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useful formulae

$\displaystyle \begin{array}{lc} E(p)&=\qquad\displaystyle\sqrt{\left(\frac{\hba... ...(\frac{p}{mc}\right)\sqrt{1+\frac{1}{4}\left(\frac{p}{mc}\right)^2} \end{array}$

(5.5)

$\displaystyle L_p = \frac{1}{mc^2} \left(E(p)-\frac{p^2}{2m} -mc^2\right)$

(5.6)

$\displaystyle \begin{array}{lc} L_p^2&\quad\displaystyle =\frac{2E(p)}{(mc^2)^2... ...\frac{2E(p)}{(mc^2)^2} \left(E(p)-\sqrt{E(p)^2+(mc^2)^2}\right) + 1 \end{array}$

(5.7)

$\displaystyle \begin{array}{lc} u_k^2&\quad\displaystyle = \frac{1}{1-L_p^2} = ... ...2} = \frac{(mc^2)^2}{2E(p)\left(\sqrt{E^2(k)+(mc^2)^2}-E(p)\right)} \end{array}$

(5.8)

$\displaystyle \begin{array}{lc} v_k^2&\quad\displaystyle = \frac{L_p^2}{1-L_p^2... ...2} = \frac{(mc^2)^2}{2E(p)\left(\sqrt{E^2(k)+(mc^2)^2}+E(p)\right)} \end{array}$

(5.9)

$\displaystyle u_k v_k = \frac{L_p}{1-L_p^2} =-\frac{mc^2}{2E(p)}$

(5.10)

$\displaystyle \frac{1+L_p}{1-L_p} = \frac{u_k+v_k}{u_k-v_k} = \frac{\hbar^2k^2}{2mE(p)}$

(5.11)