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Question on Spherical Bessel's equation.

  1. Jul 5, 2010 #1
    Spherical Bessel equation:

    [tex]r^2R''+2rR' + [kr^2-n(n+1)]R = 0 \;\hbox { where }\; k=\lambda^2 [/tex]

    Boundary condition: [itex]R(a)=0[/itex]

    Solution :

    [tex]j_n(\lambda_{n,j},r) \;\hbox { where }\; \lambda = \lambda_{n,j}= \frac{\alpha_{n+\frac{1}{2},j}}{a}[/tex]

    [tex] j_n(x)=\sqrt{\frac{\pi}{2x}}J_{n+\frac{1}{2}}(x) [/tex]

    I want to find [itex] j_n(\lambda_{n,j}r)[/itex]

    This is what I have:

    [tex] j_n(\lambda_{n,j}r)=\sqrt{\frac{\pi}{2\lambda_{n,j}r}} \; J_{n+\frac{1}{2}}(\lambda_{n,j}r) = \sqrt{\frac{\pi}{ 2 \frac{\alpha_{(n+\frac{1}{2},j)}}{a}r}} \;\; J_{n+\frac{1}{2}}(\frac{\alpha_{(n+\frac{1}{2},j)}}{a}r) [/tex]

    I think I am correct actually the confusion is the order of the Bessel function:

    [tex] p=n+\frac{1}{2} \Rightarrow\; \lambda_p = \lambda_n = \frac{\alpha_{n+\frac{1}{2}}}{a}[/tex]

    Can anyone verify this?
     
    Last edited: Jul 5, 2010
  2. jcsd
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