Dirac spinors in non-relativistic limit

[Thomas Brady]

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 doesn't 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 $\vec \sigma \cdot \vec v$?

 

[Thomas Brady]

P.S. I used n.u. for this

 

[TSny]

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.