### Heisenberg uncertainty principle

If the x-coordinate of the position of a particle is known to an accuracy of $\delta x$, then the x-component of momentum cannot be determined to an accuracy better than $\Delta P_{x}\approx \frac{\hbar }{\Delta x}$.

$\Delta P_{x}. \Delta x\approx \hbar$

The above inequality must be satisfied

 $\Delta P_{x}. \Delta x\geqslant \hbar$

Where $\hbar$ - Planck’s Constant

This is the Uncertainty principle with macroscopic objects.

Exact statement of the Uncertainty principle →

The product of the uncertainties in determining the position and momentum of the particle can never be smaller than the number of the order $\frac{\hbar }{2}$.

 $\Delta P_{x}. \Delta x\geqslant \frac{\hbar}{2}$

Where $\delta x$  and $\delta P$ are defined as the root mean square deviation from their mean values.

The Uncertainty principle can also describe by the following formula →

 $\Delta x.\Delta p_{x}\approx \frac{\hbar}{2}$

 $\Delta x.\Delta p_{x}\geqslant \frac{\hbar}{2}$

 $\Delta x.\Delta p_{x}\geqslant \frac{h}{4\pi }$

Expression for $y$ and $z$ component →

 $\Delta y.\Delta p_{y}\geqslant \frac{h}{4\pi }$

 $\Delta z.\Delta p_{z}\geqslant \frac{h}{4\pi }$

The uncertainty relation between energy and time →

 $\Delta E.\Delta t\geqslant \frac{h}{4\pi }$

 $\Delta E.\Delta t\geqslant \frac{\hbar }{2 }$

The uncertainty relation between momentum and Angular Position→

 $\Delta L.\Delta \theta \geqslant \frac{h }{4\pi }$

 $\Delta L.\Delta \theta \geqslant \frac{\hbar}{2 }$