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Solutions of the free one-particle Klein Gordon equation

  1. Jan 17, 2013 #1
    In the book "Wachter, relativistic quantum mechanics", in page 5, the KG eq. is introduced as follows:

    [tex] -\hbar^2 \frac{\partial^2 \phi(x)}{\partial t^2} = (-c^2 \hbar^2 \nabla^2 + m^2_0 c^4) \phi(x). [/tex]

    Now I tried to solve this equation using the separation ansatz (product ansatz).
    I get:

    [tex] \phi (ct) = exp [\pm \frac{ i p_0 ct}{\hbar} ], [/tex]
    and
    [tex] \psi (x) = exp[\pm \frac{i \vec{p} \cdot \vec{x}}{\hbar}]. [/tex]
    Where the usual relativistic four-momentum conservation equation holds.
    Now, this amounts to four combinations of solutions. But in the book, only two are written, namely:

    [tex] \phi^{(1)}_p= e^{-i (c p_0 t - \vec{p} \cdot \vec{x})/ \hbar},[/tex]

    [tex] \phi^{(2)}_p= e^{+i (c p_0 t - \vec{p} \cdot \vec{x})/ \hbar}.[/tex]

    I really, really want to solve the equation; I feel a little bit frustrated because I don't get only the two solutions. Basically I am halfway there but I don't know what to do next.

    So any help would be greatly appreciated!!
     
  2. jcsd
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