Using the correspondence principle, how does one show that in the classical limit, the expectation value of $H$ is the classical energy?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on:

So far, $$H$$ is only an abstract Hermitian operator in the equation $$H\Psi = i\hbar\dfrac{\partial\Psi}{\partial t}$$. However using the correspondence principle it is possible to show that, in the classical limit, the expectation value of $$H$$ is indeed the classical energy. The correspondence principle does not completely fix the form of the quantum Hamiltonian due to the uncertainty principle and therefore the precise form of the quantum Hamiltonian must be fixed empirically.

How can the correspondence principle be used to show that, in the classical limit, the expectation value of $$H$$ is indeed the classical energy in the derivation used in that link?

• Maybe this will help hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html – anna v Mar 4 '18 at 8:32
• @anna v But there isn't any proof there. – PhyEnthusiast Mar 4 '18 at 8:34
• At some level the axioms, postulates of physics models are arbitrary, in the sense they are chosen to see if they fit the data, and if successful carry on in the theory. As in mathematics, axioms cannot be proven. They are either considered succesful choices or are removed. The link gives logical analogies for postulating an operator instead of a variable, but it is a logical leap that was successful, imo. – anna v Mar 4 '18 at 12:07
• @anna v Yeah of course you are right. That is why stopped chasing for derivations of Schrodinger's equation long ago. But still when I saw this derivation, I thought if using correspondence principle, part of the form of the quantum hermitian Hamiltonian, I thought it was awesome. – PhyEnthusiast Mar 4 '18 at 12:22
• It will be mathematics. You are aware that theorems are proven starting from axioms, but also that a theorem can become an axiom to be used to prove the former assumes as axiom as a theorem. Similarly using the correspondence principle is shifting the assumptions of the postulates to another mathematical form, but in the end it is just circular. – anna v Mar 4 '18 at 14:32