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Quantum mechanics (math question)

  1. Feb 4, 2014 #1
    i'm trying to do the following integral:

    $$\int{e^{\frac{-2amx^2}{ħ}}dx}$$ (in case this is hard to see, the exponent is $$\frac{-2amx^2}{ħ}$$)

    where a, m are real constants

    but inside the integral can't i split this up into two exponentials?

    $$\int{e^{\frac{-2am}{ħ}}e^{x^2}dx} = e^{\frac{-2am}{ħ}}\int{e^{x^2}dx}$$


    if not, then why not?..
     
  2. jcsd
  3. Feb 4, 2014 #2

    strangerep

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    No, you can't split the exponential in that way. This is due to basic properties of the exponential function.

    Alternatively, you could perform a change of variable $$x \to x' = x \sqrt{2am/\hbar} ~.$$
    Maybe you first try to do the "easier" integral $$\int e^{-x^2} dx$$ (though perhaps this will still be quite difficult since you're apparently unfamiliar with the properties of the exponential function).
     
  4. Feb 4, 2014 #3

    SteamKing

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    Even QM can't get around the math.
     
  5. Feb 4, 2014 #4
    Since that integral is non-trivial unless you know the trick, you may want to read up on it at http://en.wikipedia.org/wiki/Gaussian_integral. It's well worth getting very comfortable with this type of integral, too, as it comes up again and again in QM. There's a reason for the old saying that the only integral a theoretical physicist knows how to do is a Gaussian.
     
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