In the same year that Erwin Schrödinger published the equation that now bears his name, the nonrelativistic theory was completed by Max Born's insight that the Schrödinger wave function is actually nothing but a tool for calculating probabilities, and that the probability of detecting a particle "described by" in a region of space is given by the volume integral
— provided that the appropriate measurement is made, in this case a test for the particle's presence in . Since the probability of finding the particle somewhere (no matter where) has to be 1, only a square integrable function can "describe" a particle. This rules out which is not square integrable. In other words, no particle can have a momentum so sharp as to be given by times a wave vector, rather than by a genuine probability distribution over different momenta.
Given a probability density function , we can define the expected value
To check that the two expressions are in fact equal, we plug into the latter expression:
Next we replace by and shuffle the integrals with the mathematical nonchalance that is common in physics:
The expression in square brackets is a representation of Dirac's delta distribution the defining characteristic of which is for any continuous function (In case you didn't notice, this proves what was to be proved.)
Heisenberg spoke of Unschärfe, the literal translation of which is "fuzziness" rather than "uncertainty". Since the relation is a consequence of the fact that and are related to each other via a Fourier transformation, we leave the proof to the mathematicians. The fuzziness relation for position and momentum follows via . It says that the fuzziness of a position (as measured by ) and the fuzziness of the corresponding momentum (as measured by ) must be such that their product equals at least