The Kennard inequality in quantum mechanics is the most basic uncertainty relation. Within the framework of the Kennard inequality, the Gaussian wavefunction is the minimum uncertainty wavefunction. Then, is there at least one sequence of wavefunctions: such that the absolute squares of all wavefunctions that constitute the sequence are a single statistical distribution which is differentiable everywhere; such that each wavefunction constituting the sequence is a non-Gaussian wavefunction; such that we can calculate the product of the position and momentum uncertainties of each wavefunction in exact closed-form and can make a general description of all those product values; such that another sequence consisting of the product values converges to half the Dirac constant? The answer is yes! This article aims to show (probably all) such sequences and to provide two very different types of proofs of convergence. Whereas the first type of proof is accompanied with very complicated calculations, the second type of proof is not only simple but also interesting and instructive.
Haengjin Choe (Thu,) studied this question.