There are two similar questions as the comment notified:
1. Reason for the discreteness arising in quantum mechanics?
2. How does quantization arise in quantum mechanics?
I could see two points there:
compactness of the space limited by potential.
symmetry requirement such as angular symmetry of atom, Lorentz symmetry, etc.
But I don't see the explanation of why there arise the quantized quantities such as mass, charge, and magnetic moment I explained below. (If I missed the answer, guide me please.)
The free electrons are not in the compact space and not in the symmetry requirement except the translation. Lorentz invariance would not be the point as we can talk of a non-relativistic electron. But we can observe the quantized charge and mass.
So I think my question is still not answered.
When I was a physics student, I remember to solve the Schrodinger equation with the well potential. And because of the boundary condition which the equation has to satisfy, only the solutions resonating with the well can survive. And this leads to the quantization of energy level even though Hilbert space is continuous in terms of energy level.
I also remember a similar logic when I was introduced into the quantum field theory. One of my teachers explained the quantization with the resonance. He starts with the finite box and increases the box size infinite to get the quantization of free particle.
However, how about the electric charge? The electric charge of electrons can exist as the multiples of a unit charge only(I remember some particles with different charge units though.). Also, the mass of fundamental particles are fixed and can exist as the multiples of unit mass. And there is spin whose size is fixed as giving rise to the multiples of the unit magnetic moment.
In the view of classical mechanics, those quantities (mass, charge, and magnetic moment) are assumed to be the continuous number.
Do we understand the origin of these quantized quantities? If we do, could you explain the reason for the layman?
