Introduction: Power Series Again
The hydrogen radial equation is solved by power series, similar to the harmonic oscillator. The requirement that the series terminate gives energy quantization.
Dimensionless Form
Define \(\rho = r/a_0\) and \(\epsilon = -E/E_R\). The equation becomes:
\[\frac{d^2u}{d\rho^2} = \left[\frac{l(l+1)}{\rho^2} - \frac{2}{\rho} + \epsilon\right]u\]Asymptotic Behavior
As \(\rho \to \infty\): \(u \sim e^{-\sqrt{\epsilon}\rho}\)
As \(\rho \to 0\): \(u \sim \rho^{l+1}\)
Factor: \(u = \rho^{l+1}e^{-\sqrt{\epsilon}\rho}v(\rho)\)
Quantization Condition
The power series for \(v(\rho)\) must terminate. This requires:
\[\sqrt{\epsilon} = \frac{1}{n} \quad \text{where } n = l + 1, l + 2, \ldots\] \[E_n = -\frac{E_R}{n^2} = -\frac{13.6 \text{ eV}}{n^2}\]The Quantum Connection
The \(1/n^2\) energy dependence matches Bohr's formula but now derived rigorously. The principal quantum number \(n\) determines energy; \(l\) and \(m\) determine angular structure. The accidental degeneracy (\(E\) depends only on \(n\), not \(l\)) is unique to the Coulomb potential.