4.4 Lessons from the square well

The computer demonstration showed the following features:

1. If we drop the requirement of normalisability, we have a solution to the TISE at every energy. Only at a few discrete values of the energy do we have normalisable states.
2. The energy of the lowest state is always higher than the depth of the well (uncertainty principle).
3. Effect of depth and width of well. Making the well deeper gives more eigen functions, and decreases the extent of the tail in the classically forbidden region.
4. Wave functions are oscillatory in classically allowed, exponentially decaying in classically forbidden region.
5. The lowest state has no zeroes, the second one has one, etc. Normally we say that the nth state has n − 1 “nodes”.
6. Eigen states (normalisable solutions) for different eigen values (energies) are orthogonal.