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Energy eigenvalue for particle in a box

  1. Sep 24, 2006 #1
    Hello all, I'm stuck on this question, and I would appericate if someone can tell me how to start cracking the problem.

    I have a infinite square well, and is given a wavefunction that exist inside the well. The problem is to find the probability that a measurement of the energy will result in a certain given energy eigenvalue. Also I have to find teh mean energy.


    Thanks in advance.
     
  2. jcsd
  3. Sep 24, 2006 #2

    jtbell

    User Avatar

    Staff: Mentor

    The wavefunction that you're given is not one of the eigenfunctions of the inifinite square well, right?

    In general, any wave function that satisfies the boundary conditions for a physical situation (such as the infinite square well) can be written as a linear combination of the energy eigenfunctions:

    [tex]\psi = a_1 \psi_1 + a_2 \psi_2 + a_3 \psi_3 + ...[/tex]

    where [itex]\psi_1[/itex] has energy eigenvalue [itex]E_1[/itex], and [itex]a^*_1 a_1[/itex] gives the probability that the particle has energy [itex]E_1[/itex]. So your problem is to find out what the [itex]a_k[/itex] are, or at least some of them.

    The energy eigenfunctions are orthogonal, which means that

    [tex]\int {\psi^*_j \psi_k dx} = 0[/tex]

    whenever [itex]j \ne k[/itex]. Suppose that the eigenfunctions are also normalized so that

    [tex]\int {\psi^*_k \psi_k dx} = 1[/tex]

    for any k. Take the first equation above and multiply through by (say) [itex]\psi^*_2[/itex]:

    [tex]\psi^*_2 \psi = a_1 \psi^*_2 \psi_1 + a_2 \psi^*_2 \psi_2 + a_3 \psi^*_2 \psi_3 + ...[/tex]

    Then integrate both sides. Look at what happens to the integrals on the right side, and you should see how to calculate [itex]a_2[/itex], and how to generalize this method to any value of k.
     
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