Eigenstates in a potential well

Let us now consider a potential well. In the case where the energy is smaller than the well barrier (bound state), the program keeps only the physical solution of the Schrödinger equation on the left side, that is a wavefunction decaying exponentially when moving away from the well. The program then computes the unique solution of the Schrödinger equation. Note that for most values of the energy, the wavefunction magnitude increases exponentially on the right side of the well. Such mathematical solutions of the equation have no physical meaning because they cannot be normalized. Only for a few discrete values of the energy does the solution have a physical meaning. Find them with the mouse, and see how they change when changing the potential.

This is the origin of energy quantization.

Note that for bound states the number of the energy level is equal to the number of nodes plus 1. This is a general rule always true in one dimension. In particular, the ground state never changes sign.