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Homework Help: Quantum mechanics - probability of finding an electron

  1. Jan 27, 2007 #1
    1. The problem statement, all variables and given/known data
    The wave function of an electron in the lowest (that is, ground) state of the hydrogen atom is
    [tex]\psi(r) = (\frac{1}{\pi a_0^3})^{1/2} exp(-\frac{r}{a_0})[/tex]
    [tex]a_0 = 0.529 \times 10^{-10} m[/tex]
    (a) What is the probability of finding the electron inside a sphere of volume 1.0 pm3, centered at the nucleus (1pm = 10-12m)?
    (b) What is the probability of finding the electron in a volume of 1.0 pm3 at a distance of 52.9 pm from the nucleus, in a fixed but arbitrary direction?
    (c) What is the probability of finding the electron in a spherical shell of 1.0 pm thickness, at a distance of 52.9 pm from the nucleus?

    2. Relevant equations
    [tex]|\psi(r)|^2[/tex]
    3. The attempt at a solution
    (a) [tex] volume = 1.0 \times 10^{-36} m^3[/tex]
    using r = 0, the probability is 1.137 * 10-16.
    (b), (c) What equations should I use here?
    [tex]R^2|\psi(r)|^2[/tex] ????
    [tex]4\pi r^2 R^2|\psi(r)|^2[/tex] ????
    but I don't have R...
     
  2. jcsd
  3. Jan 28, 2007 #2

    siddharth

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    Homework Helper
    Gold Member

    a) The probability of finding the electron with a real wave function [tex]\psi[/tex] in a small volume element [tex] d\tau[/tex] is [tex]|\psi|^2 d\tau[/tex]. To obtain the probability of finding the electron inside the sphere, you integrate. Can you take it from here?
     
  4. Jan 30, 2007 #3
    I don't know how to integrate it, but my teacher said although the correct way is to integrate, we won't need to integrate....
     
  5. Nov 12, 2007 #4
    I would like some help on the same problem too... I'm not sure if I'm doing it correctly. The probability of finding the electron is given by ([tex]\Psi[/tex])[tex]^{2}[/tex]dV... Though I know how to integrate I don't think its necesssary (we're not supposed to use integration). I am solving it by setting r=0 in the wave function, then squaring it, and multiplying it by dV, which I am taking to be 1.0 pm^3. I'm not sure if this is the correct way of doing it. Any help appreciated.
     
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