Deriving Quantum Numbers from the Schrodinger Equation

[Mandelbroth]

Can someone explain to me how one gets the values of n, l, and m_l (principle quantum number, azimuthal quantum number, magnetic quantum number, respectively) from the Schrödinger equation for use in chemistry involving distribution of electrons in a hydrogen atom?

 

[Nugatory]

Can someone explain to me how one gets the values of n, l, and m_l (principle quantum number, azimuthal quantum number, magnetic quantum number, respectively) from the Schrödinger equation for use in chemistry involving distribution of electrons in a hydrogen atom?

Not quickly... The basic idea is easy enough, you just solve the Schrödinger equation for an electron in a fixed potential electrical field assuming that the proton is a fixed point charge; each of these quantum numbers is an eigenvalue of one of the possible eigensolutions.

But the algebraic drudgery involved can (and usually does) fill an entire chapter of a serious undergrad textbook. Very likely someone has a link to a decent online set of lecture notes...

 

[jtbell]

You can find an overview here (follow the links to subsidiary pages also):

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2

This is at the level that you might find in a second-year "introductory modern physics" textbook.

If you want the gory details of solving the radial differential equation (which leads through the associated Laguerre polynomials) and the colatitude differential equation (which leads through the Legendre polynomials), you'll have to find a full-bore QM textbook. In graduate school many years ago, I used Merzbacher's book which did that, leaving much of the algebra to the student, of course.