Question on quantum numbers

What quantum numbers are you talking about?

If you are talking about spin quantum numbers then any decent book on QM will prove from the angular momentum commutation relations why its quantized.

If you are talking about solutions to Schroedinger's equation then that is not always quantized, and when it is it depends entirely on the Hamiltonian:
http://www.physics.ox.ac.uk/Users/smithb/website/coursenotes/qi/QILectureNotes3.pdf

Thanks
Bill

 

The Schrodinger equation: [itex]H \psi = E \psi[/itex] can be solved for arbitrary values of [itex]E[/itex], but when [itex]E[/itex] is not an eigenvalue of the hamiltonian, then [itex]\psi[/itex] will be unnormalizable--it will blow up at infinity, or at the origin, or somewhere. For example, a solution to the free particle Schrodinger equation with [itex]E < 0[/itex] is: [itex]\psi = e^{\hbar K x}[/itex], which corresponds to an energy of [itex]-\hbar^2 K^2/(2 m)[/itex].

The differential equations for the wave function are forced to have discrete values by boundary conditions: to make sure that the wave function is single-valued in the whole domain, and to make sure that the integral of [itex]|\psi|^2[/itex] is finite.