Uncertainty principle
The math above looks very hard, but the math leading up to it is very much harder and is extremely hard to understand. It is given here just to show what it looked like. Heisenberg's paper is a historical landmark. Many of the physicists who read his paper said that they could not disagree with his conclusions, but that they could not follow his explanation of how he got to those conclusions. The beginning equations that Heisenberg used involved Fourier series, and involved many factors. We will come back to the equation above because it is a kind of recipe for writing out and multiplying matrices.