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Combination of wavefunctions in oscillator

  1. Mar 13, 2014 #1
    1. The problem statement, all variables and given/known data
    The equation [itex]\psi(x) = \frac{1}{sqrt(2)}\psi_0 (x) + \frac{i}{sqrt(5)}\psi_1 (x) + \gamma\psi_2 (x)[/itex]

    is a combination of the first three eigenfunctions in the 1D harmonic oscillator. So, [itex]\psi_0 = Ae^{-mωx^2 /2\hbar}[/itex] and so on for the first and second excited states. If [itex]\psi_0[/itex], [itex]\psi_1[/itex] and [itex]\psi_2[/itex] are normalised, and [itex]\psi(x)[/itex] is also normalised, determine [itex]|\gamma|[/itex]



    3. The attempt at a solution

    You can obtain the normalised functions for the ground state and first two excited states from a variety of methods, and you can then expand out [itex]\psi(x)[/itex]. I tried plugging that back into the time dependent Schrodinger equation, but that didn't help (and it also gave a messy derivative) so im at a loss as to how i can proceed.
     
  2. jcsd
  3. Mar 13, 2014 #2

    DrClaude

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    Staff: Mentor

    It is stated that ##\psi_0##, ##\psi_1##, and ##\psi_2## are taken to be normalized.

    How is that realted to normalization?

    Let's start from the beginning: what equation does ##\psi(x)## statisfy if it is normalized?
     
  4. Mar 13, 2014 #3
    Hmm it satisfies [itex]\int^{∞}_{-∞} \psi^{*}\psi dx = 1[/itex] I think
     
  5. Mar 13, 2014 #4

    DrClaude

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    Staff: Mentor

    Correct. So plug in there the ##\psi(x)## of the problem. Keep the notation ##\psi_0## and so on (i.e., do not write the explicit functions of ##x##) and use the properties of the harmonic oscillator wave functions to simplify the result.
     
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