the angular momentum operators
The angular momentum operators are the machinery that makes angular momentum quantized -- and remarkably, you can extract the entire spectrum from their algebra alone, without ever solving a differential equation or knowing whether you are dealing with orbital motion, spin, or a combination. This algebraic approach is one of the most elegant results in quantum mechanics: three commutation relations, and out fall all the allowed values of j and m.
Write the three Cartesian components J_x, J_y, J_z of any angular momentum. They do not commute; instead [J_x, J_y] = i hbar J_z and cyclic permutations, which encodes the geometric fact that rotations about different axes do not commute. From these you show that J^2 = J_x^2 + J_y^2 + J_z^2 commutes with each component, so J^2 and (say) J_z can be simultaneously diagonalized, with eigenvalues J^2 = j(j+1) hbar^2 and J_z = m hbar. Now define the raising and lowering (ladder) operators J_± = J_x ± i J_y; they satisfy [J_z, J_±] = ± hbar J_±, meaning J_+ nudges a state up one rung in m and J_- nudges it down. Because the ladder must terminate (J_z cannot exceed the total), m runs from -j to +j in integer steps, forcing j to be one of 0, 1/2, 1, 3/2, 2, ... -- integer or half-integer, nothing else.
This is the same trick as the harmonic oscillator's ladder operators, and its reach is enormous: it is the physicist's first working example of a Lie algebra (the algebra of su(2)), and the very same commutation relations reappear for isospin, for the quark model's flavour symmetries, and throughout particle physics. The lesson is deep -- much of quantum mechanics is fixed not by the details of a potential but by symmetry and the algebra of the operators that generate it.
Apply the lowering operator to the top state of a spin-1 system, |j=1, m=+1>: J_- takes it to |1, 0>, then to |1, -1>, and applying J_- once more gives exactly zero -- the ladder ends. Three rungs, m = +1, 0, -1, which is 2j+1 = 3 states, all deduced from the algebra without any wavefunction.
The ladder operators generate all 2j+1 states of a multiplet from the commutation relations alone.
The algebra permits both integer and half-integer j, but only integer values can be realized by orbital angular momentum (whose wavefunctions must be single-valued spherical harmonics); half-integer j is the mathematical door through which spin enters. So the commutation relations are necessary but do not by themselves fix which values physically occur.