Commutators and uncertainty

ÂB̂ means act with B̂ first, then Â. The commutator [Â, B̂] = ÂB̂ − B̂Â records how much the answer changes when you swap that order. Because these are operators and not numbers, you evaluate one by feeding it an arbitrary differentiable f(x) and simplifying until f can be cancelled at the end. With p̂ = −iħ d/dx: x̂p̂f = x(−iħf′) = −iħxf′, while p̂x̂f = −iħ d(xf)/dx = −iħ(f + xf′). Subtract, the −iħxf′ terms cancel, and iħf survives — so [x̂, p̂] = iħ Î. Drop the f and you will be tempted to cancel the whole thing to zero, which is the single most common slip in this topic. From there the algebra runs on identities rather than integrals: [Â, B̂] = −[B̂, Â] fixes the sign, and [Â, B̂Ĉ] = B̂[Â, Ĉ] + [Â, B̂]Ĉ gives [x̂, p̂²] = 2iħp̂ in one line.

Put f = (Â − ⟨A⟩)ψ and g = (B̂ − ⟨B⟩)ψ. Hermiticity lets each operator move across the inner product, so ⟨f|f⟩ = σ_A² and ⟨g|g⟩ = σ_B². Cauchy–Schwarz then gives σ_A²σ_B² ≥ |⟨f|g⟩|². Write z = ⟨f|g⟩. Every complex number obeys |z|² = (Re z)² + (Im z)² ≥ (Im z)², and Im z = (z − z*)/2i works out to ⟨[Â, B̂]⟩/2i. Take the square root and σ_A σ_B ≥ ½|⟨[Â, B̂]⟩|. Notice what was thrown away in that one inequality: Re z, which is the covariance ½⟨ÂB̂ + B̂Â⟩ − ⟨A⟩⟨B⟩. Keeping it gives the stronger Robertson–Schrödinger form, and explains why a free packet that has been evolving for a while sits above ħ/2 rather than on it — its position and momentum have become correlated, and the discarded term is exactly that correlation.

Nothing in that derivation mentions an instrument. σ_A and σ_B are numbers computed from one state ψ — take ⟨²⟩ − ⟨Â⟩² by integration, and the same for B̂ — so the inequality is a property of the preparation and of nothing else. The experiment that tests it needs many identically prepared copies: measure position on half of them and build a histogram whose standard deviation is σₓ, measure momentum on the other half for σₚ, then multiply the two. Each individual measurement may be as sharp as the apparatus allows. Localise one electron to 10⁻¹³ m if your detector can do it, and nothing is violated, because that number is a resolution and not the spread of a state. The relation forbids exactly one thing: preparing a state whose two spreads multiply out below ħ/2.

Suppose [Â, B̂] = 0 and Âψ = aψ. Then Â(B̂ψ) = B̂(Âψ) = a(B̂ψ), so B̂ψ is still an eigenfunction of  with the same eigenvalue a. If that eigenvalue is non-degenerate, B̂ψ can only be a multiple of ψ, so ψ is automatically an eigenfunction of B̂ as well. If it is degenerate, B̂ maps the eigenspace into itself and you diagonalise B̂ inside that subspace to pick out a shared basis: the basis exists, but it has to be chosen rather than inherited. Hydrogen is the pay-off. Ĥ, L̂² and L̂_z commute pairwise, so a single state can be sharp in all three and carries the labels n, ℓ and m. L̂ₓ does not commute with L̂_z, which is precisely why no state is ever labelled by both at once.

Robertson's right-hand side depends on the state, so it can collapse even where the operators flatly refuse to commute. Take [L̂ₓ, L̂_y] = iħL̂_z in the state |ℓ = 1, m = 0⟩. There ⟨L̂_z⟩ = 0, so the bound reads σ_Lx σ_Ly ≥ 0 and says nothing whatever. The true spreads are far from zero: ⟨L̂ₓ²⟩ + ⟨L̂_y²⟩ = ⟨L̂²⟩ − ⟨L̂_z²⟩ = 2ħ² − 0, and symmetry splits that evenly, so σ_Lx = σ_Ly = ħ and the product is ħ² = 1.11 × 10⁻⁶⁸ J² s². The inequality is necessary, never sufficient. Compatibility is settled by the operator identity [Â, B̂] = 0 holding across the whole domain, not by the size of a bound in one chosen state. The x–p case is unusual precisely because iħ is state-independent, so its floor can never go soft.

ΔE Δt ≥ ħ/2 looks like the same theorem, but it cannot be, because there is no self-adjoint time operator to put inside the commutator. Pauli's argument: if T̂ satisfied [T̂, Ĥ] = iħ, then exp(−iεT̂/ħ) applied to an energy eigenstate would shift its eigenvalue by any ε you like, forcing the spectrum of Ĥ to fill the whole real line — impossible for a Hamiltonian bounded below. In quantum mechanics t is a parameter labelling the evolution, not an observable of the system. Two honest statements survive. A state that decays with lifetime τ has an energy width Γ ≈ ħ/τ: the 2p level of hydrogen, τ = 1.6 ns, gives Γ = 6.6 × 10⁻²⁶ J = 0.41 μeV. And the Mandelstam–Tamm bound, with Δt = σ_A/|d⟨Â⟩/dt|, says no expectation value can shift appreciably in a time shorter than ħ/2σ_E.