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Linear superposition of solutions is a solution of TDSE

  1. Jan 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider two normalised, orthogonal solutions of the TDSE

    (Note all my h's here are meant to be h-bar, I'm not sure how to get a bar through them).

    [itex]\Psi_1 = \psi_1 (x) e^{-E_1 it/h}[/itex]

    [itex]\Psi_2 = \psi_2 (x) e^{-E_2 it/h}[/itex]

    Consider the wavefunction

    [itex] \Phi = \sqrt{\frac{1}{3}}\Psi_1 + \sqrt{\frac{2}{3}}\Psi_2 [/itex]

    Which is also a normalised solution to the TDSE. Any linear superposition of solutions to the TDSE is also a solution.

    2. Relevant equations

    The TDSE is;

    [itex]\widehat{H}\Phi = ih\frac{\delta}{\delta t}\Phi[/itex]

    3. The attempt at a solution

    This isn't a question, I've just gone to pursue the statement that the linear superposition is also a solution. I can't see to show it though.

    Using [itex] \Phi = \sqrt{\frac{1}{3}}\Psi_1 + \sqrt{\frac{2}{3}}\Psi_2 [/itex]

    [itex]\widehat{H}\Phi = ih\frac{\delta}{\delta t}\Phi = \sqrt{\frac{1}{3}}E_1\psi_1 (x) e^{-E_1 it/h} + \sqrt{\frac{2}{3}}E_2\psi_2 (x) e^{-E_2 it/h}[/itex]

    Which I cannot manage to get in to the form

    [itex] (\sqrt{\frac{1}{3}}E_1 + \sqrt{\frac{2}{3}}E_2)\Phi [/itex]

    Which is the form is should be in if it was a solution surely?
     
  2. jcsd
  3. Jan 12, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    Any linear combination of two solutions is also a solution of the TDSE. But the linear combination of two eigenfunctions is not an eigenfunction if functions belonging to different energies are involved.

    Show that Φ is a solution by substituting into the TDSE.

    Show that it is normalized.

    ehild
     
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