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##.(adsbygoogle = window.adsbygoogle || []).push({});

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##?

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# Dirac spinors in non-relativistic limit

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