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Finding <r> of hydrogen atom

  1. May 14, 2013 #1
    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
     

    Attached Files:

  2. jcsd
  3. May 14, 2013 #2

    DrClaude

    User Avatar

    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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