The Mystery of s,p,d and f Revealed

We rewrote the Hamiltonian for a one-electron atom in terms of the operator L². Knowing that linear operators that commute share at least one set of eigenfunctions, we tested to see if the Hamiltonian and L² did commute. They do, and so there must exist a common set of eigenfunctions. Since we already know one set of eigenfunctions for angular momentum operator, the the spherical harmonics or Y_l,m, we tried a solution to the one-electron atom Schrodinger equation of the form R(r)Y_l,m(θ,φ). Such solutions do work and allow us to derive a differential equation in a single variable, r, to solve for the radial part of the wavefunction.

We talked about the origins of the familiar orbital designations, s, p, d nd f.

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See rotatable images of s, p, d, f and g orbitals.