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Inner product of two spherical functions

  1. Feb 5, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi all.

    The inner product between two functions f(x) and g(x) is defined as:

    [tex]
    <f | g> = \int f^*(x)g(x) dx,
    [/tex]

    where the * denotes the complex conjugate. Now if my functions f and g are functions of r, theta and phi (i.e. they are written in spherical coordinates), is the volume element then given as [itex]drd\theta d\phi[/itex] or [itex]r^2\sin \theta drd\theta d\phi[/itex]?

    3. The attempt at a solution

    I personally think the last, because originally we have <f(x,y,z) | g(x,y,z)>, which we convert to spherical coordinates, so we need the Jacobian. Am I correct?

    Thanks in advance.
     
  2. jcsd
  3. Feb 5, 2009 #2
    Putting [tex]h(x,y,z) = f^\ast(x,y,z)g(x,y,z)[/tex], we have

    [tex]\int\int\int h(x,y,z) dz \, dy\, dx = \int\int\int h(r\cos{\theta}\sin{\phi}, r\sin{\theta}\sin{\phi}, r\cos{\phi}) r^2{\underbrace{\sin{\phi}}_{\text{not $\theta$}} dr \, d\theta\, d\phi[/tex]

    with appropriate limits, as per usual. That is all your question appears to boil down to, Niles.
     
  4. Feb 5, 2009 #3
    Then I was correct. Thanks for replying!
     
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