13.3 The commutator algebra and uncertainty

The single fact that separates quantum mechanics from classical is that observables need not commute. Two quantities whose operators fail to commute cannot both have definite values at once — and the precise budget of that mutual indefiniteness, the uncertainty principle, is a theorem about their commutator. This lesson develops the commutator as an algebraic object and extracts the uncertainty relation from it.

The commutator

The commutator of two operators measures their failure to commute:

[\hat A, \hat B] \;\equiv\; \hat A\hat B - \hat B\hat A.

If it is zero the operators commute and can be applied in either order; if not, order matters, and $[\hat A,\hat B]$ is itself an operator quantifying the discrepancy. It obeys a small algebra used constantly:

Antisymmetry: $[\hat A,\hat B] = -[\hat B,\hat A]$ .
Linearity in each slot.
Product (Leibniz) rule: $[\hat A,\hat B\hat C] = [\hat A,\hat B]\hat C + \hat B[\hat A,\hat C]$ .
Jacobi identity: $[\hat A,[\hat B,\hat C]] + [\hat B,[\hat C,\hat A]] + [\hat C,[\hat A,\hat B]] = 0$ .

These are the axioms of a Lie algebra, and they are why commutators, angular momenta, and the generators of symmetries (13.5) all share one algebraic structure.

The canonical commutation relation

Everything specific to quantum mechanics is encoded in one commutator. From 13.1, $\hat x = \hat X$ (multiply by $x$ ) and $\hat p = -i\hbar\hat D$ , and the interactive there established $[\hat X,\hat D] = -\hat 1$ . Therefore:

▶ [x̂, p̂] = iℏ Derivation

Compute directly, using linearity to pull the constant $-i\hbar$ out of the commutator:

[\hat x, \hat p] = [\hat X,\, -i\hbar\hat D] = -i\hbar\,[\hat X,\hat D] = -i\hbar\,(-\hat 1) = i\hbar\,\hat 1.

Or, most transparently, act on a test function $\psi$ :

[\hat x,\hat p]\psi = x(-i\hbar\psi') - (-i\hbar)(x\psi)' = -i\hbar x\psi' + i\hbar(\psi + x\psi') = i\hbar\,\psi.

The $x\psi'$ terms cancel and $[\hat x,\hat p]\psi = i\hbar\psi$ for every $\psi$ , so

\boxed{\;[\hat x,\hat p] = i\hbar.\;}

where

$[\hat A,\hat B] = \hat A\hat B - \hat B\hat A$: commutator — the operator measuring non-commutativity
$[\hat x, \hat p] = i\hbar$: canonical commutation relation — the seed of quantum mechanics
$\hbar$: reduced Planck constant; the scale of quantum effects J·s

This canonical commutation relation is the mathematical content of quantisation. It is why $\hat x$ and $\hat p$ cannot share eigenfunctions — a state of definite position ( $\delta(x-x_0)$ ) is a uniform spread in momentum, and vice versa — and it propagates into every corner of the theory, including the harmonic-oscillator spectrum of the next lesson.

The uncertainty principle

That $\hat x$ and $\hat p$ do not commute is not a vague statement; it sets a hard lower bound on how sharply both can be defined in any state. Let $\Delta A$ denote the standard deviation of $\hat A$ in a state $\psi$ , $\Delta A = \sqrt{\langle \hat A^2\rangle - \langle\hat A\rangle^2}$ . Then:

▶ The general uncertainty relation ΔA ΔB ≥ ½|⟨[Â,B̂]⟩| Derivation

Work with the mean-subtracted operators $\delta\hat A = \hat A - \langle\hat A\rangle$ and $\delta\hat B = \hat B - \langle\hat B\rangle$ , both self-adjoint, with $\langle\delta\hat A^2\rangle = (\Delta A)^2$ . Apply the Cauchy–Schwarz inequality (inner products) to the vectors $\delta\hat A|\psi\rangle$ and $\delta\hat B|\psi\rangle$ :

\langle\delta\hat A^2\rangle\,\langle\delta\hat B^2\rangle \;\ge\; \big|\langle\delta\hat A\,\delta\hat B\rangle\big|^2.

Split the product into its Hermitian and anti-Hermitian parts, $\delta\hat A\,\delta\hat B = \tfrac12\{\delta\hat A,\delta\hat B\} + \tfrac12[\delta\hat A,\delta\hat B]$ (anticommutator plus commutator). The anticommutator part has real expectation, the commutator part imaginary; the imaginary part alone gives

\big|\langle\delta\hat A\,\delta\hat B\rangle\big|^2 \;\ge\; \left|\tfrac12\langle[\delta\hat A,\delta\hat B]\rangle\right|^2 = \tfrac14\big|\langle[\hat A,\hat B]\rangle\big|^2,

using $[\delta\hat A,\delta\hat B] = [\hat A,\hat B]$ (subtracting constants does not change a commutator). Taking square roots,

\Delta A\,\Delta B \;\ge\; \tfrac12\,\big|\langle[\hat A,\hat B]\rangle\big|.

For position and momentum, $[\hat x,\hat p] = i\hbar$ makes the right-hand side a constant, and out drops the Heisenberg uncertainty principle:

\boxed{\;\Delta x\,\Delta p \;\ge\; \frac{\hbar}{2}.\;}

It is a theorem about the state itself, not a statement about clumsy measurement. No wavefunction can be sharply peaked in both position and momentum, because the two are Fourier conjugates and their operators do not commute.

Δx = 1.00, Δp = 0.50, Δx·Δp = 0.50 = ℏ/2 (minimum)

position width σ = 1.00

The wavepacket's position and momentum spreads are locked together: squeezing it in x forces it to spread in p. Their product cannot fall below ℏ/2 — a direct consequence of the commutator [x̂, p̂] = iℏ. The Gaussian is the shape that reaches the floor.

The interactive is a Gaussian wavepacket — the state that saturates the bound. Narrow it in position and its momentum distribution visibly broadens, and vice versa; the product $\Delta x\,\Delta p$ stays pinned at exactly $\hbar/2$ . Every other state sits strictly above the floor. The same relation, applied to time and energy or to two components of angular momentum, follows from their commutators by the identical theorem.

⏳ The history — Heisenberg's arrays and Born's recognition

In the summer of 1925, on the island of Helgoland to escape hayfever, Werner Heisenberg built a new mechanics from observable quantities alone — arrays of transition amplitudes between states — and found that multiplying them, the product depended on the order. Back in Göttingen, Max Born recognised Heisenberg’s strange multiplication rule as ordinary matrix multiplication, and with Pascual Jordan wrote the crucial relation $\hat p\hat q - \hat q\hat p = \frac{\hbar}{i}\hat 1$ — the canonical commutation relation, appearing for the first time. Heisenberg extracted its physical meaning in 1927: the sharper a particle’s position is known, the less its momentum can be, $\Delta x\,\Delta p \gtrsim \hbar$ . That the whole of quantum mechanics could grow from the single statement that two operators fail to commute was, and remains, one of the most startling turns in the history of physics.

The commutator that makes position and momentum incompatible is also the engine that solves the harmonic oscillator exactly — by pure algebra, with no differential equation. That is the next lesson.