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PDE - Boundary value problem found in QM

  1. Sep 3, 2013 #1
    This is a quantum mechanics problem, but the problem itself is reduced (naturally) to a differential equations problem.

    I have to solve the following equation:
    [tex]\frac{\partial}{\partial t}\psi (x,t) = i\sigma \psi (x,t)[/tex]
    where [itex]\sigma > 0[/itex]
    The initial condition is:
    [tex]\psi (x,0) = \sqrt{\frac{2}{L_0}}\sin \left(\frac{n\pi x}{L_0}\right)[/tex]
    and the boundary conditions are:
    [tex]\psi (0,t) = \psi(L(t),t) = 0[/tex]
    where [itex]L(t)[/itex] is a smooth function with [itex]L(0)=L_0[/itex].
    I don't even know how to begin :uhh:. Any ideas?
  2. jcsd
  3. Sep 3, 2013 #2
    $$\frac{\partial}{\partial t}[e^{i\sigma t}]=i\sigma e^{i\sigma t}.$$
  4. Sep 4, 2013 #3
    Oh, wait I forgot to put the space derivative (it was 1-2 AM so my cognitive functions were down to minimum) :shy:

    The correct equation is:

    [tex]\frac{\partial}{\partial t}\psi(x,t) = i\sigma \frac{\partial^2}{\partial x^2}\psi(x,t)[/tex]
  5. Sep 4, 2013 #4
    My hint still helps in this case. :tongue:

    What might you think be the first step?
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