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I have a 6 question homework from my Quantum Mechanics Class and I solved most of it (or at least attempted most of it). This one however is tripping me up. Any help would be appreciated.

A 3D random quantity $\mathbf r$ (e.g. position of a particle) is uniformly distributed over a sphere of radius $R$, where $P(\mathbf r) = \delta(\mathbf r^2 - R^2)$ is given by a Dirac delta function. Find $P(r_z)$.

Attempt: I know the sphere is uniformly distributed around the surface. I need $\mathbf r$ in terms of $r_z$, so I'm guessing I should use cylindrical coordinates. I have a triple integral over the volume of the sphere in cylindrical coordinates: $$\int_0^{2\pi}\int_{-r_z}^{r_z}\int_0^{\sqrt{R^2-z^2}} r \delta(r^2 - R^2) \;\mathrm dr \;\mathrm dz \;\mathrm d\theta$$

... but that doesn't seem right. How do I change the Dirac delta function to something that fits what I am doing? I know my reasoning is right. (Well, at least my professor said so and that's all I could understand from what he said.)

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See here for formatting: meta.math.stackexchange.com/questions/5020/… –  Jonathan Y. Sep 2 '13 at 15:30
    
@waseem I've attempted to tidy it up; I've changed the limits on the $z$ integral to what I think you meant. Please check everything! –  Sharkos Sep 2 '13 at 18:12

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