# Heisenberg uncertainty principle

The Heisenberg uncertainty principle states that it is theoretically impossible to know both values of certain pairs of observable variables precisely.

The two most important pairs of variables affected by the principle are:

- Momentum and position
- Energy and time

This principle arises from the **probabilistic nature** of particles at a quantum level (due to wave-particle duality). It is counterintuitive as the position of objects in classical mechanics can be exactly determined if we ignore instrument uncertainty.

Heisenberg's uncertainty principle implies that any measurement taken is subject to an intrinsic, irreducible degree of **uncertainty** (irrespective of instrument uncertainty).

This is significant for microscopic measurements (where the measured values and corresponding uncertainties are small).

The **uncertainty principle** specifically gives the minimum uncertainty of **position-momentum** and **energy-time** measurements.

The **position-momentum uncertainty** is given by: $$$\Delta x\Delta p\geq\frac{\hbar}{2}$$$ $$\Delta x$$ is the uncertainty in position, $$\Delta p$$ is the uncertainty in momentum and $$\hbar=h/2\pi$$ (pronounced "h bar") is the **reduced Planck constant**.

The **energy-time uncertainty** is given by: $$$\Delta E\Delta t\geq\frac{\hbar}{2}$$$ $$\Delta E$$ is the uncertainty in energy, $$\Delta t$$ is the uncertainty in time.