The HVT Technique and the "Uncertainty" Relation for Central - Grypeos, M.; Koutroulos, C. G.; Oyewumi, K. J.; Petridou, Th. (HNPS Advances in Nuclear Physics, 2021)
The quantum mechanical hypervirial theorems (HVT) technique is used to treat the so-called "uncertainty" relation for quite a wide class of central potential wells, including the (reduced) Poeschl-Teller and the Gaussian one. It is shown that this technique is quite suitable in deriving an approximate analytic expression in the form of a truncated power series expansion for the dimensionless product $P_{nl}\equiv <r^2>_{nl}<p^2>_{nl}/\hbar^2$, for every (deeply) bound state of a particle moving non-relativistically in the well, provided that a (dimensionless) parameter s is sufficiently small. Numerical results are also given and discussed.