I was doing a problem today in the book that deals with Schrodinger's equation for a potential well. Basically a potential well is when the potential energy is $0$ for some interval and then $V_1$ everywhere else, which is used for trapping electrons.
The problem I am having is in calculating the average value for
$$(z-\langle z \rangle )^2$$
I know that to calculate the average value, the expectation formula must be used which is,
$$\langle A \rangle = \frac{\int_x \psi^* A \psi dx}{\int_x |\psi|^2dx}$$
I am having issues with this because the book does not say what $\psi ^*$ is. However they do differentiate between $\Psi$ and $\psi$. In the book, $\Psi$ is a solution to Schrodinger's equation while $\psi$ is the position component to Schrodinger's equation. Basically $\Psi = r(t)\psi$. Because this is a potential well, there is no reflected wave which leads to the
$$\psi = A\exp(ikz)$$
So given this, does that really mean then that the complex conjugate is
$$\psi ^* = A\exp(-ikz)$$
The problem here is that this leads to a computation simply of calculating
$$\langle A \rangle = \int_x Axdx$$
Note this comes from the fact that $$\langle x \rangle = 0$$ Given that the lower limit is $-L$ and upper limit is $L$. It is almost no surprise to me that this is the case because the midpoint would be the average in many symmetric distributions. But $z^2$ doesn't seem as obvious
I do know the full solution $\Psi$ as well although the time component is also complex.
To me this seems to not be possible. I know that in probability theory the each individual value depends on the probability density function. So how is it possible that the PDF is eliminated?
Any advice for this problem? I would like to learn more about dealing with expected values of Schrodinger's wave function.