Quantum numbers, orbitals and energies
But something new occurs here: l is also involved in the radial wave equation (2.), which will place a new constraint upon the permissible values of l, and hence ml. We won't go through the tedious steps of solving the radial equation; it will suffice to look at the solutions, which can be found for energies
| En = | Z2e4μ | n = 1, 2, 3, ... |
| 8e02h2n2 |
(20.7)
In this equation Z is the charge on the nucleus, which is 1 for the hydrogen atom, 2 for the He+ ion, etc. The corresponding radial wavefunctions depend on both n and l, and are of the form
| -ρ | ||
| Rn, l (r) = ρlLn, l (ρ) e | 2 | n = 1, 2, 3, ..., l = 0, 1, 2, ..., n-1 |
where ρ is proportional to r
| ρ = | 2μ | r |
| nmea0 |
and the constant a0 is known as the Bohr radius:
| a0 = | 4πε0ħ2 | = | ε0h2 | = 0.529177 Å |
| mee2 | πmee2 |
