## § 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 $\kappa + \lambda$.