4.4 Lessons from the square well
The computer demonstration showed the
following features:
- 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.
- The energy of the
lowest state is always higher than the depth of the well
(uncertainty principle).
- 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.
- Wave functions are
oscillatory in classically allowed, exponentially decaying in
classically forbidden region.
- The lowest state has
no zeroes, the second one has one, etc. Normally we say that
the
th state has
“nodes”.
- Eigen states
(normalisable solutions) for different eigen values
(energies) are orthogonal.