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Homework Help: Proving solutions to the schrodinger equation

  1. Feb 17, 2009 #1
    1. The problem statement, all variables and given/known data

    Verify that the following are not solutions to the Schrodinger equation for a free particle:

    (a) [tex]\Psi(x,t) = A*Cos(kx-\omega t)[/tex]

    (b) [tex]\Psi(x,t) = A*Sin(kx-\omega t)[/tex]

    2. Relevant equations

    Schrodinger equation: [tex]\frac{-hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} = \frac{i*hbar*\partial \Psi}{\partial t}[/tex]

    3. The attempt at a solution

    For part a:

    Let [tex]y=kx-\omega t[/tex]

    Calculating the derivatives gives...

    [tex]\frac{\partial^2 \Psi}{\partial x^2}=-Ak^2*Cos y[/tex]
    [tex]\frac{\partial \Psi}{\partial t} = A*\omega*Sin y[/tex]

    Substituting and rearranging, we see that:

    [tex]Sin y = \frac{hbar * k^2}{2mi\omega}Cos y[/tex]

    Let [tex]f=\frac{hbar * k^2}{2mi\omega}[/tex]

    Then [tex]Sin y = f Cos y[/tex]

    The equality holds when [tex]y=\pm ArcCos(\pm\frac{1}{\sqrt{1+f^2}})[/tex]

    If we substitute back in...

    [tex]kx-\omega t = \pm ArcCos(\pm\frac{1}{\sqrt{1+\frac{hbar^2k^4}{4m^2\omega^2}}})[/tex]

    If I assign some values to omega, m, hbar, and k, I can graph x as a function of t for one of the cases. It shows up as a straight line with a positive slope.

    This does not seem to prove that (a) is not a solution to Schrodinger's equation. I think I may be making this more complicated than it is. How should I approach this problem differently?
  2. jcsd
  3. Feb 18, 2009 #2


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    Homework Helper
    Gold Member

    This right here should tell you that [itex]\Psi(x,t) = A*Cos(kx-\omega t)[/itex] is not a solution to the schrodinger equation. If it were, you would expect it to satisfy the schodinger equation everywhere (i.e. for all values of y)--- which it clearly doesn't.
  4. Feb 18, 2009 #3
  5. May 31, 2009 #4
    a faster way would be to express cos(kx-[tex]\omega[/tex] t) in terms of ei(kx-[tex]\omega[/tex] t)
    and substitute in the time dependant equation both will never satisfy it(coz the wavefunc. is of the form Aei(kx-[tex]\omega[/tex] t )
    any ideas on this one mate..?
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