246 SOLUTIONS
The bound state for has the form
We have already imposed the constraint that be continuous at
This form satisfies the requirement that is continuous at the origin
and vanishes at infinity. Awayfrom the origin the potential is zero, and
the Schrödinger equationjust gives A relation between C
and E is found by matching the derivatives of the wave functions at
Taking the integral of (S.5.3.1) between and gives
Applying(S.5.3.3) to (S.5.3.2) gives the relations
We have found the eigenvalue for the bound state. Note that the dimensions
ofC are energy × distance, which makes the eigenvalue have units of energy.
Finally, we find the normalization coefficient A:
b) When the potential constantchanges from the eigenfunction
changes from where the prime denotes the eigenfunction with
the potential strength In the sudden approximation the probability
that the particle remains in the bound state is given by
where