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Time development wave of state

  1. Nov 1, 2009 #1
    1. The problem statement, all variables and given/known data

    free particle of mass m moving in 1d
    state: [tex]\Psi(x,0) = sin(k_{0}x)[/tex]

    2. Relevant equations


    [tex]\Psi(x,t) = \stackrel{1}{\overline{\sqrt{2\pi}}}\overline{}[/tex][tex]\int^{\infty}_{-\infty}b(k)e^{i(kx-\omega t)} [/tex]

    3. The attempt at a solution

    b(k)=[tex]\stackrel{1}{\overline{\sqrt{2\pi}}}\overline{}[/tex][tex]\int^{\infty}_{-\infty}sin(k_{0}x)e^{-ikx}[/tex]
     
  2. jcsd
  3. Nov 1, 2009 #2

    jambaugh

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    Science Advisor
    Gold Member

    The problem here is that you have sine instead of [tex]e^{ikx}[/tex] factor.
    Use the fact that:
    [tex]sin(kx) = \frac{1}{2i}(e^{ikx}-e^{-ikx})[/tex]

    Second your answer should be:
    [tex]\Psi(x,t) = ... [/tex]
    not
    [tex]b(k) = ...[/tex]
    Look up the source of your "relevant equation" to see what role [tex]b(k)[/tex] plays....also you should give the variable of integration which appears to be k.

    Your attempted solution is I suppose integrated over x? The integral you give has a known solution in terms of Dirac delta functions (which relates to my point above)

    Remember ultimately you are looking for a solution to the Schrodinger equation which a.) is properly normalized and b.) satisfies the initial condition given.
     
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