Electric Potential Normalization

[makhoma]

I was in my Electrodynamics lecture last week, still working the Laplacian and Poisson equations, when we discussed an infinite parallelpipid (infinite in the x direction, length a and b in the y and z direction respectively) with a potential of \Phi=\Phi_0 at x=0 plane and every other face having \Phi=0. Here he said that, due to boundary conditions we should expect the potential to have the form

<br /> \Phi(x,y,z)=\sum_{n_{y},n_{z}}A_{n_{y}n_{z}}\eta_{n_{y}}\sin\left[\frac{n_{y}\pi y}{a}\right]\eta_{n_{z}}\sin\left[\frac{n_{z}\pi z}{b}\right]\exp\left[-\pi\sqrt{\frac{n_{y}^{2}}{a^{2}}+\frac{n_{z}^{2}}{b^{2}}}x\right]<br />

where \eta and A are both normalization constants. I agree with this form (not sure about the normalization constants, but that's the question to come). We let the eta's equal \sqrt{2/a} and \sqrt{2/b} in order to coincide with QM. In looking at the case x=0, we find

<br /> \Phi_{0}=\sum_{n_{y}n_{z}}A_{n_{y}n_{z}}\sqrt{\frac{2}{a}}\sin\left[\frac{n_{y}\pi y}{a}\right]\sqrt{\frac{2}{b}}\sin\left[\frac{n_{z}\pi z}{b}\right]<br />

Then multiply both sides by

<br /> \sum_{n_{y}&#039;n_{z}&#039;}A_{n_{y}&#039;n_{z}&#039;}\sqrt{\frac{2}{a}}\sin\left[\frac{n_{y}&#039;\pi y}{a}\right]\sqrt{\frac{2}{b}}\sin\left[\frac{n_{z}^{&#039;}\pi z}{b}\right]<br />

Which after integrating over dy&#039; and dz&#039; from 0 to a and 0 to b, respectively, we find that n_y,n_z must be odd integers. So then the normalization constant is found to be

<br /> A_{n_{y}&#039;n_{z}&#039;}=\frac{8\Phi_{0}\sqrt{ab}}{n_{y}&#039;n_{z}&#039;\pi^{2}}<br />

All of this makes sense and does work out mathematically. The trouble I find is that my professor said we could do this without having the \eta terms in there, that A would absorb it and we should still come out with the same answer. But following the same math as above, and removing the \eta normalizations, I find the normalization constant to be

<br /> A_{n_{y}&#039;,n_{z}&#039;}=\frac{4\Phi_{0}ab}{n_{y}&#039;n_{z}&#039;\pi^{2}}<br />

My question is two-fold:
(1) Is my professor wrong and we should not expect the same normalization constant between the two (my answer is off by the factor 2/\sqrt{ab}?
(2) Is my professor right, and I screwed up somewhere?

 

[makhoma]

I think I found an error in what my professor did in class. We have a line that reads

<br /> \int_0^ady\int_0^bdz\sqrt{\frac{2}{a}}\sin\left[\frac{n_y&#039;\pi y}{a}\right]\sqrt{\frac{2}{b}}\sin\left[\frac{n_z&#039;\pi z}{b}\right]\Phi_0=A_{n_y&#039;n_z&#039;}<br />

And it should read

<br /> \int_0^ady\int_0^bdz\sqrt{\frac{2}{a}}\sin\left[\frac{n_y&#039;\pi y}{a}\right]\sqrt{\frac{2}{b}}\sin\left[\frac{n_z&#039;\pi z}{b}\right]\Phi_0=\frac{2}{\sqrt{ab}}A_{n_y&#039;n_z&#039;}<br />

This makes the normalization constant A_{n_y&#039;n_z&#039;} equal to what I get:

<br /> A_{n_y&#039;n_z&#039;}=\frac{4\Phi_0ab}{n_y&#039;n_z&#039;\pi^2}<br />

Guess the professor forgot to carry some factors from one line to the next. Thanks for your help anyway!