Angular momentum operators and eigenfunctions

[machofan]

Homework Statement

Homework Equations

The Attempt at a Solution

I have tried inserting the first wavefunction into Lz which gets me 0 for the eigenvalue for the first wavefunction. Is this correct?

For the second wavefunction, I inserted it into Lz and this gets me -i*hbar*xAe^-r/a which is not equal to an eigenvalue*Ay*e-r/a, although I am not sure if this is the correct justification.

Any help is much appreciated, thank you.

 

[dauto]

Seems right to me.

 

[machofan]

I'm also stuck on how to convert the wavefunctions into spherical polar coordinates, having tried to use the formula for spherical polar coordinates, I notice that the wavefunctions do not depend on the variables phi and theta, so I am unsure on how to approach the first problem.

 

[BvU]

Hello fan, and welcome to PF.

You get 0 for $\Psi_0$ because it is symmetric under rotations around the z axis: there is no angular momentum. So I think you are right. It is an eigenfunction.

For $\Psi_1$ you basically want to show that ${\bf L}_z \left( \Psi_1 \right ) = L_z\, \Psi_1$ with ${\bf L}_z$ the operator and $L_z$ an eigenvalue (i.e. a number, possibly complex), has no solutions.

You may assume |A| is not equal to zero (because of the normalization) so the eigenvalue zero is already excluded. So if you multiply the right hand side from the left by $\Psi_1^*$ and integrate, that yields $L_z$, which is nonzero.

Do the same thing with the lefthand side and show that it does give zero. That way you prove that the assumption that $\Psi_1$ is an eigenfunction leads to a false equation.

 

[BvU]

I'm also stuck on how to convert the wavefunctions into spherical polar coordinates, having tried to use the formula for spherical polar coordinates, I notice that the wavefunctions do not depend on the variables phi and theta, so I am unsure on how to approach the first problem.

With $z = r\cos\theta$ you sure have a theta dependence!

 

[BvU]

But the exercise wants you to write out the z-component of the operator cross product ${\bf r} \times {\bf p}$, not the wave functions!
[edit] sorry, it wants you to do both. In the first part the Lz, later on the $\Psi$

 

[machofan]

Hello fan, and welcome to PF.

You get 0 for $\Psi_0$ because it is symmetric under rotations around the z axis: there is no angular momentum. So I think you are right. It is an eigenfunction.

For $\Psi_1$ you basically want to show that ${\bf L}_z \left( \Psi_1 \right ) = L_z\, \Psi_1$ with ${\bf L}_z$ the operator and $L_z$ an eigenvalue (i.e. a number, possibly complex), has no solutions.

You may assume |A| is not equal to zero (because of the normalization) so the eigenvalue zero is already excluded. So if you multiply the right hand side from the left by $\Psi_1^*$ and integrate, that yields $L_z$, which is nonzero.

Do the same thing with the lefthand side and show that it does give zero. That way you prove that the assumption that $\Psi_1$ is an eigenfunction leads to a false equation.

That has really cleared things up, thank you!