WKB formalism and a lower limit for the energy eigenstates of bound states for some potentials

In the present work the conditions appearing in the WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the text books on quantum mechanics, whereas the second one is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the Rayleigh--Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.