For a particle in a box (length a, mass m), consider the addition of a perturbing potential: V(x) = 2x/a for x ∈ [0,a/2] and V(x) = A(1-x/a) for x ∈ [a/2,a].

1. Evaluate the first-order correction to the unperturbed eigenvalues of the particle in a box due to this perturbing potential.
2. Compare the behavior of the eigenstates with even quantum numbers against those with odd quantum numbers.
3. Discuss whether or not you expect this perturbing potential to mix states with even and odd quantum numbers, explaining why.
4. Evaluate the effect of the perturbation on the ground state wavefunction to first order by working out the coupling matrix elements.

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