Summary of the rotation matrices
We would like now to bring together in one place the various things we have learned about the rotations for particles of spin one-half and spin one—so they will be convenient for future reference. On the next page you will find tables of the two rotation matrices $R_z(\phi)$ and $R_y(\theta)$ for spin one-half particles, for spin-one particles, and for photons (spin-one particles with zero rest mass). For each spin we will give the terms of the matrix $\bracket{j}{R}{i}$ for rotations about the $z$-axis or the $y$-axis. They are, of course, exactly equivalent to the amplitudes like $\braket{+T}{\OS}$ we have used in earlier chapters. We mean by $R_z(\phi)$ that the state is projected into a new coordinate system which is rotated through the angle $\phi$ about the $z$-axis—using always the right-hand rule to define the positive sense of the rotation. By $R_y(\theta)$ we mean that the reference axes are rotated by the angle $\theta$ about the $y$-axis. Knowing these two rotations, you can, of course, work out any arbitrary rotation. As usual, we write the matrix elements so that the state on the left is a base state of the new (rotated) frame and the state on the right is a base state of the old (unrotated) frame. You can interpret the entries in the tables in many ways. For instance, the entry $e^{-i\phi/2}$ in Table 17–1 means that the matrix element $\bracket{-}{R}{-}=e^{-i\phi/2}$. It also means that $\Rop\,\ket{-}=e^{-i\phi/2}\,\ket{-}$, or that $\bra{-}\,\Rop=\bra{-}\,e^{-i\phi/2}$.  It’s all the same thing. 