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Math Help - Integrating exponential function in spherical polars

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    Integrating exponential function in spherical polars

    Hi. Just going over some past exam papers and can't work out how to answer one of the questions as the course notes only use simple examples.

    I need to use variational method to estimate the ground state energy of a physical system using a trial wavefunction:

    phi = Aexp(-Br/2a) [B and a constants]

    I'm then asked to normalise phi using spherical polar coordinates to show

    A^2 = (B^3)/(8.pi.a^3)

    I know this is done by finding the integral of phi*phi d^3r between 0<r<a, 0<theta<pi and 0<phi<2pi but I'm not sure what to do with the exponental function in this integral.

    Anyone have any ideas? Thanks for your help.
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    Quote Originally Posted by davebarrass View Post
    Hi. Just going over some past exam papers and can't work out how to answer one of the questions as the course notes only use simple examples.

    I need to use variational method to estimate the ground state energy of a physical system using a trial wavefunction:

    phi = Aexp(-Br/2a) [B and a constants]

    I'm then asked to normalise phi using spherical polar coordinates to show

    A^2 = (B^3)/(8.pi.a^3)

    I know this is done by finding the integral of phi*phi d^3r between 0<r<a, 0<theta<pi and 0<phi<2pi but I'm not sure what to do with the exponental function in this integral.

    Anyone have any ideas? Thanks for your help.
    I will get back to you sometime this evening if I can (and someone else doesn't answer it first.)

    -Dan
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    Thanks. I really haven't got a clue and can't find anything similar online.
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  4. #4
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    Gah. Just found a standard integral at the bottom of the question:

    Take A^2 out of integral as constant

    Convert integral to r^2exp[-Br/a] sin theta phi dr dtheta dphi

    Integral of r^2exp[-Br/a] = (2a^3)/B^3

    multiply this by 2 from sin theta integral

    multuply by 2pi from phi integral

    multiply by the A^2

    Equate to 1 for the normalisation and rearrange to get B^3/8pia^3

    Simple. Sorry if any of you wasted any of your time trying to help without the standard integral.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by davebarrass View Post
    Hi. Just going over some past exam papers and can't work out how to answer one of the questions as the course notes only use simple examples.

    I need to use variational method to estimate the ground state energy of a physical system using a trial wavefunction:

    phi = Aexp(-Br/2a) [B and a constants]

    I'm then asked to normalise phi using spherical polar coordinates to show

    A^2 = (B^3)/(8.pi.a^3)

    I know this is done by finding the integral of phi*phi d^3r between 0<r<a, 0<theta<pi and 0<phi<2pi but I'm not sure what to do with the exponental function in this integral.

    Anyone have any ideas? Thanks for your help.
    (Have you noticed that you have too many \phis in this problem. I'd switch the label on the wavefunction to \psi or something.)

    In case you still need help with the variational part:
    \bar{H} = \frac{< 0 | H | 0 >}{< 0 | 0 >}
    where | 0 > is the trial state. In this case we have
    < \bold{x} | 0 > = \phi (r) = Ae^{-Br/2a}

    The general method is to evaluate this, then minimize \bar{H} in terms of B and a. The problem is that you haven't mentioned what the Hamiltonian is.

    -Dan
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