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I don't understand why quantum numbers can not be divided into half integers and so on. The books I have read do not give clear explanations. Would anyone mind helping me understand this? Thanks!
That was really helpful! Thanks a lot! :)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].I don't understand why quantum numbers can not be divided into half integers and so on. The books I have read do not give clear explanations. Would anyone mind helping me understand this? Thanks!