- #1
issacnewton
- 1,026
- 36
Hi
Here's the problem I am trying to do.
a) Is the state [itex]\psi (\theta ,\phi)=e^{-3\imath \;\phi} \cos \theta [/itex]
an eigenfunction of [itex]\hat{A_{\phi}}=\partial / \partial \phi[/itex] or of
[itex]\hat{B_{\theta}}=\partial / \partial \theta [/itex] ?
b) Are [itex]\hat{A_{\phi}} \;\mbox{and} \;\hat{B_{\theta}}[/itex] hermitian ?
c)Evaluate the expressions [itex]\langle \psi \vert \hat{A_{\phi}} \vert \psi \rangle [/itex]
and [itex]\langle \psi \vert \hat{B_{\theta}} \vert \psi \rangle [/itex]
Now [itex] \hat{A_{\phi}}[/itex] has imaginary eigenvalues , so its not hermitian.
I could show that [itex]\psi[/itex] is an eigenfunction of square of [itex] \hat{B_{\theta}}[/itex]. I have been able to show that the commutator of A and B is zero.
So with this information, how do I check the hermiticity of B ?
for part c) , since there are two state variables , I am little confused about how to go
about it ? any guidance will be appreciated... thanks
Here's the problem I am trying to do.
a) Is the state [itex]\psi (\theta ,\phi)=e^{-3\imath \;\phi} \cos \theta [/itex]
an eigenfunction of [itex]\hat{A_{\phi}}=\partial / \partial \phi[/itex] or of
[itex]\hat{B_{\theta}}=\partial / \partial \theta [/itex] ?
b) Are [itex]\hat{A_{\phi}} \;\mbox{and} \;\hat{B_{\theta}}[/itex] hermitian ?
c)Evaluate the expressions [itex]\langle \psi \vert \hat{A_{\phi}} \vert \psi \rangle [/itex]
and [itex]\langle \psi \vert \hat{B_{\theta}} \vert \psi \rangle [/itex]
Now [itex] \hat{A_{\phi}}[/itex] has imaginary eigenvalues , so its not hermitian.
I could show that [itex]\psi[/itex] is an eigenfunction of square of [itex] \hat{B_{\theta}}[/itex]. I have been able to show that the commutator of A and B is zero.
So with this information, how do I check the hermiticity of B ?
for part c) , since there are two state variables , I am little confused about how to go
about it ? any guidance will be appreciated... thanks