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Uncertainty Principle

The uncertainty principle, expressed as \Delta x \Delta p ≥ ħ/2, states that there is a fundamental limit to the precision with which certain pairs of physical properties can be known simultaneously.
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The statement of the theorem

Let H\mathcal{H} be a separable Hilbert space, and let A^\hat{A} and B^\hat{B} be two self-adjoint operators acting on H\mathcal{H}. Define the expectation values ΔA=⟨ψ∣(A^−⟨A^⟩I^)2∣ψ⟩\Delta A = \sqrt{\langle \psi | (\hat{A} - \langle \hat{A} \rangle \hat{I})^2 | \psi \rangle} and ΔB=⟨ψ∣(B^−⟨B^⟩I^)2∣ψ⟩\Delta B = \sqrt{\langle \psi | (\hat{B} - \langle \hat{B} \rangle \hat{I})^2 | \psi \rangle} for a state ∣ψ⟩∈H|\psi\rangle \in \mathcal{H}. If A^=X^\hat{A} = \hat{X} (position operator) and B^=P^\hat{B} = \hat{P} (momentum operator), then the generalized uncertainty principle dictates the lower bound on the product of the standard deviations: \ΔXΔP≥12∣⟨[X^,P^]⟩∣\Delta X \Delta P \geq \frac{1}{2} | \langle [\hat{X}, \hat{P}] \rangle |. Given the canonical commutation relation [X^,P^]=iℏI^[\hat{X}, \hat{P}] = i\hbar \hat{I}, the minimum uncertainty product is established as: \ΔXΔP≥ℏ2\Delta X \Delta P \geq \frac{\hbar}{2}.
Source: Wikipedia