# Hybridzation- QM

1. Mar 25, 2008

### 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.

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