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Quantum Mechanics }Perturbation Theory
The energy for first-order perturbation theory of  ( is the known Hamiltonian and  is the perturbed Hamiltonian) is given by  , where the wave-functions are the unperturbed ones.
Thus, the problem amounts to calculating ^2 | n \rangle) . This is just raising and lowering operator mechanics.
^2=a^2+a^{\dag^2}+aa^\dag + a^\dag a) . But, after bra-ketting, one finds that the expectation value of  and  are 0, since  and  ,  are orthogonal. Thus, the problem becomes,
 | n \rangle) . Applying the given eigen-equations, one finds that  | n \rangle = (2n+1)V) . For  , one finds  , as in choice (E).
(Note that:  and |n\rangle ) and  .)
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