Quantum Measurements of L2 in a Wave Function: Calculations and Probabilities

[nmsurobert]

Homework Statement

A particle is in a state described by a wave function of the form ψ(r) = (x+y+z)f(r).
What are the values that a measurement of L² can yield? What is probability for all these results?

Homework Equations

The Attempt at a Solution

I feel this problem shouldn't be too hard but I've been struggling with quantum since last semester so any guidance would be nice. I think i should probably write the function in spherical terms to start off with. but I'm not too clear on where to go from there. I also know that i need to use Y_lm too but again, I'm pretty confused.

 

[blue_leaf77]

It will be helpful to use the spherical harmonics written in Cartesian coordinates. Your task is to find $Y_{lm}$'s in which the given $\psi(\mathbf{r})$ is expanded, $$\psi(\mathbf{r}) = f(r)\sum_{lm} c_{lm} Y_{lm}(\hat{r})$$

 

[nmsurobert]

ok ill try to do some work with that. I'm still pretty lost on what the whole Y_lm thing is about but hopefully i can make some sense of it.

 

[Megaquark]

I don't know what book you're studying out of, but most go through the steps to show that L²Ψ=ħ²l(l+1)Ψ That might help make the problem a bit easier.