User:Jannewmarch/sandbox

Derivation using ladder operators
A common way to derive the quantization rules above is the method of ladder operators. The ladder operators for the total angular momentum $$\mathbf{J} = \left(J_x, J_y, J_z\right)$$ are defined as:
 * $$\begin{align}

J_+ &\equiv J_x + i J_y, \\ J_- &\equiv J_x - i J_y \end{align}$$

Suppose $$|\psi\rangle$$ is a simultaneous eigenstate of $$J^2$$ and $$J_z$$ (i.e., a state with a definite value for $$J^2$$ and a definite value for $$J_z$$). Then using the commutation relations for the components of $$\mathbf{J}$$, one can prove that each of the states $$J_+ |\psi\rangle$$ and $$J_-|\psi\rangle$$ is either zero or a simultaneous eigenstate of $$J^2$$ and $$J_z$$, with the same value as $$|\psi\rangle$$ for $$J^2$$ but with values for $$J_z$$ that are increased or decreased by $$\hbar$$ respectively. The result is zero when the use of a ladder operator would otherwise result in a state with a value for $$J_z$$ that is outside the allowable range. Using the ladder operators in this way, the possible values and quantum numbers for $$J^2$$ and $$J_z$$ can be found.

Since $$\mathbf{S}$$ and $$\mathbf{L}$$ have the same commutation relations as $$\mathbf{J}$$, the same ladder analysis can be applied to them, except that for $$\mathbf{L}$$ there is a further restriction on the quantum numbers that they must be integers.