How the Hamiltonian of a classical system expressed in quantum mechanics? I was dealing with a problem, which said that,

Supposedly Hamiltonian of a conservative system in classical mechanics is $\omega xp$, where $\omega$ is a constant, and $x$ and  $p$ are the position and momentum operator respectively. The corresponding Hamiltonian in quantum mechanics is________.

To solve the problem, an initial step was taken
$$\omega xp= \frac{\omega}{2}(xp+px).$$
stating that the quantum operator should be Hermitian in nature.
What I don't understand is, how the above proceeded step was an important part/property of Hermitian operator, i.e., why the above step was taken?
 A: The hamiltonian of a system has to be an hermitian operator since it's associated to a measurable quantity, the energy. By symply taking the classical hamiltonian$$H=\omega xp$$ and converting it directly to an operatore $$\hat H = \omega\hat x\hat p$$ you can easily see that this operator is not hermitian since, given $\hat x$ and $\hat p$ hermitian, $$\hat H^\dagger = \omega\hat{p}^\dagger\hat{x}^\dagger = \omega \hat p\hat x \neq H$$ since the two operators are now in reverse order. Then you have to make $\hat H$ hermitian.
Classically speaking $$H = \omega xp = \frac{\omega}{2}(xp+px)$$ won't change the hamiltonian in any way since $x$ and $p$ being purely variables, and not operators, they clearly commute. This form of the classical hamiltonian is now more suited for converting it into a quantum hamiltonian by using operators instead of simple functions $$H = \frac{\omega}{2}(\hat x\hat p + \hat p \hat x)$$ and now you can easily convince yourself that this form of the hamiltonian is clearly hermitian.
Such a procedure to take a classical hamiltonian to a suitable quantum hamiltonian is called Weyl ordering.
A: This is an interesting question because in general there is no unique answer, although there is a "natural" answer for the simplest observables, which include $xp$.  The lack of uniqueness is linked to the ordering problem.
The simplest quantization is due to Dirac; he tried to implement the idea that the commutator of two operators representing observables should be $i\hbar$ times the operator corresponding to the Poisson bracket of the two respective classical quantities.
This is based on the observations that
\begin{align}
[\hat x,\hat p]=i\hbar \widehat{\{x,p\}}
\end{align}
where $\{\cdot,\cdot\}$ is the classical Poisson bracket. Dirac basically proposed to generalize this to
\begin{align}
[\hat f,\hat g]=i\hbar\widehat{\{f,g\}} \, . \tag{1}
\end{align}
subject to the additional assumptions that
\begin{align}
\widehat{f(x)}=f(\hat x)\, ,\qquad \widehat{g(p)}=g(\hat p)
\end{align}
The problem is that (1) eventually fails although it does work for $xp$.  To get the result you want you need to ask what is $\widehat{xp}$.  To get there start with
\begin{align}
\frac{1}{i\hbar}[x^2,p^2]&=2\left(\hat x\hat p+\hat p\hat x\right)
=\widehat{\{x^2,p^2\}}= 4 \widehat{xp}
\end{align}
from which you get
\begin{align}
\widehat{xp}=\frac{1}{2}\left(\hat x\hat p+\hat p\hat x\right)
\end{align}
which is the usual symmetrization rule.
Gronewold in

Groenewold, H.J., 1946. On the principles of elementary quantum mechanics, Physica 12 (1946) 405-460,

showed that (1) eventually fails, i.e. there is no rule consistent with (1) that will allow you to find, for instance, the operator corresponding to $x^2p^2$.  This is excellently reviewed in

Chernoff, P.R., 1981. Mathematical obstructions to quantization. Hadronic Journal, 4(3), pp.879-898.

The remarkable thing is that of course the rule (1) works for the ``most common'' observables like angular momentum.    However, it is possible to devise classical Hamiltonian for which the quantization is not unique, as per the example of this post.
