Hellmann–Feynman theorem and internal pressure for atoms, molecules and plasmas under pressure

Abstract

Abstract Since the beginning of the twentieth century, confinement of atoms, molecules and plasma inside impenetrable sharp (hard) and smooth (soft) cavities has been studied with utmost interest. Physically, such situations introduce the trapping of a quantum system under high pressure. The internal pressure ( P n , ℓ ) of the system under multi-megabar external pressure has not yet been explored. Also, there is a lack of Hellmann–Feynman theorem (HFT) in such a scenario, which is essential to derive a concrete analytical expression of P n , ℓ under stressed condition. Here using a scaling concept we provide a general HFT in terms of expectation values of total energy, potential and kinetic energy of a given confined system. A change in boundary condition (from smooth to sharp and vice versa) does not affect this. Applying this proposed HFT, a closed-form expression of P n , ℓ and has been obtained for confined and shell-confined quantum systems. Based on this P n , ℓ , several virial-like equations are also modeled. This HFT and P n , ℓ are demonstrated for one- and many-electron atoms, plasmas and molecules using both analytical and numerical methods. Their applicability in several other confined systems is discussed from simple arguments.