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It's pretty darn difficult to describe quantum field theory without technical details but here goes nothing. What we want is some field, $\phi$, which satisfies the equations associated with a scalar (spin 0) fields such as the Higgs. One such equation is the Klein-Gordon equation, which is related to the relationship between energy, momentum and mass found ...
The answer is Yes. Define function $g(q):= \frac{1}{f(q)}$ for later convenience. Then the classical Hamiltonian reads $$2h~=~g(q)p^2.$$ One may show that the Weyl-ordered Hamiltonian reads $$2H_W~=~ (g(q)p^2)_W ~=~ \frac{1}{4}P^2 g(Q)+\frac{1}{2} Pg(Q)P+\frac{1}{4} g(Q)P^2$$ $$~=~ Pg(Q)P - \frac{1}{4}\hbar^2g^{\prime\prime}(Q),$$ see e.g. Ref. 1 and this ...