Heisenberg's uncertainty principle states that: if the x-component of the momentum of a particle is measured with an uncertainty

$$\Delta \vec p_x$$

then its x-position cannot, at same time, be measured more accurately than $$\Delta\vec x=\frac {\hbar}{2\Delta\vec p_x},$$

$$\Delta\vec x\Delta\vec p_x \ge \frac {\hbar}{2}.$$

What is the scientific proof of this principle?