How Are Quantum Numbers Derived from the Schrödinger Equation?

[jalalmalo]

let me see if I got this one right:

By solving the S E for bound system u got the quantum number n l and ml. each set of numbers corresponds to a curtain wave function and electron. One gots fine structure when adding the spin up 1/2 and down 1/2. so every electron in the atom has a set of these quantum numbers and no two electrons can have the same set.

thanx for your patience and replies

 

[turin]

By solving the S E for bound system u got the quantum number n l and ml.

Yes, for V~1/r^2 at least.

each set of numbers corresponds to a curtain wave function ...

Yes, in particular an eigenfunction of the Hamiltonian.

... and electron.

I'm not sure. Actually, I don't think so. I haven't ever treated multi-electron atoms too rigorously, but I think that, strictly speaking, the eigenstates would be product states that combine all electrons as fermions, rather than treating them individually. Since electrons are identical, it doesn't make quantum sense to speak of individual electrons in a multi-electron system.

One gots fine structure when adding the spin up 1/2 and down 1/2.

There is an energy splitting between the spin up and spin down states of a given orbital.

so every electron in the atom has a set of these quantum numbers and no two electrons can have the same set.

Again, I think that the eigenstates of electrons in a multi-electron atom are actually products states, but I'm not sure.