#### 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.