This post can also be called
Non-commutativity and Uncertainty Principle.
It is, like Pythagoras Theorem and Time Dilation, a prequel to Quantum Integer Programming (QuIP).
To find an explanation for the double slit experiment that simply baffled physicists, something very different had to be done.
It turned out that rethinking momentum not as a vector of numbers but rather as a derivative operator in complex domain, however arbitrary, seemed to work like a charm.
This brought non-commutativity into physics. See top of the post to see how.
The involvement of i (square root of -1) is crucial.
The basic object now to study is a wave function in a complex vector space (that is conceptual, called Hilbert space), and not a particle (in physical space, like how Newton viewed our physical Universe).
An inevitable consequence of this mathematical formalism – and immediate too as you can see when you scroll right on the top of the post – was worked out by Heisenberg:
Start with Schrodinger reframing of momentum. Apply a triangle inequality.
Easily the most mis-interpreted (and most over-hyped) result in physics, Mike and Ike say it well:
One should be wary of a common misconception about the uncertainty principle, that measuring some observable to some accuracy causes the value of another observable to be disturbed by some amount such that the inequality holds. While it is true that measurements in quantum mechanics cause disturbance to the system being measured, this is most emphatically not the content of the uncertainty principle.
The correct interpretation is that if we prepare a large number of quantum systems in identical states, and then perform measurements on some of the systems of one observable, and of another observable in other systems, then the standard deviation of the first set of measurements multiplied by the standard deviation of the second set of measurements will satisfy the inequality.