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Math Help - Eigenenergy and Eigenfunctions

  1. #1
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    Eigenenergy and Eigenfunctions

    I'm trying to write a normalised wavefunction at time t, given the normalised wavefunction at t=0. I've found out that first I should write the initial wavefunction as a sum of the energy eigenfunctions, and then as time evolves, each term gets multiplied by e^(-i*E_n*t/hbar) where E_n is the eigenenergy of the eigenfunction.

    Given that the eigenfunction is

    sqrt(2/L)*sin(n*Pi*x/L)

    how do I find the eigenenergy?
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    Quote Originally Posted by Unoticed View Post
    I'm trying to write a normalised wavefunction at time t, given the normalised wavefunction at t=0. I've found out that first I should write the initial wavefunction as a sum of the energy eigenfunctions, and then as time evolves, each term gets multiplied by e^(-i*E_n*t/hbar) where E_n is the eigenenergy of the eigenfunction.

    Given that the eigenfunction is

    sqrt(2/L)*sin(n*Pi*x/L)

    how do I find the eigenenergy?
    The average energy of any state can be determined by < H >. If you have an eigenstate then this returns the eigenvalue:
    E_n = < H > = \int \psi ^* \left [ \left ( -\frac{\hbar ^2}{2m} \right ) \left ( \frac{d^2}{dx^2} \right ) + V \right ] \psi~dx

    -Dan
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