Wave functions

  1. Oct 10, 2008 #1
    Why are wave functions, e.g., Schrodinger's, based on the complex exponential function (e[tex]^{}ix[/tex]) and not trigonometric functions (sine or cosine)?

    See Euler's formula for their relationship: http://en.wikipedia.org/wiki/Euler's_formula

    Furthermore, by using the complex exponential function, the probability amplitude becomes a complex-valued function (a + bi). Were sine or cosine used, the probability amplitude of the wave function would not be a complex-valued function. Is there a reason that the probability amplitude should be a complex-valued function?
     
    Last edited: Oct 10, 2008
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  3. Oct 10, 2008 #2

    Ben Niehoff

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    You need complex-valued wavefunctions so that you can have standing waves of constant magnitude over time (such as in a potential well). Real-valued functions would have to oscillate in magnitude.
     
  4. Oct 10, 2008 #3
    What do you mean by magnitude? Is it the same as amplitude?
     
  5. Oct 10, 2008 #4
    Nothing keeps you from using sine and cos. That what Euler's identity says.

    The time evolution of a state vector/ wave function must conserve the norm of the state vector, and it always has to be one.

    Note e[tex]^{it}[/tex] e[tex]^{it}[/tex]=1.
     
  6. Oct 10, 2008 #5
    My post https://www.physicsforums.com/showpost.php?p=1539835&postcount=20 and some other posts in that thread may be relevant.
     
  7. Oct 10, 2008 #6

    dx

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    The simple answer is that standard quantum mechanics as we understand it requires a complex wavefunction [tex] \psi [/tex] defined on the configuration space. You can replace this with a function to [tex] \mathbb{R} \times \mathbb{R} [/tex] and change the equations accordingly, since [tex] \mathbb{C} [/tex] and [tex] \mathbb{R} \times \mathbb{R} [/tex] are isomorphic. But you can't replace [tex] \psi [/tex] with a function to just [tex] \mathbb{R} [/tex].

    Also, Euler's formula doesn't turn a complex number into a real number. [tex] \cos \theta + i \sin \theta [/tex] is still a complex number, and the probability amplitude will still be a complex valued function.
     
  8. Oct 12, 2008 #7
    What properties of standard quantum mechanics require a complex wavefunction?
     
  9. Oct 12, 2008 #8

    malawi_glenn

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  10. Oct 12, 2008 #9

    dx

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    That's like asking what property of Newtonian physics requires the mechanical state of a particle to be position and momentum. It is possible to construct a theory where the time evolution of a particle depends only on initial position, but that's not the way nature is. It just happens to be so that the mechanical state of a quantum system needed to predict future probabilities is a complex function.
     
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