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B How to Derive Wave Function

  1. Sep 3, 2016 #1
    Everybody knows what is the Wave Function is.
    $$\Psi=\space e^{i(kx-\omega t)}$$
    $$\Psi=\space cos{(kx-\omega t)} \space - \space isin{(kx-\omega t)}$$

    But can anyone tell me how it is derived. Since Schrodinger Equation is derived so easily using this Wave Function. I think it is necessary to understand how the Wave Function is derived.
  2. jcsd
  3. Sep 3, 2016 #2


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    The wave function of what? You have just given the wave function for a free particle. In many other cases the wave function will be completely different and behave different wrt time.
  4. Sep 3, 2016 #3
    This is the wave function I found when proving the Schrodinger equation. I think this is the wave function of the electron in Hydrogen atom.
  5. Sep 3, 2016 #4
    No, it's definitely not a wave function of electron in hydrogen. You need to read more carefully your sources.
  6. Sep 3, 2016 #5
    Alright, let us think this is a Wave function of a particle for the time being, can you tell me how it is derived.
  7. Sep 3, 2016 #6


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    Classically, it is the solution to the EOM of the simple harmonic oscillator.
  8. Sep 3, 2016 #7


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    It's the other way around. One derives the wave function for a particular system (e.g. a free particle) by solving Schrödinger's equation for that system.

    Many introductory textbooks "justify" the Schrödinger equation or "motivate" it or "make it plausible" by assuming that a free particle with a definite momentum must be represented by a simple harmonic wave ##\Psi(x,t) = Ae^{i(kx - \omega t)}##, but fundamentally, the Schrödinger equation comes first. Many or most elementary treaments of QM simply present the SE as a fundamental assumption of the theory.

    (I predict that we will now have a long debate about what is the "real" logical starting point for non-relativistic quantum mechanics. :-p)
    Last edited: Sep 3, 2016
  9. Sep 3, 2016 #8
    I think the best derivation is founded on Universty Physics With Modern Physics by Sears and Zemansky. I did not find any derivation even in more advanced textbooks on QM, because they usually apresents the wave function only as an postulate.
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