Hidden potentials in classical theorems

Some exact equations are derived which clarify some potentials that are hidden in the classical theorems such as the Hellmann–Feynman (H–F) and the integral Hellmann–Feynman (I-H–F) theorems. The differential form of the density equation given previously includes not only the classical force operator but also the force operator associated with the quantum-mechanical potential introduced by Bohm. The latter arises essentially only from the noncommuting property of coordinates and momenta in quantum mechanics. However, after integration only the classical force term survives and results in the H–F theorem. The important role of the quantum force term is completely hidden in the H–F theorem. This fact would be closely related to the nondeterminicity and the classical interpretation of the H–F theorem. A similar role of the quantum potential is also shown for the I-H–F theorem. We have also investigated the origin of force density (the integrand of the H–F theorem) and isolated the roles of the generalized exchange and correlation effects which are also hidden in the H–F theorem.