Concept

Commutator — where it appears

The difference AB − BA between two operators taken in the two orders, which vanishes exactly when the two observables can have definite values at once. Its size sets the lower bound in the uncertainty relation for that pair.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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

Which pairs have anything between them. The commutator of every pair of 4 observables of a spin-½, multiplied out in 2×2 complex arithmetic and shown by the size of AB − BA. The diagonal is exactly zero: an observable commutes with itself, which is why measuring the same thing twice gives the same answer. S², the total angular momentum, is a multiple of the identity here and so commutes with everything — its row and its column are zero, and a spin can have a definite total angular momentum and a definite component at the same time. The largest entry is 0.7071 in units of ħ². Sz: 0.000 with Sz, 0.707 with Sx, 0.707 with Sy, 0.000 with S²; Sx: 0.707 with Sz, 0.000 with Sx, 0.707 with Sy, 0.000 with S²; Sy: 0.707 with Sz, 0.707 with Sx, 0.000 with Sy, 0.000 with S²; S²: 0.000 with Sz, 0.000 with Sx, 0.000 with Sy, 0.000 with S². A zero cell is a promise that the two quantities can be sharp together; a non-zero one is an obstruction whose size sets how badly they cannot.

The questions that can be asked together

Two quantities can have definite values at once exactly when their operators commute. That is a piece of arithmetic about matrices, and everything the uncertainty principle forbids follows from it — including the fact that most of the time it forbids nothing at all.

quantum · Uncertainty

A sharp angle costs a spread of turning. Left: the probability of finding a rotor at each angle, for three states. Right: the probability of each value of its angular momentum, in units of ħ, for the same states. A state bunched to a circular spread of 0.025 has an angular-momentum spread of 2.21ħ; a state bunched to a circular spread of 0.302 has an angular-momentum spread of 0.59ħ; a state with no preferred angle has an angular-momentum spread of 0.00ħ. The angular momentum can only take whole-number values, so its probabilities are bars; the angle is continuous. The state with no preferred angle has exactly one value of angular momentum, and its angle is spread evenly round the circle — where the naive rule would say that its angle's spread times ħ/2 should be infinite, it is π/√3.

The angle the uncertainty principle cannot be written for

Position and momentum trade exactly: squeeze one and the other spreads, with a product never below ħ/2. Angle and angular momentum look like the same pair and are not. A rotor in a state of definite angular momentum has zero spread in angular momentum and a finite spread in angle, π/√3, and their product is zero. Nothing is wrong with the rotor. The relation was written with a commutator that does not hold, because an angle lives on a circle and a line has no ends to wrap round. The relation that does hold measures the angle's spread by how far its average position on the circle falls short of the rim.

quantum · Uncertainty

What an amplifier does to a signal's uncertainty. A signal drawn in the plane of its two quadratures — the in-phase and out-of-phase parts of a wave — as a disc one standard deviation across: a coherent input (black), with the vacuum's own uncertainty; and three outputs at a gain of 4: what a noiseless amplifier would give, the input disc scaled up by √G (dashed), which quantum mechanics forbids; what the least noisy phase-insensitive amplifier gives (blue), the same centre with a disc 1.323 times wider again; and what a phase-sensitive amplifier gives (red), the in-phase quadrature stretched by √G and the other squeezed by 1/√G, with no noise added. Referred back to the input, the phase-insensitive amplifier has added 0.375 of a quantum of noise; the signal's distance from the origin, measured in its own spread, has fallen by 0.756.

The noise every amplifier must add

An amplifier that magnifies a signal without caring about its phase must also add noise, and not because of hot transistors or bad design. However perfect it is, it adds at least half a quantum of noise referred to its input — exactly as much as the vacuum fluctuations the signal already carried — so a faint coherent signal always leaves with half its signal-to-noise ratio. The reason is the same rule that makes position and momentum uncertain, applied to the two halves of a wave, and an amplifier that broke it would be a machine for copying quantum states.

quantum · Uncertainty

Named alongside it

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

Uncertainty principle Coherent state Amplifier Angular momentum Anharmonicity Circular statistics Classical limit Correspondence Decoherence Degeneracy Ehrenfest theorem Eigenvalue

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