1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    gabbagabbahey

    User Avatar
    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..?
    https://www.physicsforums.com/showthread.php?p=2218109#post2218109
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proving solutions to the schrodinger equation
Loading...