The correspondence principle

The correspondence principle states that, in order to obtain quantum mechanical operators, we should replace the position vectors \(\require{physics} \qty{ \vb{r}_i }\) and the momentum vectors \(\qty{ \vb{p}_i }\) inside the classical operators with their quantum mechanical operator analogues. In coordinate representation (which we will denote by using the superscript \(c\)), this amounts to the following: \[\begin{align} \vb{r} &\rightarrow \vb{r}^c \\ \vb{p} &\rightarrow \vb{p}^c \thinspace , \end{align}\] in which the momentum operator in coordinate representation is \[\begin{equation} \vb{p}^c = -i \grad \thinspace . \end{equation}\]

These substitutions are consistent with the canonical commutator \[\begin{equation} \comm{ \vb{r}^c }{ \vb{p}^c } = i \thinspace . \end{equation}\]

The angular momentum operator is then given by \[\begin{equation} \vb{L}^c = \vb{r}^c \cross \vb{p}^c % \thinspace . \end{equation}\]