The second postulate of QMquantum mechanics says that all physical observables are described by their corresponding operators acting on quantum states. I assume this is why we can change from classical x$x$ and p$p$ to quantum operators. But I'm not logically convinced we can simply say that their cross products (for angular momentum) will also produce the quantum operator for angular momentum. Is there a way to prove that cross products (or other operations) of quantum operators will always give a new viable Hermitian operator corresponding to a physical observable? Or is this just part of a postulate?
Edit: Nobody is in disagreement that theoretical physics consists of theories that must be shown to be good models. To give a simple example to help clarify my question. F=ma$F=ma$ is a postulate that is a great model for everyday physics problems that freshman do. But we can use definitions such as F = qE$F = qE$ to find the acceleration of a particle in an electric field by making the substitution qE = ma --> a = qE/m$qE = ma \rightarrow a = qE/m$. If this did not predict the acceleration of a particle in a field, but it did predict the acceleration of a ball rolling down a hill, then we know that there is a hole somewhere that we are missing theoretically.
My original question is: starting with the postulate of classical x and p corresponding to quantum operators $\hat{x}$ and $\hat{p}$ can we show derive that classical L = r x p$\vec{L} = \vec{r} \times\vec{p}$ will also correspond to quantum operator $\hat{L}$ = $\hat{x}$ x $\hat{p}$$\hat{x} \times \hat{p}$?
