It was almost thirty years ago when I recall leaving the great lecture hall at the University of Waterloo Math Building and telling my co-student,
Getting Started with Quantum
"I can accept that I can never know where the electron is, but I CAN'T accept that the electron itself doesn't know where it is!" That in fact turns out to be the crucial issue of Quantum Physics.
From the time we are infants we learn that things either exist, or don't exist, and everything that does exists, exists in some specific, and in theory, precisely measureable location.
In quantum mechanics, objects, actually particles, either exist or don't exist but their location may be undetermined, even to them. What we can know about these particles is the probability density function which describes our chances of our finding them within a specific range of locations.
The act of making a measurement has the magic effect of 'collapsing' the probability function to a point. So for the instant of making our measurement we actually do know where the electron is, but during the moments following, the probability density function again expands over a larger range.
Now what is really disturbing to the uninitiated is that during the periods when we are 'not looking' the particle actually becomes its probability function, or rather a function usually referred to as , whose magnitude squared is the actual probability density function of the particle in question. is determined by solving Schrodinger's Equation for the particle, subject to the existing boundary conditions.
This fact is demonstrated by Young's Double Slit Experiment where single photons, or single electrons, pass through two slits simultaneously in order to create an interference pattern with themselves on the other side.
The above is the essense of Quantum Mechanics ... the rest ... is merely math.