For a particle of mass m with a total energy E and subject to a potential V(x), the Time-Independent Schrodinger equation in one dimension is as follows:

  (1)22md2Ψ(x)dx2+V(x)Ψ(x)=EΨ(x)

Placing the particle in a potential well of length L in which the potential inside is zero the well and infinite outside the well, otherwise known as the particle in an infinite box, is a standard example in introductory textbooks of Quantum Mechanics used to explore the properties of the solutions to Eq.(1) for bound states.

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 E when placed in an arbitrary position (see Quantization of 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 ψ1(x)=Aeıkx or ψ2(x)=Beıkx. This set of solutions is shown to give values of probability, momentum and position that are consistent with those expected by observers in difference frames. Orthogonality is preserved as well for certain pairs of states. However, it is shown that there is no requirement for continuity of the solutions at the boundary. Furthermore, the solutions require the use of the Bohr-Sommerfeld quantization rules. In addition, it is shown that the infinite well is an exception to the general rule prohibiting degeneracy in one-dimensional systems. Finally, the derived equations are correlated with the traditional solutions. As such, a didactic distinction is made between the mathematical principle that demands the addition of the two solutions to Schrodinger’s Equation and the Quantum Mechanical superposition principle that allows – but does not demand – that two states can be added together.