I am reading the following paper by Slater:

https://journals.aps.org/pr/pdf/10.1103/PhysRev.81.385

On page 5 they write above equation (12) the following:

"If we now average over all wave functions, we find that the properly weighted average of $F(\eta)$ is 3/4."

Now, $F(\eta)=1/2 + \frac{1-\eta^2}{4\eta}\ln((1+\eta)/(1-\eta))$.

I don't understand what does it mean to average over wave functions, I thought that they calculated: $\lim_{T\to \infty} \frac{1}{T}\int_0^T F(x)dx$, but I have given maple to calculate this limit (for the additive part without 1/2), and it didn't gave me 1/4.

So I don't understand which average of this function did they calculate?

ANyone knows?

Thanks!