How Do You Write the Hamiltonian in the Basis |\theta>?

[TeddyYeo]

Homework Statement

$H = \frac{2e^2}{\hbar^2 C} \hat{p^2} - \frac{\hbar}{2e} I_c cos\hat\theta$,
where $[\hat\theta , \hat{p}] = i \hbar$
How can we write the expression for the Hamiltonian in the basis $|\theta>$

Homework Equations

The Attempt at a Solution

I have already solved most part of the question and this is just one part of it that I am not sure how to convert into the basis form.
Is it that I just now need treat
$\hat{p}] = -i \hbar ∇ which is means that it is -i \hbar frac{\partial }{ \partial \theta}$
and put
$H = \frac{2e^2}{C} \frac{\partial^2}{\partial\theta^2} - \frac{\hbar}{2e} I_c cos\hat\theta$
then this is the final form??

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[ehild]

Homework Statement

$H = \frac{2e^2}{\hbar^2 C} \hat{p^2} - \frac{\hbar}{2e} I_c cos\hat\theta$,
where $[\hat\theta , \hat{p}] = i \hbar$
How can we write the expression for the Hamiltonian in the basis $|\theta>$

Homework Equations

The Attempt at a Solution

I have already solved most part of the question and this is just one part of it that I am not sure how to convert into the basis form.
Is it that I just now need treat
$\hat{p}] = -i \hbar ∇$ which is means that it is $-i \hbar frac{\partial }{ \partial \theta}$

No. $\hat{p^2}=-i \hbar ∇(-i \hbar ∇) =-h^2 \Delta$. Use the Laplace operator written in spherical polar coordinates.