This is, as mentioned by Piotr Migdal, in the comments, standard textbook material.

In Matrix Mechanics, observables like position and momentum are promoted to operators. Obviously, multiplication of operators need not commute. Specifically, the difference is given by the commutator bracket:

\(\left[ X, P\right] = XP-PX=i\hbar I \)

where *I* is the identity matrix.

Therefore, there have to be uncertainties (standard deviation after taking an infinite number of measurements) in* X* and *P* and it should be intuitive that:

\( {\sigma _x}{\sigma _p} \ge k\hbar \)

For some scalar (constant) value of *k*. So far, everything applies to any pair of such dynamical variables, like position and momentum. It happens to be, however, that specifically for position and momentum, *k* is \(\frac12 \). This can be derived from special cases, c.f. Feynman, Hibbs and Styer, who derive this in the first chapter through special cases.

One may also derive it through the wave interpretation, through Fourier transforms, and stuff; see Wikipedia.