Classical Energy vs Quantum Energy

[eep]

Hi,
If we find an expression for the total energy of a system in terms of classical mechanics, can we replace the observables with their quantum-mechanical operators and state that this new equation acting on the wave function should give you the energy eigenvalues? My gut reaction is to say no, because there must be some quantum mechanical effects which just simply can't be accounted for in classical mechanics, however I noticed when working on solving a rigid rotor that it is indeed the case. Moreover, isn't the Hamiltonian in QM derived by following this prescription?

 

[zhangpujumbo]

Hi,
If we find an expression for the total energy of a system in terms of classical mechanics, can we replace the observables with their quantum-mechanical operators and state that this new equation acting on the wave function should give you the energy eigenvalues? My gut reaction is to say no, because there must be some quantum mechanical effects which just simply can't be accounted for in classical mechanics, however I noticed when working on solving a rigid rotor that it is indeed the case. Moreover, isn't the Hamiltonian in QM derived by following this prescription?

I've seen some arguments about your question in the Dirac's classic QM textbook.

Any more opinions?

 

[masudr]

If there is a dynamical variable on classical phase space $\omega(x, p)$ then we can make the replacement

$$x\rightarrow X, p \rightarrow P,$$

where $X, P$ are the position and momentum operators, to get the quantum operator

$$\Omega(X, P).$$

There may be ordering ambiguities etc. which can be resolved by comparison with experiment.