## § 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 $\kappa + \lambda$.