Following Professor’s Shankar’s Physics II was quite difficult, however, since I foresee quantum computing is going to be trending and hence need to pick up quantum computing skills, it’s inescapable a task to return to Prof.Shankar’s virtual class.
There are three pillars leading to the quantum physics early 20th century. The very first is proposed by Planck purely based on experiment. He described his theory an act of desperation since it’s so anti-intuition to fill below equation with his Planck constant, equal to approximately 6.626×10-34 joule-seconds, so the behavior – red line – fits to what experiments tell the physicists.
In the act of locating an electron using microscope, using wave theory, horizontal momentum and vertical momentum, uncertainty principle. observation causes collapse. In another langue, Prof. Shankar convey the point of uncertain principle, performing an experiment, to ensure no vertical photons, however, due to wave property of photons, p.sintheta = lymbda, it fans out, there is a y momentum generated.
(note often they use denotation planck constant bar = planck constant/2*pie for the sake of brevity)
Prof. Shankar also mentioned the “Compton Scattering” effect where high energy photon collides with an electron,
With blizzard observation of photon, De Brogli proposed a bold hypothesis that electron also display the same particle/wave properties. He conducted similar double slit experiments with astonishingly same result.
This is the famous Heisenberg uncertainty principle.
No matter how weird it is, it’s been observed in experiments over and over again.
Given a particle, we are often asked to spot the future of x and p in Newtonian mechanics, one can just apply Newton’s law, however, in quantum world, wave function sai Y(x), the square of which is the odds of finding it at that location. 3Blue1Brown has a good video explaining it.
Now how to mathematically describe wave? Physicists tried the traditional sin or cos function, it violates the uncertainly principle, so they postulate the below equation after mathematicians’ exploration of imaginary/complex numbers.
Note that the square of sai/wave function is the probability. As to why so, there is a quora thread.