Transforming amplitudes
In the last chapter, using a system of spin one as an example, we outlined the general principles of quantum mechanics: Any state $\psi$ can be described in terms of a set of base states  			by giving the amplitudes to be in each of the base states. The amplitude to go from any state to another can, in general, be  			written as a sum of products, each product being the amplitude to go  			into one of the base states times the amplitude to go from that base  			state to the final condition, with the sum including a term for each  			base state: \begin{equation} \label{Eq:III:6:1} \braket{\chi}{\psi}=\sum_i\braket{\chi}{i}\braket{i}{\psi}. \end{equation} 			The base states are orthogonal—the amplitude to be in one if you are in the 			other is zero: \begin{equation} \label{Eq:III:6:2} \braket{i}{j}=\delta_{ij}. \end{equation} 			The amplitude to get from one state to another directly is the complex conjugate 			of the reverse: \begin{equation} \label{Eq:III:6:3} \braket{\chi}{\psi}\cconj=\braket{\psi}{\chi}. \end{equation}  We also discussed a little bit about the fact that there can be more than one base for the states and that we can use Eq. (6.1) to convert from one base to another. Suppose, for example, that we have the amplitudes $\braket{iS}{\psi}$ to find the state $\psi$ in every one of the base states $i$ of a base system $S$, but that we then decide that we would prefer to describe the state in terms of another set of base states, say the states $j$ belonging to the base $T$. In the general formula, Eq. (6.1), we could substitute $jT$ for $\chi$ and obtain this formula: \begin{equation} \label{Eq:III:6:4} \braket{jT}{\psi}=\sum_i\braket{jT}{iS}\braket{iS}{\psi}. \end{equation} The amplitudes for the state $(\psi)$ to be in the base states $(jT)$ are related to the amplitudes to be in the base states $(iS)$ by the set of coefficients $\braket{jT}{iS}$. If there are $N$ base states, there are $N^2$ such coefficients. Such a set of coefficients is often called the “transformation matrix to go from the $S$-representation to the $T$-representation.” This looks rather formidable mathematically, but with a little renaming we can see that it is really not so bad. If we call $C_i$ the amplitude that the state $\psi$ is in the base state $iS$—that is, $C_i=\braket{iS}{\psi}$—and call $C_j'$ the corresponding amplitudes for the base system $T$—that is, $C_j'=\braket{jT}{\psi}$, then Eq. (6.4) can be written as \begin{equation} \label{Eq:III:6:5} C_j'=\sum_iR_{ji}C_i, \end{equation} where $R_{ji}$ means the same thing as $\braket{jT}{iS}$. Each amplitude $C_j'$ is equal to a sum over all $i$ of one of the coefficients $R_{ji}$ times each amplitude $C_i$. It has the same form as the transformation of a vector from one coordinate system to another.  In order to avoid being too abstract for too long, we have given you some examples of these coefficients for the spin-one case, so you can see how to use them in practice. On the other hand, there is a very beautiful thing in quantum mechanics—that from the sheer fact that there are three states and from the symmetry properties of space under rotations, these coefficients can be found purely by abstract reasoning. Showing you such arguments at this early stage has a disadvantage in that you are immersed in another set of abstractions before we get “down to earth.” However, the thing is so beautiful that we are going to do it anyway.  We will show you in this chapter how the transformation coefficients can be derived for spin one-half particles. We pick this case, rather than spin one, because it is somewhat easier. Our problem is to determine the coefficients $R_{ji}$ for a particle—an atomic system—which is split into two beams in a Stern-Gerlach apparatus. We are going to derive all the coefficients for the transformation from one representation to another by pure reasoning—plus a few assumptions. Some assumptions are always necessary in order to use “pure” reasoning! Although the arguments will be abstract and somewhat involved, the result we get will be relatively simple to state and easy to understand—and the result is the most important thing. You may, if you wish, consider this as a sort of cultural excursion. We have, in fact, arranged that all the essential results derived here are also derived in some other way when they are needed in later chapters. So you need have no fear of losing the thread of our study of quantum mechanics if you omit this chapter entirely, or study it at some later time. The excursion is “cultural” in the sense that it is intended to show that the principles of quantum mechanics are not only interesting, but are so deep that by adding only a few extra hypotheses about the structure of space, we can deduce a great many properties of physical systems. Also, it is important that we know where the different consequences of quantum mechanics come from, because so long as our laws of physics are incomplete—as we know they are—it is interesting to find out whether the places where our theories fail to agree with experiment is where our logic is the best or where our logic is the worst. Until now, it appears that where our logic is the most abstract it always gives correct results—it agrees with experiment. Only when we try to make specific models of the internal machinery of the fundamental particles and their interactions are we unable to find a theory that agrees with experiment. The theory then that we are about to describe agrees with experiment wherever it has been tested—for the strange particles as well as for electrons, protons, and so on.  One remark on an annoying, but interesting, point before we proceed: It is not possible to determine the coefficients $R_{ji}$ uniquely, because there is always some arbitrariness in the probability amplitudes. If you have a set of amplitudes of any kind, say the amplitudes to arrive at some place by a whole lot of different routes, and if you multiply every single amplitude by the same phase factor—say by $e^{i\delta}$—you have another set that is just as good. So, it is always possible to make an arbitrary change in phase of all the amplitudes in any given problem if you want to.  Suppose you calculate some probability by writing a sum of several amplitudes, say $(A+B+C+\dotsb)$ and taking the absolute square. Then somebody else calculates the same thing by using the sum of the amplitudes $(A'+B'+C'+\dotsb)$ and taking the absolute square. If all the $A'$, $B'$, $C'$, etc., are equal to the $A$, $B$, $C$, etc., except for a factor $e^{i\delta}$, all probabilities obtained by taking the absolute squares will be exactly the same, since $(A'+B'+C'+\dotsb)$ is then equal to $e^{i\delta}(A+B+C+\dotsb)$. Or suppose, for instance, that we were computing something with Eq. (6.1), but then we suddenly change all of the phases of a certain base system. Every one of the amplitudes $\braket{i}{\psi}$ would be multiplied by the same factor $e^{i\delta}$. Similarly, the amplitudes $\braket{i}{\chi}$ would also be changed by $e^{i\delta}$, but the amplitudes $\braket{\chi}{i}$ are the complex conjugates of the amplitudes $\braket{i}{\chi}$; therefore, the former gets changed by the factor $e^{-i\delta}$. The plus and minus $i\delta$’s in the exponents cancel out, and we would have the same expression we had before. So it is a general rule that if we change all the amplitudes with respect to a given base system by the same phase—or even if we just change all the amplitudes in any problem by the same phase—it makes no difference. There is, therefore, some freedom to choose the phases in our transformation matrix. Every now and then we will make such an arbitrary choice—usually following the conventions that are in general use. 