Say we have a classical Hamiltonian for a conservative system given by $H=\omega xp$, $\omega$ being a constant, $x$ and $p$ are the position and its conjugate momentum respectively.
What should its quantum mechanical counterpart be, in coordinate representation?
I saw somewhere that first we need to symmetrize the classical operator. So \begin{align} H=\omega xp \rightarrow \frac{\omega}{2}(xp + px) \end{align} And only then we can transform the classical observables into operators. So then in QM, we shall have \begin{align} &\hat{H}=\frac{\omega}{2}(\hat{x}\hat{p}+\hat{p}\hat{x})\\ \Rightarrow &\hat{H}\equiv-i\hbar\omega(x\frac{\partial}{\partial x}+\frac{1}{2}) \end{align} in coordinate representation.
My questions are-
Am I correct?
Why am I correct, i.e, why do we have to symmetrize a product of classical observables before raising them as operators in quantum mechanics?
