References B. D. Anderson, J. Chem. Additional references . The PIB is an idealized physical system consisting of a particle constrained to move in one dimension between infinite potential walls. Thus, the particle is subject to no forces between the walls but infinite force at the walls to prevent its escape from the box. With these changes, the Schroedinger equation becomes .
Elementary calculus is enough to find the solutions as shown here .
Here is a plot to show the energy levels with wave functions superimposed. Wave functions are normalized, but their vertical scale on the graph is arbitrary. Not counting the nodes at the walls of the box, n is the number of interior nodes in the wave function. Link to Numerical Solution Analytical solutions of PIB are not always possible when the potential inside the box is modified. In such cases numerical solutions can be found.