Time independent perturbation - potential with inner function

1. Nov 17, 2013

ffia

1. The problem statement, all variables and given/known data
I need to find the corrected ground state for perturbed harmonic oscillator (1D) with perturbation of the form V(x) = λ sin(κx), κ>0.

My problem is I have no idea how to handle a potential that has its operator as an inner function.

2. Relevant equations
The perturbed Hamiltonian of the system:
H = H$_{0}$ + λV = $\frac{1}{2}$mω$^{2}$x$^{2}$ + λsin(κx).

At least the first correction to the ground state is defined
$\sum\frac{<m| λV |0>}{E_{m}-E_{n}}$ where the sum is taken over m, m goes from 1 to infinity? (m and 0 are states, sorry I couldn't get the latex notation working there)

3. The attempt at a solution
I tried to use the ladder operators.
In terms of the ladder operators
x = $\sqrt{\frac{h}{2mω}}{}(a+a^\dagger)$
But the problem is I don't know how to operate the states when the operators are inside sines argument. I could use the taylor series of sin, but it would leave me with (a+a^\dagger)$^{2n+1}$ and that doesn't seem good either.

Thank you in advance, I'm really stuck here.