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Homework Help: Solutions to Schrodinger's Wave Equation

  1. Sep 5, 2007 #1
    1. The problem statement, all variables and given/known data
    Assume that [tex] \psi_{1}(x,t) [/tex] and [tex] \psi_{2}(x,t) [/tex] are solutions of the one-dimensional time-dependent Schrodinger's wave equations.
    (a) Show that [tex] \psi_{1} + \psi_{2} [/tex] is a solution.

    (b) Is [tex] \psi_{1} \cdot \psi_{2} [/tex] a solution of the Schrodinger's equation in general?



    2. Relevant equations
    Is this the "One-Dimensional Time-Dependent Schodinger's Wave Equation":
    [tex] \eta = \imath \hbar \cdot \frac{1}{\phi(t)} \cdot \frac{\partial \phi(t)}{ \partial t}[/tex]

    If so, it says in my book that the solution is [tex] \phi(t) = e^{- \imath (\frac{E}{\hbar})t [/tex]



    3. The attempt at a solution
    I have a feeling that all I have to do is show that these solutions are linear, then use the superposition technique.
     
    Last edited: Sep 5, 2007
  2. jcsd
  3. Sep 5, 2007 #2

    Avodyne

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    Your relevant equation is *half* of the time-dependent Schrodinger equation in the special case that there is no potential energy, and *after* separation of variables has been performed in x and t. (The other half involves x, and not t.)

    Yes, linearity and superposition is the key point.
     
  4. Sep 5, 2007 #3
    So I don't really even need to know what the solutions are? All I need to do is some sort of "proof" that the sum of the two solutions to the linear P.D.E. is also a solution?

    If that is the case, do you think you could help me get started with working that out?
     
  5. Sep 5, 2007 #4
    Could you please write the full time-dependent schodinger equation?
     
  6. Sep 5, 2007 #5
    Anyone?
     
  7. Sep 5, 2007 #6

    Avodyne

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    In one space dimension, the full time-dependent Schrodinger equation is

    [tex]i\hbar{\partial\over\partial t}\psi(x,t) = \left[-{\hbar^2\over2m}{\partial^2\over\partial x^2}+V(x)\right]\psi(x,t)[/tex]

    Edit: the derivative on the right-hand side is wrt x, now fixed and correct.
     
    Last edited: Sep 5, 2007
  8. Sep 5, 2007 #7
    So how do I show that [tex] \psi_{1}(x,t) [/tex] and [tex] \psi_{2}(x,t) [/tex] have linearity and superposition can be used to create a third solution?
     
  9. Sep 5, 2007 #8

    Avodyne

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    You know that [tex] \psi_{1}(x,t) [/tex] and [tex] \psi_{2}(x,t) [/tex] obey this equation. You want to show that [tex] \psi_{1}(x,t) + \psi_{2}(x,t) [/tex] does as well. So, plug [tex] \psi_{1}(x,t) + \psi_{2}(x,t) [/tex] into the equation. Can you used what you know to show that the result is true?
     
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