Last night in physics, we were diving further and further into Quantum Mechanics, and my professor gave us a simple derivation of Heisenberg's Uncertainty Principle. However, there is something I don't get. Maybe you guys can help me out. I get all the stuff at the beginning, but its pretty much the last bit where I'm lost...

$\displaystyle \sin(\vartheta)=\frac{\lambda}{W}$

$\displaystyle \tan(\vartheta)=\frac{\Delta p_y}{p_x}$

Due to Taylor Approximations, for small values of $\displaystyle \vartheta , \ \sin(\vartheta)\approx\tan(\vartheta)\approx\varth eta$

Thus, $\displaystyle \frac{\lambda}{W}=\frac{\Delta p_y}{p_x}$

From deBroiglie's Equation, we discover that $\displaystyle p_x=\frac{h}{\lambda}$.

Therefore, $\displaystyle \frac{\lambda}{W}=\frac{\Delta p_y}{\frac{h}{\lambda}}\implies \frac{\lambda}{W}=\frac{\Delta p_y \lambda}{h}$.

Using the idea that $\displaystyle \sin(\vartheta)\approx\tan(\vartheta)\approx\varth eta$, We can see that $\displaystyle \frac{\lambda}{W}=\frac{\lambda}{\Delta y}\implies W=\Delta y$.

Thus, $\displaystyle \frac{1}{\Delta y}=\frac{\Delta p_y}{h}\implies \Delta p\Delta y=h{\color{red}\geqslant\frac{\hbar}{2}}$

I understand numerically that $\displaystyle h\geqslant\frac{\hbar}{2}\implies 1\geqslant\frac{1}{4\pi}$, but why is it that we say in the Uncertainty Principle that $\displaystyle h\geqslant\frac{\hbar}{2}$???

I'd appreciate any input.

--Chris