The uncertainty principle, properly
18 min read · Tier toggle changes depth (T1 · T2 · T3)
You should be comfortable with Hermitian operators, expectation values, and the commutator [A,B] = AB − BA — see PU-202 / L24.
Why this matters
Uncertainty is not about clumsy measuring apparatus jostling the particle. It is a statement about what a quantum state is.
The popular slogan — "you can't know position and momentum at the same time" — smuggles in a wrong picture: that the particle has a definite position and momentum which the act of looking spoils. The honest statement is quieter and stranger. A quantum state simply does not, in general, assign sharp values to two quantities whose operators fail to commute. The spread you measure over many identically prepared systems is built into the state before anyone measures anything. This lecture composes the general result from parts already established elsewhere; we teach and interpret it here rather than re-derive it.
What the principle really says
For any two observables A and B and any state, the product of their standard deviations is bounded below by the size of their commutator in that state. When the operators commute the bound is zero and both can be sharp together; when they do not, sharpening one distribution necessarily widens the other. Position and momentum are just the most famous instance, because [x̂, p̂] = iħ is a constant — the bound never vanishes, for any state whatsoever.
The derivation
The generalised uncertainty relation
From the Cauchy–Schwarz inequality applied to the state vectors Â|ψ⟩ and B̂|ψ⟩, for any two Hermitian operators:
Reading the result
Substitute A = x̂ and B = p̂. Since [x̂, p̂] = iħ, the expectation ⟨[x̂,p̂]⟩ = iħ regardless of the state, so |⟨[x̂,p̂]⟩| = ħ and the general relation collapses to the familiar σx σp ≥ ħ/2. The content is entirely in the commutator: the number ħ/2 is not a measurement error budget, it is the algebraic shadow of non-commutativity. Equality holds only for Gaussian states — the minimum-uncertainty wavepackets.
The bound is state-dependent whenever the commutator is not a c-number. For A = L̂x, B = L̂y the commutator is iħL̂z, so the right-hand side is ½ħ|⟨L̂z⟩| — it can genuinely vanish in states with ⟨L̂z⟩ = 0, meaning Lx and Ly can both be arbitrarily sharp there. Treating "uncertainty relations" as universal constant bounds is the error; only the canonical pair inherits a state-independent floor. C
Three quick tests
Why does σxσp ≥ ħ/2 hold for every state but σLxσLy does not? What makes a state saturate the bound? What would [x̂,p̂] = 0 imply physically?
Problem set · PU-202
Derive the number–phase and energy–time relations, and identify which are genuine operator inequalities and which are not.
Derivations cited: D-009 (the generalised uncertainty relation) · Threads: chance