- 67

- 0

Hi,

So I am given linear combination state:

[tex] |\psi> = \cos \theta |0> + \sin \theta |1> [/tex]

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

I did that and got as far as

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

Substitute in for [tex] |\Psi> [/tex]

[tex] \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>) ) [/tex]

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 [tex] \sqrt{\frac{\hbar}{2m\omega}} \sin 2\theta [/tex]

So I am given linear combination state:

[tex] |\psi> = \cos \theta |0> + \sin \theta |1> [/tex]

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

I did that and got as far as

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

Substitute in for [tex] |\Psi> [/tex]

[tex] \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>) ) [/tex]

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 [tex] \sqrt{\frac{\hbar}{2m\omega}} \sin 2\theta [/tex]

Last edited: