Hydrogen-like Atom - Mathematical Characterization

Mathematical Characterization

The atomic orbitals of hydrogen-like ions are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's law:

where

• ε₀ is the permittivity of the vacuum,
• Z is the atomic number (number of protons in the nucleus),
• e is the elementary charge (charge of an electron),
• r is the distance of the electron from the nucleus.

After writing the wave function as a product of functions:

(in spherical coordinates), where are spherical harmonics, we arrive at the following Schrödinger equation:

where is, approximately, the mass of the electron. More accurately, it is the reduced mass of the system consisting of the electron and the nucleus.

Different values of l give solutions with different angular momentum, where l (a non-negative integer) is the quantum number of the orbital angular momentum. The magnetic quantum number m (satisfying ) is the (quantized) projection of the orbital angular momentum on the z-axis. See here for the steps leading to the solution of this equation.