Planck Length

The length at which measurements of positions of particles smaller than this will create a blackhole

$l_P=\sqrt{\frac{\hslash G}{c^3}}\cong 1.616\cdot 10^{-35}\mathrm{m}$

Black Hole Derivation

$$\begin{array}{rl}\Delta x\Delta p& \sim \hslash \\ \Delta p& \sim \frac{\hslash }{L}\\ E& =pc\\ & =\frac{\hslash c}{L}\\ R_s& =\frac{2Gm}{c^2},E=m{c}^2\to m=\frac{E}{c^2}=\frac{\hslash }{Lc}\\ & =\frac{2G\hslash }{c^{3}L},\text{ let }{R}_s=L\\ L^2& =\frac{2G\hslash }{c^3}\\ l_P& \sim \sqrt{\frac{\hslash G}{c^3}}\end{array}$$

• Start with Heisenberg’s Uncertainty Principle