### 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 to for economy.) We will have 4 equations in 4 unknown coefficients.

At we get

At we get

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

The and terms show up on both sides of the equation and cancel. What's left is

Either or , 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.

k If we set , the even states have the constraint on the energy that

and, if we set , the odd states have the constraint

Jim Branson 2013-04-22