Quantum observables that do not commute cannot generally have arbitrarily sharp values in the same state. For Hermitian operators $A$ and $B$, define standard deviations $$\Delta A=\sqrt{\langle A^2\rangle-\langle A\rangle^2},\qquad \Delta B=\sqrt{\langle B^2\rangle-\langle B\rangle^2}.$$ They obey the Robertson uncertainty relation $$\Delta A,\Delta B\ge\frac12\left|\langle[A,B]\rangle\right|,$$ where $$[A,B]=AB-BA$$ is the commutator.

Position and momentum satisfy $$[x,p]=i\hbar,$$ so $$\Delta x,\Delta p\ge\frac{\hbar}{2}.$$ This is not primarily a statement about experimental clumsiness. It says that no quantum state can make both position and momentum distributions arbitrarily narrow.

A Gaussian wavepacket can attain the minimum product $\Delta x\Delta p=\hbar/2$. Making the packet more localized in position requires a broader range of spatial frequencies and therefore a broader momentum distribution.

The relation does not say that measuring position literally 'creates' a fixed amount of momentum error, nor that every pair of observables is uncertain. Compatible observables with vanishing commutator can share simultaneous eigenstates. The deeper principle is algebraic: the noncommutative structure of quantum observables limits how sharply certain properties can coexist in a state.