Bound States of a 1D Potential Well *
In the two outer regions we have solutions
In the center we have the same solution as before.
(Note that we have switched from
We will have 4 equations in 4 unknown coefficients.
Divide these two pairs of equations to get two expressions for
Factoring out the
, we have two expressions for the same quantity.
If we equate the two expressions,
and cross multiply, we have
terms show up on both sides of the equation and cancel. What's left is
, but not both, must be zero.
We have parity eigenstates, again, derived from the solutions and boundary conditions.
This means that the states separate into even parity and odd parity states.
We could have guessed this from the potential.
Now lets use one equation.
If we set
the even states have the constraint on the energy that
and, if we set
, the odd states have the constraint