(Solved): EXERCISE 5.3 Calculate the =+21 helicity eigenspinor of an electron of momentum p=(psin,0 ...

EXERCISE 5.3 Calculate the $\lambda =+\frac{1}{2}$ helicity eigenspinor of an electron of momentum ${\mathbf{p}}^{\prime }=(p\sin \theta ,0,p\cos \theta )$. EXERCISE 5.4 Confirm the desired result that the Dirac equation describes "intrinsic" angular momentum ( $\equiv$ spin)- $\frac{1}{2}$ particles. Hint We are clearly interested in angular momentum, so first we should explore the commutation of the orbital angular momentum $\mathbf{L}=\mathbf{r}\times \mathbf{P}$ with the Hamiltonian. Use $\left[x_i,P_j\right]=i{\delta }_{ij}$ to show that $[H,\mathbf{L}]=-i(\mathbf{\alpha }\times \mathbf{P})\text{. }$ So $\mathbf{L}$ is not conserved! There must be some other angular momentum. Show that $[H,\mathbf{\Sigma }]=+2i(\mathbf{\alpha }\times \mathbf{P}),\text{ where }\mathbf{\Sigma }\equiv \left(\begin{array}{ll}\mathbf{\sigma }& 0\\ 0& \mathbf{\sigma }\end{array}\right).$ Clearly, neither $\mathbf{L}$ nor $\mathbf{\Sigma }$ are conserved. The combination $\mathbf{J}=\mathbf{L}+\frac{1}{2}\mathbf{\Sigma },$ which is nothing other than the total angular momentum, is however conserved, as now $[H,\mathbf{J}]=0.$ The eigenvalues of $\frac{1}{2}\mathbf{\Sigma }$ are $\pm \frac{1}{2}$.