How to prove Δx and Δp obey the Heisenberg uncertainty principle?, chemistry homework help

The spread in an observable is ΔA =(A^2 – (a)^2)^1/2 where A is the expectation value for the operator A hat.

How do I evaluate knowing this Δx and Δp for the particle in a box to show it obeys Heisenbergs uncertainty principle?

Also, why is the expectation value P^2 not equal to p^2 and the same for the expectation value of X^2 not equal to x^2?

 
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