Why Angular momentum is important?

Why (linear) momentum is important? Angular momentum is just the rotational analog of that. If you move something in a straight line, you get momentum. If you rotate it, you get angular momentum. If you want to describe an object, you should include both.

In quantum mechanics, angular momentum is particularly important, because the same mathematical formalism that describes "real" angular momentum (a particle spinning around an axis) also turns out to describe some other internal degree of freedom (which we, confusingly, call spin) which is also extremely important.

 

Although, care needs be taken over that characterisation. Spin certain does contribute to the angular momentum of the system --- see the Einstein-de Hass effect. Furthermore, we know that neither spatial angular momentum (L) or spin (S) alone is a symmetry of the universe, but that the combined J is.

 

Another reason is that the angular momentum of charged particles relates to the concept of magnetic moment from electrodynamics, and so is responsible for creating magnetic fields within the atom (which is how you explain paramagnetism, for example) and governs how the atom interacts with external magnetic fields.

 

The answer is a group theoretical one. A system generally has one or more wavefunctions with the same energy, and so a general wavefunction with that energy is a linear combination of those wavefunctions. Different wavefunctions with the same energy are thus related to eachother by matrix multiplication, and these matrices form groups (in the group theory sense).

For consistency, these groups are the symmetry groups of the system (e.g., the rotation group), and one of the most important features steps in the development of quantum mechanics was the (re)discovery of groups. If you knew the symmetries of a system, you could already tell a lot about it, like selection rules, degeneracy, etc., without really "solving" anything. Different systems with the same symmetries look very much the same.

For systems with rotational symmetry, states come in representations of the rotation group SO(3), and these representations are labeled by quantum numbers which also happen to correspond to angular momentum eigenvalues. But they are very general. Try to solve for the electromagnetic field of some localized current distribution (a problem that has nothing to do with quantum mechanics) and the same "quantum numbers" will appear in the spherical harmonic functions.

 

Adding: Energy levels of certain systems, look for instance on the Nuclear Shell model.