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Normalizing the wave function of a free particle

  1. Apr 19, 2007 #1
    Hello!

    Can somebody tell me, how it is possible to normalize the wave function of a free particle using the Dirac delta function?

    Thanks!
     
  2. jcsd
  3. Apr 19, 2007 #2

    Meir Achuz

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    There are two methods that are commonly used:
    1. Box normalization. Space is assumed to be contained in an LXLXL box.
    After calculating, say, a scattering amplitude, taking the limit L-->\infty
    gives a ifntie result if done carefully.

    2. Delta function normalization <x|x'>=\delta(x-x')/(2\pi)^{3/2}.
     
  4. Apr 24, 2007 #3

    How can i use the 2nd one this in this case? I have the wave function: psi(x) = A*e^ikx + B*e^-ikx with k = sqrt(2mE/h^2).
    I think I can take A = 1, but then i don't know how to continue.
     
  5. Apr 24, 2007 #4
    normailization is simple.
    u have the wavefunction, all u do is square it and integrate, setting equal to 1
    so in ur case, int[-inf to inf] A*e^ikx=A^2*e^2ikx=1
    pull A^2 from the integral to get A^2 int[-inf to inf]e^2*ikxdx=1 for the first
    1/A^2
     
  6. Apr 24, 2007 #5
    Thx, but I know how normalization normally works :rolleyes:

    But not in this case: Free Particle and I HAVE TO use the DELTA FUNCTION.
     
  7. Apr 24, 2007 #6
    Okay, so let me ask you what [tex]\int_{-\infty}^{\infty} dx e^{\imath (k - k') x}[/tex] is. Once you figure that one out, I think you could probably normalize the wave function pretty well.
     
  8. Apr 24, 2007 #7

    Meir Achuz

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    What are you going to do with the wave function. If you are going to calculate reflection and transmission coefficients, you odn't have to normalize it.
     
  9. Apr 24, 2007 #8
    I only want to know how I can normalize it using the Dirac delta function. That is all.
    I never said that i want to calculate the reflection or transmission coefficient.
     
  10. Apr 25, 2007 #9

    Meir Achuz

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    Your \int |psi|^2 will have four terms. Four each term use
    \int exp{ikx-ik'x}=(2pi)^3\delta(k-k').
     
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