Now comes to the most difficult portion for me. It took me three times watching the video to feel more comfortable of the theory of inferring the state of momentum of a quantum particle.
In the previous session, Prof. Shankar demonstrated quantum particle’s peculiarity can only be described in a sai/wave function, note not a sai function commonly used in Newtonian world, it is a e raised by complex number combination.
As is in Classic world, to describe an object, you need to locate position x and momentum p, in the same vein, the next, natural question for a quantum particle is to calculate the momentum.
Deducing from the position formula, one can try to depict a particle at that instant with the momentum of, say 5,
Hence, the integral has to be L, enclosed universe, and if so, sai(x, L) = sai(x), furthermore, the conclusion is planck constant/discrete momentum phenomena observed:
Then, with the aid of Fourier transformation and concept of Kronecker delta , Prof.Shankar gave the equation to compute state of momentum, that in the following forms, m = n.
so if you were given a sai function as N*cos() as below, applying above equations is just procedual:
For this particular sai equation, one can eyeball it and circumvent the complex integral calculation by using Euler’s equations
To make it valid, m has to be either positive 3 or negative 3. So Sai_P(3) = 50%, Sai_P(-3) = 50%.
In essence, when you measure a particular position x at an instant, the momentum p has multiple dimensions until a set wave sai is given and using the above math tool, you can compute the actual state of P, or the integer m.