1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Volume Effect of the Proton in Hydrogen Atom

  1. Apr 23, 2010 #1
    This Prob is from Shankar, 17.2.3
    "we assumed that the proton is a point charge e. If the proton is a uniformly dense charge distribution of radius R, the interaction is modified as
    V(r)= -2(e)^2/(2R) + (er)^2/(2(R)^3) r<R
    = -e^2/r r>R

    Calculate 1st Order shift in the ground-state energy of H, due to this modification
    Assume Exp[-R/a0]~1. (Correct answer is E(1)=2(eR)^2/(5(a0)^3))"

    I try ro make perturbated Hamiltonian term to use |nlm>

    H` is V+e^2/r (this H` gives zero when r>R)

    and calculate 1st order purtubation. but it doesn't give correct answer

    I think the method I used is something wrong.(because calculation has no error)

    Please show me the way~
  2. jcsd
  3. Apr 23, 2010 #2
    Well I am unable to get the same shift as your answer. I get:


    So I have no idea where they get the 2/5 coefficient. I could be doing something wrong as well.
  4. Apr 24, 2010 #3
    how could you calculate it? please explain it to me
  5. Apr 24, 2010 #4
    Well integrate your H' from 0 to R in spherical coords. Then make the approximation for your exponentials at the end.
  6. Apr 25, 2010 #5
    Hm.. using Hydrogen atom's e.ft, |100>, and calculating <100|H`|100> in spherical coords. I tried that way before I ask. but It doesn't gives same answer.

    main Integration is
    [tex]I_n= \int {r^n}{e^(\frac{r}{a_0})}[/tex]

    and n=4,2,1. with some coeff. It doesn't give R^2 and a_0^3

    Is that wrong?

    may be I tried 6~10 times... I'm tired..................................
  7. Apr 25, 2010 #6
    Well you need to write out the whole integral. And carefully solve for it. Then expand out the exponentials with the 'R' term in it. I expanded them out to first order in R, to get a similar answer they got.
  8. Apr 28, 2010 #7
    to get higer order of R, I expand exponential term to 1st order

    but they doesn't give correct order. I_4's coefficient is R^(-3) and Integral gives R^(5) but this order vanished and R^(3) and R^(2) remains. that's the problem

    let C=2R/a_0, x=2r/a_0,

    (dummy r zero to n)
    I_n gives -exp[-C]*[tex]\sum[nPr*C^(n-r)][/tex]+n!

    and I_1+I_2+I_4 with some coeff is the wrong answer I solved.
    Hm.............. I don't know what's wrong with this...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook