In this work the conditions appearing in the so-called 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 textbooks on quantum mechanics, whereas the other 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 so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.
Charged particles in higher dimensional homogeneous gravitational field: self-energy and self-force