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Thread: Normalizatoin Constant and a Gaussian Integral.

  1. #1
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    Normalizatoin Constant and a Gaussian Integral.

    Hello! I'm attempting to compute the normalization constant N for the following function $\displaystyle f(x)=N e^{-(x-x_0)/2k^{2})}$. This is generally done by evaluating $\displaystyle \int |f(x)|^{2}dx=1$, where the limits of integration are -inf to +inf.
    I can get this right up until the point where I need to make a u substitution.
    $\displaystyle 1=\int |N e^{-(x-x_0)/2k^{2})}|^{2}dx$
    $\displaystyle 1=\int N^2 e^{-(x-x_0)^2/k^2} dx$
    $\displaystyle 1=N^2 \int e^{-(x-x_0)^2/k^2} dx$
    making the substitution $\displaystyle u=x-x_0, du=dx$
    $\displaystyle 1=N^2 \int e^{-u^2/k^2}du$
    This integral is only slightly more clear to me. It looks like a Gaussian integral, but, how do I go about solving it? Any help is much appreciated.
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    Re: Normalizatoin Constant and a Gaussian Integral.

    I would actually make the subtitution

    $\displaystyle u=\frac{x-x_{0}}{k}.$

    That should make the integral a more standard integral.
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    Re: Normalizatoin Constant and a Gaussian Integral.

    So, that would yield: $\displaystyle 1=N^{2} \int e^{-u^{2}}du=N^{2}\sqrt{\pi}$ ?
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    Re: Normalizatoin Constant and a Gaussian Integral.

    Quote Originally Posted by atomicpedals View Post
    So, that would yield: $\displaystyle 1=N^{2} \int e^{-u^{2}}du=N^{2}\sqrt{\pi}$ ?
    I don't think you've quite got it yet. The substitution I suggested requires a change in the differential as well. The final answer had better have a 'k' in it somewhere.
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  5. #5
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    Re: Normalizatoin Constant and a Gaussian Integral.

    Woops! $\displaystyle 1=N^{2}\frac{\sqrt{\pi}}{k}$ which then yeilds a normalization constant $\displaystyle N=\pm\frac{\sqrt{k}}{\pi^{1/4}}$ .
    Last edited by atomicpedals; Oct 23rd 2011 at 09:49 AM.
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    Re: Normalizatoin Constant and a Gaussian Integral.

    Better, but still not quite there yet. Be careful as to numerator and denominator. Can you show your working, please?
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    Re: Normalizatoin Constant and a Gaussian Integral.

    So my line of thinking was as follows:

    $\displaystyle u=\frac{(x-x_{0})}{k}$ which I think yields $\displaystyle du=\frac{1}{k}dx$ and so $\displaystyle 1=N^{2}\int \frac{1}{k}e^{-u^{2}}du=1=N^{2}\frac{1}{k}\int e^{-u^{2}}du=N^{2}\frac{1}{k}\sqrt{\pi}$
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  8. #8
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    Re: Normalizatoin Constant and a Gaussian Integral.

    Wait... I should have $\displaystyle kdu=dx$ shouldn't I...
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    Re: Normalizatoin Constant and a Gaussian Integral.

    Quote Originally Posted by atomicpedals View Post
    So my line of thinking was as follows:

    $\displaystyle u=\frac{(x-x_{0})}{k}$ which I think yields $\displaystyle du=\frac{1}{k}dx$
    Fine so far.

    and so $\displaystyle 1=N^{2}\int \frac{1}{k}e^{-u^{2}}du$
    The problem here is that what you start with is a dx, not a du. Solve du = dx/k for dx. That's what you must plug in.

    $\displaystyle =1=N^{2}\frac{1}{k}\int e^{-u^{2}}du=N^{2}\frac{1}{k}\sqrt{\pi}$
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