If you can only measure either position and momentum in quantum mechanics how would one find the average value of $x^{2}p^{2}$ for an infinite square well?
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What you are asking for can indeed be found. It is simply $\int\psi^*(x)\ \left(-x^2\hbar^2\frac{\partial^2}{\partial x^2}\right)\psi(x)dx$. Substitute whichever state of the infinite square well you like and work out the integral.
The problem is that this is a different result from say $p^2x^2$ or $xpxp$ because the order of $x$ and $p$ cannot be swapped.
J.G.
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