How can the Schrodinger equation be used to solve a potential symmetry problem?

[whitegirlandrew]

Homework Statement

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Homework Equations

The Attempt at a Solution

I Have no clue where to start for b), i need major help! For a I said that you can argue since the potential is symetric along the x-axis that this operator will go to 0. However I don't know where to start for b) or c).

 

[blue_leaf77]

First write out the formula for $\sigma_x$ and $\sigma_p$ and then use the result from a) to simplify them.