Finding <r> of hydrogen atom

1. May 14, 2013

Verruto

1. The problem statement, all variables and given/known data
Hydrogen is in n=2, l=1, and m=0.
Wave function is ψ(r,θ,∅)=(1/4(√2pi)ab3/2)(r/ab)(e-r/2ab)(cos(θ)

Find <r> for this state.

2. Relevant equations

P(r) = 4pir2|R(r)|2

<r> is equal to the integral from 0 to ∞ of P(r)dr

3. The attempt at a solution

I understand that you need to go from the wave function to R(r) and then P(r) to put that in the <r> equation. Im just not sure how you do the first step.
I put the equations in the picture below to better visualize it. Thanks in advance!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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2. May 14, 2013

Staff: Mentor

The equation for $P(r)$ already takes into account the fact that you have integrated the spherical harmonic over $\theta, \phi$, and this is where the factor $4 \pi$ comes from. So you would have to remove the angular part from the initial wave function.

That said, I believe that you will learn more if you don't take that ready-made equation, but calculate it from first principles:
$$\langle r \rangle = \int_{0}^{2 \pi} \int_{0}^{\pi} \int_{0}^{\infty} \psi^* r \psi r^2 dr \sin \theta d\theta d\phi$$

Note that you have a factor $r$ missing in your description of the integration step, it should be
$$\langle r \rangle = \int_{0}^{\infty} P(r) r^3 dr$$

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