There are a few good reasons why the dependence in the solution is
on ka,
a and
a: These are all dimensionless
numbers, and mathematical relations can never depend on
parameters that have a dimension! For the case of the even solutions,
the ones with B2 = 0, we find that the number of bound states is
determined by how many times we can fit 2
into
a.
Since
is proportional to (the square root) of V0,
we find that increasing V0 increases the number bound states,
and the same happens when we increase the width a.
Rewriting
a slightly we find that the governing
parameter is
, so that a factor
of two change in a is the same as a factor four change in V0.
If we put the two sets of solutions on top of one another we see that after every even solution we get an odd solution, and vice versa.
There is always at least one solution (the lowest even one), but the
first odd solution only occurs when
a =