Why Must \( A_n \) for Even \( n \) Vanish in Example 3-5?

[Xyius]

Homework Statement

The problem is referring to an example in the chapter.
Use Parity Arguments to show that in Example3-5 the A_n for n is even must vanish.

Here is the example:
http://imageshack.us/a/img28/5664/qmproblem.gif

The Attempt at a Solution

I honestly do not know where to start with this. The parity operator simply switches the sign of "x" in the wave function. It is easy enough to SEE that they go to zero without using parity arguments. Any push in the right direction would be appreciated!

~Matt

 

[vela]

I think the problem intends for you to look at the symmetry about the line x=a/2 and that it's not using the word parity in the way you've interpreted it. (But I could be wrong.)

 

[Xyius]

I think the problem intends for you to look at the symmetry about the line x=a/2 and that it's not using the word parity in the way you've interpreted it. (But I could be wrong.)

I still don't understand how I would even do that. If I draw it, it is symmetric about x=a/2. But I do not know how I would link this to the fact that all odd n's go to zero.

I know that for n=odd, the solution to the wave function for a particle in a box are even functions. (Cosine) Don't know if this helps...

 

[vela]

Look at the symmetry of the eigenstates about the line x=a/2 as well.

 

[usmcguardian]

Basically to use parity you have to be symmetrical across the y axis. So a simple shift of (defining our new x as x')

x' = x - \frac{a}{2}

will get such a symmetry. You can then apply the Parity argument to show that when n is odd you get a 1 at your parity (since parity can only have the value of \pm1. However, when n is an even function this shows a parity of -1 which we know is not an eigenstate.