Quantum orbital angular momentum

Angular momentum is an important physical quantity because it is a conserved quantity. For a point mass, it is classically defined as the cross product between its position vector with its linear momentum. First, we derive the basic commutation relations between the different angular momentum components, and that of the total angular momentum. We then discuss compatible versus incompatible observables. Second, we introduce the ladder operators and their algebra with the angular momentum operators, and their utility in raising and lowering eigenvalues and obtaining eigenstates for the next rung. Third, we show how these angular momentum algebra, or their commutator relations, allows us to derive the angular momentum eigenvalues in a very elegant way. Lastly, we obtain the angular momentum eigenfunctions in spherical coordinates. Before we begin, let’s first develop some intuitions from simple arguments.