Separating a wave function into radial and azimuthal parts

[Nitram]

Homework Statement

Shankar's 'Principles of Quantum Mechanics' Exercise 12.3.4.

A particle is described by a wave function $$\psi (\rho, \phi) = Ae^{-\rho^2/2\Delta^2} \left(\frac \rho\Delta cos\phi+sin\phi\right) $$

Show that

$$ P(l_{z} = \hbar) = P(l_{z} = -\hbar) = \frac 1 2$$

Relevant Equations

see above

I know how to work through this problem but I have a question on the initial separation of the wave function. Assuming $\psi(\rho, \phi) = R(\rho)\Phi(\phi)$ then for the azimuthal part of the wavefunction we have $\Phi(\phi)=B\left(\frac \rho\Delta cos\phi+sin\phi\right)$, but this function still contains a $\rho$ term. Is this correct?

 

[TSny]

Homework Statement:: Shankar's 'Principles of Quantum Mechanics' Exercise 12.3.4.

A particle is described by a wave function $$\psi (\rho, \phi) = Ae^{-\rho^2/2\Delta^2} \left(\frac \rho\Delta cos\phi+sin\phi\right) $$

Show that

$$ P(l_{z} = \hbar) = P(l_{z} = -\hbar) = \frac 1 2$$
Relevant Equations:: see above

I know how to work through this problem but I have a question on the initial separation of the wave function. Assuming $\psi(\rho, \phi) = R(\rho)\Phi(\phi)$ then for the azimuthal part of the wavefunction we have $\Phi(\phi)=B\left(\frac \rho\Delta cos\phi+sin\phi\right)$, but this function still contains a $\rho$ term. Is this correct?

The wavefunction for this problem cannot be written in the form $\psi(\rho, \phi) = R(\rho)\Phi(\phi)$. But, that's ok. The problem can be worked similarly to the previous problem 12.3.3. That is, try expressing $\cos \phi$ and $\sin \phi$ in terms of the eigenfunctions $\Phi_m(\phi)$ as given in equation (12.3.9).

 

[Nitram]

The wavefunction for this problem cannot be written in the form $\psi(\rho, \phi) = R(\rho)\Phi(\phi)$. But, that's ok. The problem can be worked similarly to the previous problem 12.3.3. That is, try expressing $\cos \phi$ and $\sin \phi$ in terms of the eigenfunctions $\Phi_m(\phi)$ as given in equation (12.3.9).

Thanks for clarifying this!