Hybridzation- QM

[Biest]

Hi,

So I am given linear combination state:

|\psi> = \cos \theta |0> + \sin \theta |1>

Now we are supposed to apply: x = \sqrt{\frac{\hbar}{2m\omega}} (\hat{A} + \hat{A^\dagger}) such that <\Psi|x|\Psi> so I can find the angle \theta maximizes the expected value.

I did that and got as far as

\sqrt{\frac{\hbar}{2m\omega}} (<\psi|\hat{A^\dagger}|\psi> + <\psi|\hat{A}|\psi>)

Substitute in for |\Psi>

\sqrt{\frac{\hbar}{2m\omega}} ( (<0|\cos \theta +<1| \sin \theta) \hat{A^\dagger} (\cos \theta |0> + \sin \theta |1>) + (<0|\cos \theta +<1| \sin \theta)\hat{A^\dagger} (\cos \theta |0> + \sin \theta |1>) )

Now I am getting confused as to how to apply the operator.

Thanks in advance.

Cheers,

Biest

EDIT:

I took an educated guess and got out \sqrt{\frac{\hbar}{2m\omega}} \sin 2\theta