Commutators and the Generalized Uncertainty Principle

The canonical commutator $[\overset{x}{^}, \overset{p}{^}] = i ℏ$ says position and momentum have no common eigenbasis. This lesson makes that qualitative statement quantitative: for any two observables the product of their spreads in a state is bounded below by the expectation of their commutator. The bound is exact, saturated by identifiable states, and it applies to every pair of incompatible observables, not only position and momentum. The derivation is a direct application of the Cauchy–Schwarz inequality from the first lesson.

The commutator as obstruction

Two observables can be simultaneously sharp only if the state is a common eigenvector, and commuting operators share an eigenbasis. When $[\hat{A}, \hat{B}] \neq = 0$ no state is a joint eigenstate, so at least one of the two spreads is nonzero in every state. The commutator quantifies that residual disturbance. Two algebraic facts about it are used repeatedly.

The commutator of Hermitian operators is anti-Hermitian. If $\hat{A}^{†} = \hat{A}$ and $\hat{B}^{†} = \hat{B}$ , then $[\hat{A}, \hat{B}]^{†} = (\hat{A} \hat{B} - \hat{B} \hat{A})^{†} = \hat{B} \hat{A} - \hat{A} \hat{B} = - [\hat{A}, \hat{B}]$ . Its expectation value is therefore purely imaginary.
The anticommutator is Hermitian. The symmetric product ${\hat{A}, \hat{B}} = \hat{A} \hat{B} + \hat{B} \hat{A}$ satisfies ${\hat{A}, \hat{B}}^{†} = {\hat{A}, \hat{B}}$ , so its expectation value is real. Together the two decompose the product $\hat{A} \hat{B} = \frac{1}{2} {\hat{A}, \hat{B}} + \frac{1}{2} [\hat{A}, \hat{B}]$ into Hermitian and anti-Hermitian parts, real and imaginary expectations respectively.

Because incompatible observables share no eigenbasis, the order of two measurements matters: preparing a sharp value of one and then measuring the other leaves the first no longer sharp, the physical face of a nonzero commutator.

The generalized uncertainty relation

Fix a normalized state $∣ ψ ⟩$ and define the mean-subtracted operators, whose spreads are the variances,

Δ \hat{A} = \hat{A} - ⟨ \hat{A} ⟩, σ_{A}^{2} = ⟨ ψ ∣ (Δ \hat{A})^{2} ∣ ψ ⟩ = ∥ Δ \hat{A} ∣ ψ ⟩ ∥^{2} .

Both $Δ \hat{A}$ and $Δ \hat{B}$ are Hermitian, and they share the commutator of the originals, $[Δ \hat{A}, Δ \hat{B}] = [\hat{A}, \hat{B}]$ , since subtracting constants does not change a commutator.

The bound keeps only the imaginary part of $⟨ f ∣ g ⟩$ ; discarding the real part, which is $\frac{1}{2} ⟨ {Δ \hat{A}, Δ \hat{B}} ⟩$ , is why generic states do not saturate the relation. The stronger Robertson–Schrödinger inequality retains that term,

σ_{A}^{2} σ_{B}^{2} \geq (\frac{1}{2} ⟨ {Δ \hat{A}, Δ \hat{B}} ⟩)^{2} + (\frac{1}{2 i} ⟨ [\hat{A}, \hat{B}] ⟩)^{2},

and reduces to the Heisenberg form when the symmetric correlation vanishes.

Schwarz-inequality geometry: the two mean-subtracted vectors have a bracket whose magnitude is bounded by the product of their lengths, and the imaginary part of that bracket carries the commutator, giving the uncertainty bound.

Position and momentum

The archetype recovers Heisenberg's original relation. With $\hat{A} = \overset{x}{^}$ , $\hat{B} = \overset{p}{^}$ , and $[\overset{x}{^}, \overset{p}{^}] = i ℏ$ , the commutator expectation is the constant $⟨ i ℏ ⟩ = i ℏ$ , so

σ_{x} σ_{p} \geq \frac{1}{2} ∣ i ℏ ∣ = \frac{ℏ}{2} .

Position and momentum cannot both have vanishing spread in any state: driving $σ_{x} \to 0$ forces $σ_{p} \to \infty$ . The bound is a property of the state space, not of the apparatus — no cleverness in measurement evades it, because it constrains the spreads of outcomes over an ensemble of identically prepared systems, before any single measurement is made. The physical readings of the relation collected in the matter-waves treatment — zero-point energy, the size of the hydrogen atom, natural line widths — are all consequences of this one inequality applied to a confined particle.

The bound is not always a fixed constant. When the commutator is itself an operator, the right side depends on the state. Angular momentum is the standard case: with $[\hat{J}_{x}, \hat{J}_{y}] = i ℏ \hat{J}_{z}$ , the theorem gives

σ_{J_{x}} σ_{J_{y}} \geq \frac{ℏ}{2} ⟨ \hat{J}_{z} ⟩ .

A state with $⟨ \hat{J}_{z} ⟩ = 0$ places no lower bound on the transverse spreads, which is why an eigenstate of $\hat{J}_{z}$ with eigenvalue zero can have arbitrarily small $σ_{J_{x}}$ and $σ_{J_{y}}$ simultaneously, whereas a state aligned along $z$ ( $⟨ \hat{J}_{z} ⟩ = j ℏ$ ) forces a large transverse spread — the algebraic origin of the vector-model cone on which angular momentum cannot point exactly along an axis.

Minimum-uncertainty states

The states that saturate $σ_{x} σ_{p} = ℏ/2$ can be found from the two equality conditions inside the proof. Saturation requires both the Cauchy–Schwarz inequality and the discard of the real part to be equalities.

Schwarz equality demands $∣ g ⟩ = c ∣ f ⟩$ for some complex $c$ , i.e. $Δ \overset{p}{^} ∣ ψ ⟩ = c Δ \overset{x}{^} ∣ ψ ⟩$ .
Vanishing real part demands $⟨ {Δ \overset{x}{^}, Δ \overset{p}{^}} ⟩ = 0$ , which with the proportionality forces $c$ to be purely imaginary, $c = iλ$ with $λ$ real.

In the position representation, with $⟨ \overset{x}{^} ⟩ = x_{0}$ and $⟨ \overset{p}{^} ⟩ = p_{0}$ , the condition $(\overset{p}{^} - p_{0}) ψ = iλ (\overset{x}{^} - x_{0}) ψ$ becomes a first-order differential equation,

- i ℏ \frac{d ψ}{d x} - p_{0} ψ = iλ (x - x_{0}) ψ,

whose solution is a Gaussian modulated by a plane wave,

ψ (x) = N exp [- \frac{λ}{2ℏ} (x - x_{0})^{2}] exp [\frac{i p _{0} x}{ℏ}], λ > 0.

A state saturates the uncertainty bound if and only if it is a Gaussian wave packet. Their width is set by $λ$ : $σ_{x}^{2} = ℏ/2 λ$ and $σ_{p}^{2} = ℏ λ /2$ , so $σ_{x} σ_{p} = ℏ/2$ for every $λ$ . This is why the Gaussian recurs as the most classical state — it is the closest a quantum state comes to a sharp point in phase space — and it is the ground state of the harmonic oscillator and the coherent state for exactly this reason.

A Gaussian packet saturates the uncertainty bound: its position and momentum spreads multiply to exactly

ℏ/2

, and the error box in phase space has the smallest area the theory permits.

The energy–time relation

The relation $Δ E Δ t ≳ ℏ/2$ resembles the position–momentum bound but is not an instance of the generalized theorem, because time is not an operator in quantum mechanics — there is no $\hat{t}$ to commute with $\hat{H}$ . Its correct meaning comes from the rate of change of observables. For any observable $\hat{Q}$ with no explicit time dependence, the Heisenberg equation of motion gives

\frac{d ⟨ Q ^ ⟩}{d t} = \frac{i}{ℏ} ⟨ [\hat{H}, \hat{Q}] ⟩,

derived in the time-evolution lesson. Applying the generalized uncertainty principle to $\hat{H}$ and $\hat{Q}$ ,

σ_{H} σ_{Q} \geq \frac{1}{2} ⟨ [\hat{H}, \hat{Q}] ⟩ = \frac{ℏ}{2} \frac{d ⟨ Q ^ ⟩}{d t} .

Define $Δ t = σ_{Q} / ∣ d ⟨ \hat{Q} ⟩ / d t ∣$ , the time for the expectation of $\hat{Q}$ to shift by one standard deviation — the time scale over which the state changes appreciably as registered by $\hat{Q}$ . Then, writing $Δ E = σ_{H}$ ,

Δ E Δ t \geq \frac{ℏ}{2}

The relation quantifies the natural line width of a decaying state: an excited level with lifetime $τ$ has an energy uncertainty $Δ E \sim ℏ/ τ$ , broadening its emission line by $Δ ν \sim 1/ (2 π τ)$ . A state decaying as $∣ c (t) ∣^{2} = e^{- t / τ}$ has amplitude $c (t) \propto e^{- i E_{0} t /ℏ} e^{- t /2 τ}$ , whose Fourier transform is a Lorentzian in energy,

∣ c (E) ∣^{2} \propto \frac{1}{( E - E _{0} ) ^{2} + ( ℏ/2 τ ) ^{2}},

with full width at half maximum $Γ = ℏ/ τ$ . The finite lifetime and the spectral width are Fourier conjugates, exactly as position width and momentum width are. It also explains why a truly stationary state ( $τ \to \infty$ ) has an exactly defined energy, and why rapid processes require access to a wide band of energies. The interpretation is the recurring pitfall: the relation is about how fast states evolve, not about an uncertainty principle between two simultaneously measured quantities.

A finite state lifetime and its energy width are Fourier conjugates: a long-lived level is sharp in energy, a short-lived one broad, with the Lorentzian width set by

Γ = ℏ/ τ

The commutator has now been developed from an obstruction to a quantitative limit. The final lesson of the module puts the commutator in charge of dynamics: an observable that commutes with the Hamiltonian is conserved, and the algebra of commutators generates time evolution itself.