For a particle of mass
Placing the particle in a potential well of length
When the well is centered around the origin, it is referred to as the symmetric well (see The Symmetric Well). Other authors place the well with its left-hand edge at the origin and refer to it as the asymmetric well (see The Asymmetric Well).
The traditional approach however involves setting the boundary conditions of the solutions in the interior of the well to zero in order to be continuous with the solutions outside the well. This approach, however, leads to an inconsistency in the well’s energy
In this manuscript, a set of conditions is proposed that allow for derivation of solutions that are consistent regardless of the well’s position within the coordinate system. The final solutions are of the form