Is it possible to calculate the average value of $x^{2}p^{2}$ for an infinite square well? [duplicate]

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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.