Hydrogen-like orbitals are the foundation of atomic and molecular theory
Determining the properties of atoms starts with the exact solutions to the hydrogen atom. Shapes of the s, p, d and f orbitals are the same for all atoms. There are differences in atomic radius that arise from the particular electron configuration. THe orbital approximation, which is also known as the Hartree approximation is the hypothesis that the multi-electron wave function can by factored into one electron orbitals. We have no hope of solving the mult-electron wave function. However, once it is composed of a product of one-electon orbitals we can proceed to solve the mult-electron problem approximately. We assume that the energy of each electron can be solved in the field of all of the other electrons (self-consistent field).
The role of angular moment operators is to provide a means to determine which aspects of agnular momentum in atoms can be measured. The commutation operation tells us that two observables can be measured simultaneously if the result of [a,b] = 0 for two operators a and b. For the angular momentum we learn that Lz and L2 commute and therefore we can know both the total angular momentum and the z-component of the angular momentum simultaneously.
