My professor, introducing Heisenberg uncertainty principle, started from the Fourier transform and the classical uncertainty for waves.
He told about the localized impulsive wave $\delta(x)$ which has defined position but total uncertainty of impulse (its Fourier transform is composed of every possible momentum). On the other hand, a wave of defined impulse is a monochromatic wave, which spreads over the entire position axis and doesn't have a proper localization.
I'm perfectly comfortable those considerations, but then, out of noting, he writes
$$\Delta x \: \Delta k \geq 1/2$$
From this it's easy to derive the Heisenberg principle, but I can't understand where the previous formula comes from.
Does it come from Fourier transform properties, from the properties of optical waves, or from something else?