Schrödinger's Equation and the Hydrogen Atom — Question 1

Question 1

The hydrogen quantum-number restrictions determine spatial degeneracy

Treat hydrogen with a fixed nucleus, a nonrelativistic Coulomb potential and no spin, fine structure or external field. Write the Bohr length as a. Allowed orbital quantum numbers satisfy $n=1,2,\ldots$, $\ell=0,\ldots,n-1$ and $m_\ell=-\ell,\ldots,\ell$.

1. List the allowed $\ell$ values for $n=4$ and count the spatial states.

2. Assess the proposed triples $(3,3,0)$ and $(3,2,3)$, and derive the general spatial count at fixed n.

Question 1 - Solution

The hydrogen quantum-number restrictions determine spatial degeneracy

(a) Method and calculation

$$n=4:\quad\boxed{\ell=0,1,2,3},\qquad g_4=1+3+5+7=\boxed{16}.$$$$\ell=0,1,2,3\Rightarrow 2\ell+1=1,3,5,7\quad\text{allowed }m_\ell\text{ values respectively}.$$

(b) Interpretation and further calculation

$$(3,3,0):\ \ell\not\le n-1;\qquad(3,2,3):\ |m_\ell|>\ell\Rightarrow\boxed{\text{both invalid}}.$$$$\boxed{g_n=\sum_{\ell=0}^{n-1}(2\ell+1)=n^2}.$$

Physical check

This counts spatial Coulomb eigenstates only. Spin and the ways corrections split the ideal degeneracy belong to the following chapter.