# What is the only quantum number that does not derive from the Schrodinger equation?

##### 1 Answer

That would be the spin quantum number,

Here's where the rest of them came from. The **wave function**

#psi_(nlm_l)(r,theta,phi) = R_(nl)(r)Y_(l)^(m_l)(theta,phi)#

#= R_(nl)(r)Theta(theta)Phi(phi)#

- The
**magnetic quantum number**#m_l# came from solving the#phi# part of the angular portion, i.e. for#Phi(phi)# , and using a cyclic boundary condition. - The
**azimuthal quantum number**#l# came from solving the#theta# part of the angular portion, i.e. for#Theta(theta)# , using the associated Legendre polynomials, which are functions of#costheta# . - The
**principal quantum number**#n# came from solving the radial portion, i.e. for#R_(nl)(r)# , but it is one of the more difficult and esoteric parts of solving the Schrodinger equation for the hydrogen atom, and few textbooks show it in full.

It basically comes from a convergence condition when solving for a coefficient in a power series, though I've never had to know that myself.

I did show a good portion of solving the Schrodinger equation for the hydrogen atom here, if you want more context:

https://socratic.org/questions/solve-schr-dinger-equation-for-hydrogen-atom

Or, a more condensed version is found here:

http://users.aber.ac.uk/ruw/teach/237/hatom.php