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Quantum mechanics, free particle normalization question

  1. Nov 23, 2013 #1
    1. The problem statement, all variables and given/known data

    A free particle has the initial wave function

    ψ(x,0)=Ae^(-a|x|)

    Where A and a are positive real constants.

    a) Normalize ψ(x,0)



    2. Relevant equations

    1= ∫|ψ|^2 dx

    3. The attempt at a solution

    I attempted to normalize using 1= ∫|ψ|^2 dx from -∞ to ∞. When doing this i obtained

    1=(A^2)∫e^(-2a|x|) dx from -∞ to ∞. doing this integral between these limits i get 0 as the value of the integral, which is obviously wrong.

    I looked up the answer online as this is a problem from Griffiths, introduction to quantum mechanics. (problem 2.21 in the link)

    http://www.thebestfriend.org/wp-content/uploads/IntroductiontoQuantumMechanics2thEdition.pdf

    The solution on there says you integrate from 0 to ∞, this is then my question, why do you integrate from 0 to ∞ instead of from -∞ to ∞?

    Thanks, any help would be appreciated.

    John
     
  2. jcsd
  3. Nov 23, 2013 #2
    Last edited: Nov 23, 2013
  4. Nov 23, 2013 #3
    I think I get it, is it like the integral of x^3 from -1 to 1 = 0? I'm guessing that's why the factor of 2 appears in the model answer. Anyway thank you, I believe I get it now.
     
  5. Nov 24, 2013 #4
    Wait! :eek:

    It's not a divergent integral - notice the absolute value sign in the exponent.

    Since it's an even function that's being integrated, you can integrate from [itex]0[/itex] to [itex]\infty[/itex] and then double the answer.
     
  6. Nov 25, 2013 #5

    vela

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    You didn't evaluate the integral correctly if you got 0. You shouldn't get different answers which depend on the method you choose to integrate. If you're getting inconsistent results, it means you're making a mistake.
     
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