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Showing that the Schrödinger equation implies the de Broglie relation when PE=0

  1. Nov 1, 2012 #1
    Hello,

    In the book physical chemistry (P. Atkins & Julio de Paula, 2009, 5 ED) the authors derive a justification of the Schrödinger equation.

    1.) [tex]\frac{-\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}}+V(x)\psi=E \psi[/tex]

    The derivation goes as follows:

    Derivation:
    We can justify the form of the Schrödinger equation to a certain extent by showing that it implies the de Broglie relation for a freely moving particle.
    By free motion we mean motion in a region where the potential energy is zero (V=0 everywhere).

    If V=0, equation 1 simplifies to:

    2.) [tex]\frac{-\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}}=E \psi[/tex]

    So far all good, however they then present a solution to equation 2. without showing how they obtained it.

    The solution is:

    [tex]\psi=sin(kx)[/tex]
    [tex]k=\frac{(2mE)^{2}}{\hbar}[/tex]

    I have no problem understanding that this is a valid solution however i would like to derive it myself.

    Could you provide me with the derivation to the solution of equation 2?

    /Thanks in advance,

    Linus.
     
  2. jcsd
  3. Nov 1, 2012 #2
    I think you can find it here:

    http://www.cliffsnotes.com/study_guide/Constant-Coefficients.topicArticleId-19736,articleId-19720.html [Broken]
     
    Last edited by a moderator: May 6, 2017
  4. Nov 1, 2012 #3
    :confused: :smile:
     
  5. Nov 1, 2012 #4
    Lol, yes I can see how what I wrote is contradicting. That is the kind of help that I was looking for actually.

    /Thank you torquil :)
     
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