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Basic wave equation question

  1. Jul 30, 2008 #1
    This is a fairly simple question, but the first such question I have done. Inorder to check my work I was hoping somone could show me how to normalize the following.

    [tex]\Psi(x,t) = Ae^{-a[(mx^{2}/\hbar)+it][/tex]
    where m is the particles mass

    And also that the expectation values of x and x2 would be.

    Don't wory, this is not for a class, I am studying this on my own
  2. jcsd
  3. Jul 30, 2008 #2


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    Science Advisor

    Psi^2 leads to a Gaussian integral, which is done by completing the square in the exponent.
    <x> is zero bly symmetry.
    <x^2>is found by integrating by parts.
  4. Jul 31, 2008 #3
    After playing with this I found

    [tex]A = \sqrt[4]{\frac{2am}{\hbar\pi}}*e^{ait}[/tex]


    [tex]\Psi = \sqrt[4]{\frac{2am}{\hbar\pi}}*e^{-amx^{2}/\hbar}[/tex]

    Can annyone confirm this for me because I am realy uncomfortable with my answer.
  5. Jul 31, 2008 #4
    that looks alright, but have you lost your time component along the way? when calculating [tex]\left|A\right|^{2}[/tex] the time-dependence drops off, but you need to be sure to attach your value for [tex]A[/tex] to the full wavefunction. i think it should look like this? [tex]\Psi\left(x,t\right)=\left(2ma/\pi\hbar\right)^{1/4}e^{-amx^{2}/\hbar}e^{-iat}[/tex]
  6. Jul 31, 2008 #5
    the only think was that [tex]e^{iat}[/tex] from the second part of A canceld with [tex]e^{-iat}[/tex] from the wave function, or is that wrong.
  7. Jul 31, 2008 #6
    You would be correct, but technically you're [tex]A[/tex] is wrong. The [tex]e^{-iat}[/tex] term cancels with its conjugate in the process of calculating [tex]A[/tex] through normalization. [tex]A[/tex] should be just [tex]\left(2ma/\pi\hbar\right)^{1/4}[/tex]
  8. Jul 31, 2008 #7
    Looking back my mistake was simply squaring the wave function without taking the modulus first
  9. Aug 7, 2008 #8
    I was working with this a little more, and came up with a corisponding potential energy function of:

    V(x) = [tex]2a^{2}mx^{2}[/tex]

    Could anyone run it and verify that I have this right (My text has no answer key)?

    Here is the wave function again.
    [tex]\Psi(x,t) = Ae^{-a[(mx^{2}/\hbar)+it][/tex]
    where m is the particles mass
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