Question

What is Heisenberg uncertainty principle, and how does it work for position and momentum measurements?

Answer 1

Heisenberg uncertainty principle states that for two non-commuting operators, A and B,

$$\sigma_A \sigma_B \ge \frac{1}{2}\left\lt[A,B]\right\gt$$

where

$$\sigma_A = \sqrt{\left\lt A^2\right\gt-\left\lt A\right\gt ^2} -standard\space deviation$$

For momentum and position:

$$\sigma_p \sigma_x \ge \frac{1}{2}[p,x]=\frac{\hbar}{2}$$

that means that we cannot measure momentum and position with an infinity precision simultaneously.