(Solved): Consider the state of a spin-up electron moving along the z-axis u(1)(p)=(E+mc2)/c0 ...

Consider the state of a spin-up electron moving along the $z$-axis $u^{(1)}(p)=?\begin{array}{c}\sqrt{\left(E+m{c}^2\right)/c}\\ 0\\ \sqrt{\left(E?m{c}^2\right)/c}\\ 0\end{array}?$ Calculate expressions for the following chiral-projected states: i. $u_L^{(1)}=P_L{u}^{(1)}$, and ii. $u_R^{(1)}=P_R{u}^{(1)}$. Furthermore, show that in the ultra-relativistic limit $E>>m{c}^2,u_L^{(1)}$ becomes zero, and $u_R^{(1)}$ and $u^{(1)}$ become equal. The concept: the quantities $u_L$ and $u_R$ are defined to be the left- and righthanded "chiral" parts of the electron wavefunction, but the analogy to left-and right-handed helicity eigenstates only becomes exact for $E>>m{c}^2$.