Q:

Coherent state evolution for a given non usual Hamiltonian

I am trying to compute the temporal evolution of a coherent state $|\alpha\rangle$ using a given hamiltonian of the form:$$\hat{H}=\hbar \omega(\hat{a}^{\dagger}\hat{a}-\alpha(\hat{a}+\hat{a}^{\dagger})).$$
 - My first attempt has been to find the action of the $\hat{H}$ over a Fock state $|n\rangle$ due to the fact that I can express the coherent states in terms of the Fock states: $$|\alpha\rangle=e^{\frac{-\alpha^{2}}{2}}\sum_{n=0}{\frac{\alpha^{n}}{n!}}|n\rangle,$$
and using that $f(\hat{H})|n\rangle=f(h_{n})|n\rangle$ attack the $e^{-i\hat{H}t/\hbar}$ that appears when computing the time evolution. 
My problem is:  I have found $\hat{H}|n\rangle$ to be:  $$\hat{H}|n\rangle=\hbar \omega (n|n\rangle-\alpha\sqrt{n}|n-1\rangle-\alpha\sqrt{n+1}|n+1\rangle,$$
where I am stucked as it is not of the form $\hat{H}|n\rangle\propto |n\rangle $ so i don't know what to do with the exponential: $$e^{i\hat{H}t/\hbar}|n\rangle=e^{-i\omega t (\hat{a}^{\dagger}\hat{a}-\alpha(\hat{a}+\hat{a}^{\dagger}))}|n\rangle=?$$
I have attemped to manipulate the exponential term by using the BCH formula but I achieved nothing but a messed result
EDIT: 
Here I add my achieved answer using the BCH formula: 
$$|\alpha(t)\rangle=e^{\frac{-\alpha ^{2}}{2}}e^{\frac{-\omega^{2}t^{2}(1+\alpha ^{2})}{2}}\sum_{n=0}\frac{\alpha ^{n}}{n!}e^{-i \omega t (n-\alpha \sqrt{n+1}-\alpha \sqrt{n})}|n\rangle,$$ which seems, to my unexperienced eye, to be wrong. 
Thanks in advance. 

A:

Yes you need to use a BCH but you can see your way if you instead recognize that your Hamiltonian can be written as
$$
\hat H=\frac{\hat p^2}{2m}+\frac{1}{2}m\omega^2 (\hat x- \kappa)^2
$$
for some $\kappa$, i.e. as a shifted harmonic oscillator.  What you can then do is consider the translated Hamiltonian
$$
\hat H= e^{\beta(\hat a-\hat a^\dagger)} \,\hat a^\dagger \hat a\,e^{-\beta(\hat a-\hat a^\dagger)}=T(\beta)\hat a^\dagger \hat a T^{-1}(\beta)
$$
for suitable choice of $\beta$ and remember that the coherent state is itself a translate of the ground state so as to get something like
$$
\hat H\vert \alpha \rangle=T(\beta) \hat a^\dagger \hat a \,T^{-1}(\beta)T(\alpha)\vert 0\rangle\, .
$$

