Heisenberg Uncertainty Principle

$({\Delta }_{\psi }X)({\Delta }_{\psi }P)\ge \frac{\hslash }{2}$

The above shows that the product of Measurement uncertainty between the Position Operator and the Momentum Operator is bounded below. The relation applies to any pair of Conjugate variables, and follows from the Fourier Uncertainty Principle, since Conjugate variables are related to each other by being the Fourier Transform of each other, up to a constant defined in terms of Planck’s reduced constant $\hslash$.

• $X$ is the Position Operator
• $P$ is the Momentum Operator
• $\psi \in H$ is a Quantum state
• ${\Delta }_{\psi }$ is the Measurement uncertainty of an Operator for the Quantum state $\psi$.
• $\hslash$ is Planck’s reduced constant