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Probability Density

  1. Feb 16, 2010 #1
    Hello I am trying to find the probability density function for a particle in a potential well of a harmonic oscillator. (My question is about complex conjugates).

    I know the formula. I have to multiply [tex]\Psi^{*} (x, t) \Psi (x, t)[/tex]

    The wave function is a linear combination of stationary states, i.e.

    [tex]\Psi (x, t) = \frac{1}{\sqrt{2}} [ \psi_{0} (x) e^{\frac{-i \omega_{0} t}{2}} + \psi_{1} (x) e^{\frac{- 3i \omega_{0} t}{2}} ][/tex]

    [tex] \psi_{0} (x)[/tex]and [tex] \psi_{1} (x)[/tex] are real

    the conjugates of [tex] \psi_{0} (x)[/tex]and [tex] \psi_{1} (x)[/tex] are

    [tex] \psi_{0}^{*} (x)[/tex]and [tex] \psi_{1}^{*} (x)[/tex] but since the eigenfunctions are real (are the conjugates the same as the eigenfunctions), what happens when I multiply them together?

    I thought that, for example,
    [tex] \psi_{0}^{*} (x) \psi_{0} (x) = \psi_{0} (x) \psi_{0} (x)= |\psi_{0} (x) |^{2}[/tex]
  2. jcsd
  3. Feb 16, 2010 #2


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    Science Advisor

    Yes, that is correct, so the problem boils down to the correct handling of the time-dependent complex phases ...
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