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Homework Help: Normalising simplified proton distribution

  1. Oct 5, 2006 #1
    I tried asking this question in the maths help, but am still stuck.

    Q. For a simplified model of a proton's charge distribution, (where R can be considered as some characteristic "size" of the proton):


    [tex] \rho [/tex](r) [tex] \propto [/tex] [tex](1/r)[/tex]Exp(- r / R) where R is some characteristic size of the proton.

    With some help I've figured the integral to this solution is the exponentialintegral: Ei(-r/R) in the past when I'm asked to normalise something I'd take the limits from infinity to -infinity and find N so the probability is 1, here though the exponential integral can't be normalised from what I can see. Any pointers would be appreciated.
     
    Last edited: Oct 5, 2006
  2. jcsd
  3. Oct 5, 2006 #2

    Meir Achuz

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    d^3r in spherical coords with no angular dependence is 4 pi r^2dr so the integral rho r^2 dr converges.
     
  4. Oct 5, 2006 #3
    So

    [tex] \rho [/tex](r) = (4pi r^2 dr).(1/r)Exp(-r / R)

    Integral from infinity to 0 of |[tex] \rho [/tex](r)|^2 = N^2 . [(1/r^2).Exp(-2r / R)]

    is 1 = N^2. (4pi) [1 / (-2/R)] = N^2 .2piR => N = 1 / sqrt(2.pi.R)
     
    Last edited: Oct 5, 2006
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