"△x△p>h/2" is a simple consequence of the fundamental principle of using wavefunctions ("Amplitudes") to determine the probability of finding a particle.
A plane wave is evenly spread over all space and is the eigenfunction of one precisely known value of p.
In order to get anything other than such complete indeterminacy of position x, one must add several plane waves with different p, forming a wave packet which tails out at the spacial extremes and becomes more and more localised at one value of x, the more different p are added to the superposition.
In the limit you get an infinitely narrow wave packet (Dirac impulse), which is the eigenfunction of a precisely known value of x, which contains all possible p values (p is completely indeterminate).
Reality always lies in between these two extreme situations, and △x△p>h/2 follows from a Fourier analysis of the wave superposition (see e.g. Schiff: Quantum Mechanics).