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Cant understand integral tranasition to spherical coordinates

  1. Dec 13, 2009 #1
    there is a function [tex]\Psi =\frac{c}{\sqrt{r}}e^{\frac{-r}{b}}[/tex]

    find the probaility in [tex]\frac{b}{2}<r<\frac{3b}{2}\\[/tex] region

    the rule states [tex]\int_{-\infty}^{+\infty}|\Psi|^2dv=1\\[/tex]


    [tex]1=\int_{-\infty}^{+\infty}|\frac{c}{\sqrt{r}}e^{\frac{-r}{b}}|^2dv[/tex]



    then they develop it as



    [tex]c^2\int _{all space}\frac{1}{r}e^{\frac{-2r}{b}}2\pir^2dr=4\pic^2\int_{0}^{+\infty}re^{\frac{-2r}{b}}dr\\[/tex]


    they as it because of spherical coordinates

    but i cant see here the jacobian of spherical coordinates.

    i cant see here the x,y,z transition to r ,theta,phi



    i cant see it in the last equation
     
  2. jcsd
  3. Dec 13, 2009 #2

    tiny-tim

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    Welcome to PF!

    Hi nhrock3! Welcome to PF! :smile:

    (have a pi: π and an infinity: ∞ and an integral: ∫ and try using the X2 and X2 tags just above the Reply box :wink:)

    I'm a little confused by what you've written, but basically you start with

    ∫ c2/r e-2r/b dxdydz

    and because it's spherically symmetric, you can divide the region into spherical shells of radius r and volume 4πr2dr,

    which gives you ∫0 c2/r e-2r/b 4πr2dr

    = 4πc20 r e-2r/b dr
     
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