Arbitrary rotations
Now we can see how to do any angle at all. First, notice that any relative orientation of two coordinate frames can be described in terms of three angles, as shown in Fig. 6-9. If we have a set of axes $x'$, $y'$, and $z'$ oriented in any way at all with respect to $x$, $y$, and $z$, we can describe the relationship between the two frames by means of the three Euler angles $\alpha$, $\beta$, and $\gamma$, which define three successive rotations that will bring the $x$, $y$, $z$ frame into the $x'$, $y'$, $z'$ frame. Starting at $x$, $y$, $z$, we rotate our frame through the angle $\beta$ about the $z$-axis, bringing the $x$-axis to the line $x_1$. Then, we rotate by $\alpha$ about this temporary $x$-axis, to bring $z$ down to $z'$. Finally, a rotation about the new $z$-axis (that is, $z'$) by the angle $\gamma$ will bring the $x$-axis into $x'$ and the $y$-axis into $y'$.6 We know the transformations for each of the three rotations—they are given in (6.19) and (6.34). Combining them in the proper order, we get \begin{equation} \begin{alignedat}{3} C_+'&=\phantom{i}&\cos\frac{\alpha}{2}&e^{i(\beta+\gamma)/2}C_+\!+ i&\sin\frac{\alpha}{2}&e^{-i(\beta-\gamma)/2}C_-,\\ \\ C_-'&=i&\sin\frac{\alpha}{2}&e^{i(\beta-\gamma)/2}C_+\!+ \phantom{i}&\cos\frac{\alpha}{2}&e^{-i(\beta+\gamma)/2}C_-. \end{alignedat} \label{Eq:III:6:35} \end{equation}  So just starting from some assumptions about the properties of space, we have derived the amplitude transformation for any rotation at all. That means that if we know the amplitudes for any state of a spin one-half particle to go into the two beams of a Stern-Gerlach apparatus $S$, whose axes are $x$, $y$, and $z$, we can calculate what fraction would go into either beam of an apparatus $T$ with the axes $x'$, $y'$, and $z'$. In other words, if we have a state $\psi$ of a spin one-half particle, whose amplitudes are $C_+=\braket{+}{\psi}$ and $C_-=\braket{-}{\psi}$ to be “up” and “down” with respect to the $z$-axis of the $x$, $y$, $z$ frame, we also know the amplitudes $C_+'$ and $C_-'$ to be “up” and “down” with respect to the $z'$-axis of any other frame $x'$, $y'$, $z'$. The four coefficients in Eqs. (6.35) are the terms of the “transformation matrix” with which we can project the amplitudes of a spin one-half particle into any other coordinate system.  We will now work out a few examples to show you how it all works. Let’s take the following simple question. We put a spin one-half atom through a Stern-Gerlach apparatus that transmits only the $(+z)$ state. What is the amplitude that it will be in the $(+x)$ state?  The $+x$ axis is the same as the $+z'$ axis of a system rotated $90^\circ$ about the $y$-axis. For this problem, then, it is simplest to use Eqs. (6.32)—although you could, of course, use the complete equations of (6.35). Since $C_+=1$ and $C_-=0$, we get $C_+'=1/\sqrt{2}$. The probabilities are the absolute square of these amplitudes; there is a $50$ percent chance that the particle will go through an apparatus that selects the $(+x)$ state. If we had asked about the $(-x)$ state the amplitude would have been $-1/\sqrt{2}$, which also gives a probability $1/2$—as you would expect from the symmetry of space. So if a particle is in the $(+z)$ state, it is equally likely to be in $(+x)$ or $(-x)$, but with opposite phase.  There’s no prejudice in $y$ either. A particle in the $(+z)$ state has a $50$–$50$ chance of being in $(+y)$ or in $(-y)$. However, for these (using the formula for rotating $-90^\circ$ about $x$), the amplitudes are $1/\sqrt{2}$ and $-i/\sqrt{2}$. In this case, the two amplitudes have a phase difference of $90^\circ$ instead of $180^\circ$, as they did for the $(+x)$ and $(-x)$. In fact, that’s how the distinction between $x$ and $y$ shows up.  As our final example, suppose that we know that a spin one-half particle is in a state $\psi$ such that it is polarized “up” along some axis $A$, defined by the angles $\theta$ and $\phi$ in Fig. 6-10. We want to know the amplitude $C_+$ that the particle is “up” along $z$ and the amplitude $C_-$ that it is “down”  along $z$.  We can find these amplitudes by imagining that $A$ is the $z$-axis of a system whose $x$-axis lies in some arbitrary direction—say in the plane formed by $A$ and $z$. We can then bring the frame of $A$ into $x$, $y$, $z$ by three rotations. First, we make a rotation by $-\pi/2$ about the axis $A$, which brings the $x$-axis into the line $B$ in the figure. Then we rotate by $\theta$ about line $B$ (the new $x$-axis of frame $A$) to bring $A$ to the $z$-axis. Finally, we rotate by the angle $(\pi/2-\phi)$ about $z$. Remembering that we have only a $(+)$ state with respect to $A$, we get \begin{equation} \label{Eq:III:6:36} C_+=\cos\frac{\theta}{2}\,e^{-i\phi/2},\quad C_-=\sin\frac{\theta}{2}\,e^{+i\phi/2}. \end{equation}  We would like, finally, to summarize the results of this chapter in a form that will be useful for our later work. First, we remind you that our primary result in Eqs. (6.35) can be written in another notation. Note that Eqs. (6.35) mean just the same thing as Eq. (6.4). That is, in Eqs. (6.35) the coefficients of $C_+=\braket{+S}{\psi}$ and $C_-=\braket{-S}{\psi}$ are just the amplitudes $\braket{jT}{iS}$ of Eq. (6.4)—the amplitudes that a particle in the $i$-state with respect to $S$ will be in the $j$-state with respect to $T$ (when the orientation of $T$ with respect to $S$ is given in terms of the angles $\alpha$, $\beta$, and $\gamma$). We also called them $R_{ji}^{TS}$ in Eq. (6.6). (We have a plethora of notations!) For example, $R_{-+}^{TS}=\braket{-T}{+S}$ is the coefficient of $C_+$ in the formula for $C_-'$, namely, $i\sin\,(\alpha/2)\,e^{i(\beta-\gamma)/2}$. We can, therefore, make a summary of our results in the form of a table, as we have done in Table 6–1.  It will occasionally be handy to have these amplitudes already worked out for some simple special cases. Let’s let $R_z(\phi)$ stand for a rotation by the angle $\phi$ about the $z$-axis. We can also let it stand for the corresponding rotation matrix (omitting the subscripts $i$ and $j$, which are to be implicitly understood). In the same spirit $R_x(\phi)$ and $R_y(\phi)$ will stand for rotations by the angle $\phi$ about the $x$-axis or the $y$-axis. We give in Table 6–2 the matrices—the tables of amplitudes $\braket{jT}{iS}$—which project the amplitudes from the $S$-frame into the $T$-frame, where $T$ is obtained from $S$ by the rotation specified. 