Exam 2 overview
- problem 1
Inside the well we should expect plane wave solutions:
Where
Applying the BC this gives:
At this point you have two equatins and two unknowns. I don't care how you solve them. Here is one way. Add the two equations to get:
now subtract to get:
These are nice since we can factor out the common sine and cosine terms:
There are only two ways we can satisfy both equations simultaneously. If A=B then the second equatino is
taken care of, but then A+B cannot = 0, so we have to insist that . This requires
On the other hand, if we choose A = -B, then the first equation is automatically satisfied and then we must insist that . This requires
So we have two families of solutions:
The cosine states are even (have even parity), the sine states are odd (have odd parity)
Now since we have
for the odd parity states and
for the even parity states.
In numerical order these are:
- problem 5
is an operator equation. By definition it is: . In order to work this out it is helpful to apply the operator to a test function.