Is the Expectation Value of x Zero for an Even Potential Energy Function?

[FourierX]

Homework Statement

Its a problem from a foreign book. It sounded simple to me but I am confused now.


If V(x), a potential energy function, is known to be an even function, what can you say about wave function for any stationary state? What shall be the expectation value of x for any stationary state ?

Homework Equations

The Attempt at a Solution

I grabbed a book by Griffith from my library and figured that if V(x) is even, the time independent wave function can be taken to be either even or odd. I know that expectation value of x for a stationary state has to be 0. But can some help me see this with the standpoint of V(x) being even ?


thanks in advance

 

[lanedance]

V(x) is even, so it is symmetric around the origin

the solutions to TISE will be either symmetric or anti-symmetric around the origin

note if $$\psi$$ is a solution, so will $$e^{i \phi}\psi$$ for any phi, ie the solution is only unique upto an overall phase

so in the anti-symmetric solution, there is no meaning in one side being negtive and teh other positive positive, as it is equivalent to any solution with phase shifted by phi. the only point is that they are out of phase by p.

when you look at the probabilty distribution given by
$$P(r)dr = \psi \psi*$$
the overall phase cancels and P(r)dr will be a non-negative symmetric function, thus the expectation position will be zero