I'd like to understand the relationship between quantum mechanics and the shape of the orbitals, (s, p, d and f) in terms of the letters m, l and n.

But I can give a qualitative treatment.

The orbitals s, p, d and f are calculated from the Schrödinger Equation for the hydrogen atom, which is the only atom for which we can solve the S.E. analytically. One of the assumptions of qualitative and semi-quantitative molecular orbital theory is that we can use "hydrogen-like wave functions"--those self-same orbitals--to describe the way electrons behave in atoms.

The consequence is that we think of atoms as having not only shells, described by the principle quantum number (n), but also subshells, described by the azimuthal quantum number (l). Within each subshell are one, three, five,... orbitals, described by the magnetic quantum number (m). The subshell is assigned a letter (s, p, d or f) based partly on tradition. The number of orbitals in each subshell is defin ed by the allowed values of m.

The orbitals within each subshell must have no net overlap with each other, but must combine to make a sphere. This means that, for l=0 (the s-subshell) the single orbital must be spherical, and it also dictates (for example) the shape of the three p-orbitals (l=1). An example of what I mean by "no net overlap" is shown at right; the two p-orbitals each have two lobes, one lobe with a positive sign and one with a negative sign. If the overlap of positive with negative equals that of positive with positive or negative with negative, there is no net overlap.

Of course, with the third p-orbital (perpendicular to the screen) the p-subshell of any atomic shell combines to make a sphere.

The combination of "no net overlap" (a mathematical property called orthogonality) with "forming a sphere" dictates the shapes of the atomic orbitals. And if you work through the mathematics, you'll find that my qualitative description fits.