashed curve is a classical ensemble started from the same distribution, each member orbiting at the frequency its own energy gives: it collapses in the same way and never returns, its swing at the half and full revival times 0.1% and 0.1% of the start.

The return a classical cloud never makes

Put a swinging quantum packet in a well whose frequency depends a little on the amplitude and it spreads round its orbit until its swing has vanished. That part is not quantum at all: a cloud of classical oscillators does exactly the same. What no classical cloud can do is come back — and the quantum packet reassembles whole, on schedule, splitting into copies on the way, because its energies are discrete.

quantum · Correspondence A quasi-probability that goes negative. The Wigner function of the oscillator's state at a quarter of its revival time, with 4 quanta on average: two copies of the packet, at x = ±2.83, in equal superposition. It is computed from the wavefunction by the Wigner transform on a lattice, and it integrates to one. The two copies are the two positive blobs. Between them lies a pattern of stripes with no classical counterpart, running from 0.289 down to −0.289, where the shaded warm regions and their outlines mark negative values. A negative probability is not a probability, so this distribution cannot describe a cloud of classical particles. The stripes are also taller than the blobs, so the interference carries more structure than the copies themselves: the highest stripe reaches 0.289 and the centre of a blob 0.159.

The probability that goes below zero

Classical mechanics describes an uncertain state as a cloud of points in the plane of position and momentum. Quantum mechanics has an exact counterpart, the Wigner function, whose shadows are the true position and momentum distributions — and which goes negative. It goes negative for a single photon, for every superposition of two packets, and for every pure state that is not a Gaussian. Where it is negative no classical cloud can imitate the state, and losing energy to the surroundings erases the negative regions first.

quantum · Correspondence An atom made of powers of n. Five properties of a hydrogen atom in a level of principal quantum number n, each divided by its value in the ground state, against n from 1 to 300 on logarithmic axes. The radius of a circular orbit rises as n², to 0.53 µm at n = 100; the orbital period as n³, to 1.5 × 10⁻¹⁰ s; the radiative lifetime of the circular state, from the Larmor loss on its orbit, as n⁵ — slope 5.000 — to 0.9 s. The binding energy falls as n⁻², to 1.36 meV, and the electric field that pulls the electron away falls as n⁻⁴, to 3.2 V/cm. Across n = 1 to 300 the lifetime changes by 2 × 10¹² and the ionising field by 8 × 10⁹.

The atom the size of a bacterium

Excite hydrogen's electron to the hundredth level and the atom is a micrometre across, bound by a thousandth of an electronvolt, pulled apart by the field of a torch battery, and — in its most circular state — able to live for nearly a second. Every one of those numbers is a power of n, and together they make the most exaggerated atom there is into one of the most useful.

quantum · Atomic spectra Fringes that identical nuclei add. The cross-section for two identical spinless nuclei repelling each other, divided by the classical answer — the Rutherford counts for the nucleus hit and the nucleus thrown, added — against the scattering angle in the centre-of-mass frame, for Sommerfeld parameters η = 1, 4 and 16 (η is half the closest approach in units of the wavelength over 2π). Because the detector cannot tell which nucleus it caught, the two amplitudes add before squaring, and the cross term oscillates in angle: at 90° the two paths are mirror images and the rate is exactly twice the classical one at every energy. Between 30° and 90° the curve completes about 2 oscillations for η = 4 and 7 for η = 16 — their number grows in proportion to η, which is to say in proportion to 1/ħ.

The scattering quantum mechanics returns unchanged

Rutherford worked out how alpha particles bounce off a nucleus by treating them as tiny planets on hyperbolic orbits, two years before Bohr's atom and fifteen before quantum mechanics. He had no right to be exactly correct, and he was: the full quantum calculation returns his formula without a single correction, at every energy. Coulomb's force is the one force for which that happens. Make the two nuclei identical, though, and quantum mechanics adds fringes no orbit can produce — doubling the rate at right angles for one kind of particle and halving it for the other.

quantum · Correspondence A chaotic rotor heats without limit, and its quantum version stops. The mean kinetic energy of a rotor kicked once a period by a potential of strength K = 5, against the number of kicks, starting at rest. Classically (an ensemble of 4,000 rotors with random angles) the motion is chaotic and the momentum random-walks, so the energy grows in a straight line, 6.3 per kick, near the K²/4 = 6.25 a random walk of kicks K sin θ gives. Quantum mechanically, with an effective Planck constant ƛ = 1 and 0.5, the energy follows the classical line for a while and then stops, fluctuating about 58 and 123: the smaller ƛ lets it follow the classical growth for longer and climb further before it stops. Nothing in the quantum rotor's equations damps it; the growth stops by interference.

The heating a quantum rotor refuses

Kick a freely turning rotor hard, at regular intervals, and classical mechanics says its motion becomes chaotic and its energy grows without limit, the momentum random-walking upward with every kick. Do the same to a quantum rotor and it follows the classical heating for a while, then stops — not because anything damps it, but because the waves of its momentum states interfere so as to cancel further spreading. The correspondence principle, which promises classical behaviour when quantum numbers are large, is broken here in the most direct way: by a quantum system that refuses to get hot.

quantum · Correspondence

Named alongside it

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

Classical limitCoherent stateDecoherenceInterferenceSuperpositionFractional revivalWave packetZero-point energyAnderson localisationAnharmonicityBlackbody radiationBohr model

All concepts