A Universe of Sorts

§ Why commutator is important for QM

Suppose we have an operator $L$ with eigenvector $x$ , eigenvalue $\lambda$ . So $Lx = \lambda x$ .
Now suppose we have another operator $N$ such that $[L, N] = \kappa N$ for some constant $\kappa$ .
Compute $[L, N]x = \kappa Nx$ , which implies:

\begin{aligned} &[L, N]x = \kappa Nx \\ &(LN - NL)x = \kappa Nx \\ &L(Nx) - N(Lx) = \kappa Nx \\ &L(Nx) - N(\lambda x) = \kappa Nx \\ &L(Nx) - \lambda N(x) = \kappa Nx \\ &L(Nx) = \kappa Nx + \lambda Nx \\ &L(Nx) = (\kappa + \lambda)Nx \\ \end{aligned}

So $Nx$ is an eigenvector of $L$ with eigenvalue $\kappa + \lambda$ .
This is how we get "ladder operators" which raise and lower the state. If we have a state $x$ with some eigenvalue $\lambda$ , the operator like $N$ gives us an "excited state" from $x$ which eigenvalue $\kappa + \lambda$ .