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Heisenberg Uncertainty Principle

States that it is impossible to simultaneously know both the position and momentum of a particle with perfect accuracy; a fundamental limit in quantum mechanics applied to optics.
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The statement of the theorem

Let H\mathcal{H} be the Hilbert space of the quantum system, and let ∣ψ⟩∈H|\psi\rangle \in \mathcal{H} be the state vector. Define the position operator x^\hat{x} and the momentum operator p^\hat{p} acting on H\mathcal{H} such that their commutator is [x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar. The uncertainty in an observable A^\hat{A} is defined by σA2=⟨A^2⟩−⟨A^⟩2\sigma_A^2 = \langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2. The Heisenberg Uncertainty Principle states that for any normalized state ∣ψ⟩|\psi\rangle, the product of the variances satisfies: σx2σp2≥14∣⟨[x^,p^]/iℏ⟩∣2 \sigma_x^2 \sigma_p^2 \ge \frac{1}{4} |\langle [\hat{x}, \hat{p}] / i\hbar \rangle|^2 Substituting the canonical commutation relation yields the fundamental bound: σxσp≥ℏ2 \sigma_x \sigma_p \ge \frac{\hbar}{2}