Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Heisenberg Uncertainty Principle

A fundamental limitation on the precision with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known.
📜

The statement of the theorem

For any pair of non-commuting Hermitian operators A^\hat{A} and B^\hat{B} representing physical observables, the generalized uncertainty principle states: σAσB≥12∣⟨[A^,B^]⟩∣\sigma_{A} \sigma_{B} \ge \frac{1}{2} | \langle [\hat{A}, \hat{B}] \rangle | Specifically, for position x^\hat{x} and momentum p^\hat{p}: σxσp≥ℏ2\sigma_{x} \sigma_{p} \ge \frac{\hbar}{2} where σA=⟨A^2⟩−⟨A^⟩2\sigma_{A} = \sqrt{\langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2} is the standard deviation.