When writing the quantum Hamiltonian from a classical hamiltonian, we cannot simply replace position and momentum variables with their respective operators. We have to make sure that the resultant Hamiltonian is also hermitian.

In R.Shankar book he mentions that this is ambiguous but usually symmetrization is used. So a product r.p in the Hamiltonian becomes: \begin{equation} r.p \rightarrow \hat{r}\hat{p} + \hat{p}\hat{r} \end{equation}

My question was why can't we use: \begin{equation} r.p \rightarrow i(\hat{r}\hat{p} - \hat{p}\hat{r}) \end{equation} which would also keep the Hamiltonian hermitian. What are the physical consequences of choosing such a method of quantising the Hamiltonian?

When writing the quantum Hamiltonian from a classical hamiltonian, we cannot simply replace position and momentum variables with their respective operators. We have to make sure that the resultant Hamiltonian is also hermitian.

In R.Shankar book he mentions that this is ambiguous but usually symmetrization is used. So a product r.p in the Hamiltonian becomes: \begin{equation} r.p \rightarrow \frac{\hat{r}\hat{p} + \hat{p}\hat{r}}{2}. \end{equation}

My question was why can't we use: \begin{equation} r.p \rightarrow i(\hat{r}\hat{p} - \hat{p}\hat{r}) \end{equation} which would also keep the Hamiltonian hermitian. What are the physical consequences of choosing such a method of quantising the Hamiltonian?