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Homework Help: Probability of finding an electron betweeen two spheres of radii?

  1. May 3, 2009 #1
    1. The problem statement, all variables and given/known data

    For a hydrogen atom in a state with n=2 and l=0, calculate the probability of finding the electron between two spheres of radii r = 5.00a0 and r = 5.01 a0.

    2. Relevant equations

    As stated above, calculate the probability of finding the electron in the given spheres of radii.

    3. The attempt at a solution

    Maybe it's something like this?

    ∫ psi^2 (r) dr, from r = 5.00a to r = 5.01a

    = ∫ (1/32π(a^3)) * ((2 - ((r/a)^2))^2) * (e ^ (-r/a) dr, where π means pi.

    = (after complicated processes of trying to find the integral) some unsolvable identity! D=
  2. jcsd
  3. May 3, 2009 #2
    Short answer: yes, that's the way to do it :)

    Some hints:
    First off all, apart from an overal constant you have to find the primitive function of:
    [2-(r/a)^2]^2 exp(-r/a) = [4 - 4(r/a)^2 + (r/a)^4]exp(-r/a)

    To find this you can use the following.
    You know that:
    ∫ exp(-y b) dy = -(1/b) * exp(-y b)
    Now differentiate left and right with respect to b, twice. This gives the following:
    ∫ y^2 exp(-y b) dy = -[2(1/b)^3+y(1/b)^2] exp(-y b) + (y/b)(1/b+y)exp(-y b)

    which you can simplify a bit further, but I'll leave this up to you (also, my answer might be wrong, cause I'm being a bit sloppy with one eye on the tv ;) ). You can perfrom this trick again (differentiating twice) and that will give you an expression for the integral of y^4 exp(-yb)dy. Hopefully this puts you in the right direction.
  4. May 3, 2009 #3
    Thanks! But...I'm still a bit confused. So in order to find the probability of finding the electron between the two spheres of radii, I have to differentate the integral? Doesn't that cancel it out?
  5. May 3, 2009 #4
    No, no! That's just a trick to calculate the integral! You integrate over y, but you differentiate with respect to b. It's a neat little trick, but quite weird if you haven't seen it before. Try to work it out. You can post your results here.
  6. May 3, 2009 #5
    Okay, I'll try it out. But just for an extra clarification, when you put exp(-y b), you substituted (-y b) (which equals -y/b?) in for r/a, correct?

    All right, I'm going to try it now....thanks!
  7. May 3, 2009 #6
    All right, I tried it. =.= I came out in circles. My ending was:

    (1/(32pi(a^2))) * int((4-4u+(u^2))*(e^-u)) du, where u = r/a

    And then I got stuck. :P
  8. May 4, 2009 #7
    Can anyone please help by directing me in some direction? Thanks! (and I understand that xepma kindly helped me out, so thank you! But I'm still lost on where to go, because I have tried the integral, but I just can't seem to get it. =.=)
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