Similarly, a constant arises in the colatitude equation which gives the orbital quantum number: Finally, constraints on the azimuthal equation give what is called the magnetic quantum number: The solution to the radial equation can exist only when a constant which arises in the solution is restricted to integer values. Quantum Numbers from Hydrogen EquationsThe hydrogen atom solution requires finding solutions to the separated equations which obey the constraints on the wavefunction. The starting point is the form of the Schrodinger equation:
Solving it involves separating the variables into the form The expanded form of the Schrodinger equation is shown below. The potential energy is simply that of a point charge: The electron in the hydrogen atom sees a spherically symmetric potential, so it is logical to use spherical polar coordinates to develop the Schrodinger equation. The separation leads to three equations for the three spatial variables, and their solutions give rise to three quantum numbers associated with the hydrogen energy levels. The solution is managed by separating the variables so that the wavefunction is represented by the product: The solution of the Schrodinger equation for the hydrogen atom is a formidable mathematical problem, but is of such fundamental importance that it will be treated in outline here. Hydrogen Schrodinger Equation The Hydrogen Atom Spherical polar coordinates
