H atom electron in combined spin/position state

[outhsakotad]

Homework Statement

An electron in a H atom occupies the combined spin and position state: R21{(sqrt(1/3)Y10χ+) + (sqrt(2/3)Y11χ-)} If you measured both the z component of spin and the distance from the origin, what is the probability density for finding the particle with spin up and at radius r?

Homework Equations

The Attempt at a Solution

The answer should just be |R21|^2*(1/3)*|Y10|^2*|χ+|^2 = (r^2)/(96πa^5) * exp(-r/a) * (cosθ)^2, right? Or do I need to do an integral? The theta dependence of my answer is bugging me, but I'm not entirely sure if I need to integrate over theta and phi to just get an r dependent answer? Could somebody please help me think through this? Thanks very much.

 

[Matterwave]

If the question just asks for "radius r", then, yes, you do have to integrate out the theta and phi dependence.

 

[outhsakotad]

Sorta makes sense, but if the probability density is not isotropic in theta, it still seems wrong to me to integrate that dependence away.

 

[vela]

If you don't integrate over the angles, the expression you have doesn't give the probability density that the electron is at a distance r; it's the probability density that the electron is at a distance r, angle θ, and angle φ.