Energy–time uncertainty and the classical limit

Robertson's bound σ_A σ_B ≥ ½|⟨[A, B]⟩| needs two Hermitian operators acting on the state. Energy supplies one: H. Time supplies nothing. Pauli's argument shows the gap is not an oversight. Suppose a self-adjoint T existed with [T, H] = iħ. Then the unitary family U(ε) = e^(−iεT/ħ) obeys U(ε) H U(ε)† = H + ε for every real ε, so if E lies in the spectrum of H then so does E + ε for all real ε — the spectrum is the entire real line. Every Hamiltonian in this course is bounded below: hydrogen bottoms out at −13.6 eV, the oscillator at ½ħω. So T does not exist, and ΔE Δt ≥ ħ/2 cannot be read off the commutator machinery. What travels under that name is a family of theorems sharing a shape, not one theorem. Any symbol you write as Δt must be defined before the inequality means anything at all.

Take an observable A with no explicit time dependence. Robertson still applies to the pair (H, A): σ_E σ_A ≥ ½|⟨[H, A]⟩|. Ehrenfest supplies d⟨A⟩/dt = (i/ħ)⟨[H, A]⟩, so |⟨[H, A]⟩| = ħ|d⟨A⟩/dt|. Substitute and the bound becomes σ_E σ_A ≥ (ħ/2)|d⟨A⟩/dt|, that is σ_E τ_A ≥ ħ/2 with τ_A = σ_A / |d⟨A⟩/dt|. Read τ_A as the time ⟨A⟩ needs to move by one standard deviation of A — the time after which the change is statistically visible above the state's own spread. Notice what has not appeared: no clock, no apparatus, no duration of an interaction. A stationary state has d⟨A⟩/dt = 0 for every A, so τ_A is infinite and the bound holds trivially, which is exactly right, since σ_E = 0 there and nothing about the state ever changes. Different observables give different τ_A for the same state, and the relation holds separately for each.

An excited level coupled to a continuum has no sharp energy, because it is not an eigenstate of the full atom-plus-field Hamiltonian. Model its amplitude as ψ(t) = e^{(−iE₀t/ħ)} e^(−t/2τ) for t > 0, so that |ψ|² = e^(−t/τ) reproduces the observed exponential decay. Its Fourier transform gives an energy distribution |ψ̃(E)|² ∝ 1/[(E − E₀)² + (ħ/2τ)²] — a Lorentzian of full width at half maximum Γ = ħ/τ. Note the equals sign: the profile is fixed and Γτ = ħ exactly, with Δν = 1/(2πτ) in frequency. Nothing here is saturated and nothing is claimed about a clock; the width is a property the level carries before it decays and after it has gone. The relation also runs backwards, which is how it earns its keep at the short end: the Z boson's measured width Γ = 2.50 GeV implies τ = ħ/Γ = 2.6×10⁻²⁵ s, a lifetime no apparatus could ever time directly, read instead off the shape of a resonance curve. One warning about Γ. It is built from the total rate at which the level empties, so every open channel adds — collisions, quenching and photoionisation broaden a line exactly as spontaneous emission does, and a measured width is only an upper bound on the radiative part.

Push Mandelstam–Tamm to its extreme case by letting A be the projector onto the initial state. The bound becomes a floor on t⊥, the time for a state to become orthogonal to where it started: t⊥ ≥ πħ/(2σ_E). Margolus and Levitin found a second, independent floor with σ_E replaced by ⟨E⟩ − E₀, and whichever is larger is the real limit. An equal superposition of two levels split by ΔE saturates the first exactly: σ_E = ΔE/2, the overlap is |cos(ΔE t/2ħ)|, and its first zero sits at t⊥ = πħ/ΔE. This is the sense in which energy spread is a resource. A state with no energy spread is a stationary state and does nothing at all; the more spread you are willing to carry, the faster the state can travel to a distinguishable one. The same inequality bounds gate times in a quantum computer — and being a bound, it says nothing about whether any particular Hamiltonian gets anywhere near it.

Ehrenfest's theorem gives d⟨x⟩/dt = ⟨p⟩/m and d⟨p⟩/dt = −⟨V′(x)⟩. That is Newton's second law for the means only when ⟨V′(x)⟩ = V′(⟨x⟩), which is exact for a free particle, a uniform field and a harmonic oscillator, and otherwise carries a leading correction −½σₓ²V‴(⟨x⟩) that stays negligible while the packet is narrow on the scale over which the force curves. The other half of the check is counting quanta. A 10 g bob swinging at 1.0 Hz with 5.0 cm amplitude holds E = ½mω²A² = 4.9×10⁻⁴ J, which is n = E/ħω = 7.4×10²⁹ quanta at a level spacing of ħω = 6.6×10⁻³⁴ J, or 4.1×10⁻¹⁵ eV — a discreteness no instrument will resolve. Both statements are correspondence checks: the quantum description reproduces the classical one where the classical one already worked. Neither derives classical mechanics, and neither explains why the bob has a definite position in the first place.

Suppose the bob were prepared in a superposition of two positions 1 mm apart. Ehrenfest returns the mean, which sits between them, where the bob is not; the interference terms remain in the state, and no smallness of ħ removes them. What removes them is the environment. Every air molecule and thermal photon that scatters off the bob carries away a record of which position it took, and once that record exists elsewhere the two branches can no longer interfere locally: the off-diagonal elements of the object's reduced density matrix decay, and they decay faster for a wider separation and for a shorter scattering wavelength. When that wavelength is short compared with the separation, one scattering event already carries a complete which-path record, so the coherence time is simply the mean time between collisions — and in air at room pressure that is shorter than any dynamical time the object has. Lower the pressure and the same superposition survives proportionally longer, which is why matter interferometry lives in ultra-high vacuum. Two cautions. This is ordinary unitary evolution of object plus environment, so it accounts for why interference becomes unobservable and which basis survives, not for why one outcome rather than another is the one you see; that last question stays open and interpretation-dependent. And ħ never moved.