Introduction: The Radial Wavefunctions
The terminated power series solutions for hydrogen are associated Laguerre polynomials. Combined with exponential and power factors, they give the complete radial wavefunctions.
The Radial Wavefunctions
\[R_{nl}(r) = \sqrt{\left(\frac{2}{na_0}\right)^3 \frac{(n-l-1)!}{2n[(n+l)!]^3}} e^{-r/na_0} \left(\frac{2r}{na_0}\right)^l L_{n-l-1}^{2l+1}\left(\frac{2r}{na_0}\right)\]First Few Radial Functions
- \(R_{10} = 2\left(\frac{1}{a_0}\right)^{3/2}e^{-r/a_0}\) (1s)
- \(R_{20} = \frac{1}{2\sqrt{2}}\left(\frac{1}{a_0}\right)^{3/2}\left(2 - \frac{r}{a_0}\right)e^{-r/2a_0}\) (2s)
- \(R_{21} = \frac{1}{2\sqrt{6}}\left(\frac{1}{a_0}\right)^{3/2}\frac{r}{a_0}e^{-r/2a_0}\) (2p)
Energy Spectrum
\[E_n = -\frac{13.6 \text{ eV}}{n^2} \quad n = 1, 2, 3, \ldots\]Degeneracy at each \(n\): \(\sum_{l=0}^{n-1}(2l+1) = n^2\) (without spin)
The Quantum Connection
The hydrogen wavefunctions are chemistry's foundation. The 1s orbital is compact, 2s larger with a node, 2p dumbbell-shaped. Understanding these shapes explains covalent bonding, molecular geometry, and the entire periodic table.