The uncertainty principle tells us that there are limits to our ability to measure conjugate variables with simultaneous accuracy. Conjugate variables are position and momentum or time and energy. Why should such a limitation exist? Did some decide that this was part of quantum mechanics? The answer is that it is the necessary consequence of the wave equation. The wave equation tells us that we cannot know the precise location of a particle, but rather we obtain a wave function that gives us the probability amplitude of the particle. Thus, the wave function itself contains uncertainty in the form of the statistical interpretation of matter. The conjugate variables position and momentum are implied since the Hamiltonian is proportional to the square of the momentum and the coordinates of the wave function are coordinates in units of position. In the time dependent Schrodinger equation a similar comment applies to time and energy. We can see the necessity of teh uncertainty principle in the zero-point motion of nuclei. Even at absolute zero of temperature the vibrational hamiltonian tells us that nuclei will have a positional uncertainty equal to their root-mean-square zero point motion. There is nothing we can do to change that. And it must be the case since if the nuclei were completely frozen in one location we could know both their position and momentum with arbitrary accuracy.

The uncertainty principle