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Dirac spinors in non-relativistic limit

  1. May 4, 2017 #1
    So, I have to show that in the non-relativistic limit the lower two components of the positive energy solutions to the Dirac equation are smaller than the upper two components by a factor of ##\beta##.

    I started with the spinor $$\psi = \begin{pmatrix} \phi \\ \frac {\vec \sigma \cdot \vec p} {E + m} \phi \end{pmatrix}$$ (##\phi## is a 2-component spinor and this doesnt include the normalization factor or the exponential)
    The ##\sigma## being the Pauli matrices. Then I noted that in the non-relativistic limit ##E = \gamma m## and ##\gamma \rightarrow 1## so the denominator of the lower component is ##2m## and ##\vec p = m\vec v## in the non-relativistic limit so the m's cancel in the numerator and the denominator and I'm left with

    $$\psi = \begin{pmatrix} \phi \\ \frac {\vec \sigma \cdot \vec v} {2} \phi \end{pmatrix}$$

    so now I'm confused as to what to do with the ##\vec \sigma \cdot \vec v##. It seems like what I could try to do would leave the lower two components a factor of ##\frac \beta 2## smaller than the numerator.rather than just ##\beta##. So how do I approach this ##\vec \sigma \cdot \vec v##?
     
  2. jcsd
  3. May 4, 2017 #2
    P.S. I used n.u. for this
     
  4. May 4, 2017 #3

    TSny

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    I think that the question is meant to ask you to show that the lower two components are smaller than the upper two components by a factor of order ##\beta## (even though it wasn't stated that way explicitly.) So, you don't need to distinguish between a factor of ##\beta## and a factor of ##\beta/2##. Hope I'm not misinterpreting things here.
     
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