Amplitudes and vectors
Before we begin the main topic of this chapter, we would like to describe a number of mathematical ideas that are used a lot in the literature of quantum mechanics. Knowing them will make it easier for you to read other books or papers on the subject. The first idea is the close mathematical resemblance between the equations of quantum mechanics and those of the scalar product of two vectors. You remember that if $\chi$ and $\phi$ are two states, the amplitude to start in $\phi$ and end up in $\chi$ can be written as a sum over a complete set of base states of the amplitude to go from $\phi$ into one of the base states and then from that base state out again into $\chi$: \begin{equation} \label{Eq:III:8:1} \braket{\chi}{\phi}= \sum_{\text{all $i$}}\braket{\chi}{i}\braket{i}{\phi}. \end{equation}  We explained this in terms of a Stern-Gerlach apparatus, but we remind you that there is no need to have the apparatus. Equation (8.1) is a mathematical law that is just as true whether we put the filtering equipment in or not—it is not always necessary to imagine that the apparatus is there. We can think of it simply as a formula for the amplitude $\braket{\chi}{\phi}$. We would like to compare Eq. (8.1) to the formula for the dot product of two vectors $\FLPB$ and $\FLPA$. If $\FLPB$ and $\FLPA$ are ordinary vectors in three dimensions, we can write the dot product this way: \begin{equation} \label{Eq:III:8:2} \sum_{\text{all $i$}}(\FLPB\cdot\FLPe_i)(\FLPe_i\cdot\FLPA), \end{equation}  with the understanding that the symbol $\FLPe_i$ stands for the three unit vectors in the $x$, $y$, and $z$-directions. Then $\FLPB\cdot\FLPe_1$ is what we ordinarily call $B_x$; $\FLPB\cdot\FLPe_2$ is what we ordinarily call $B_y$; and so on. So Eq. (8.2) is equivalent to \begin{equation*} B_xA_x+B_yA_y+B_zA_z, \end{equation*}  which is the dot product $\FLPB\cdot\FLPA$. Comparing Eqs. (8.1) and (8.2), we can see the following analogy: The states $\chi$ and $\phi$ correspond to the two vectors $\FLPB$ and $\FLPA$. The base states $i$ correspond to the special vectors $\FLPe_i$ to which we refer all other vectors. Any vector can be represented as a linear combination of the three “base vectors” $\FLPe_i$. Furthermore, if you know the coefficients of each “base vector” in this combination—that is, its three components—you know everything about a vector. In a similar way, any quantum mechanical state can be described completely by the amplitude $\braket{i}{\phi}$ to go into the base states; and if you know these coefficients, you know everything there is to know about the state. Because of this close analogy, what we have called a “state” is often also called a “state vector.” Since the base vectors $\FLPe_i$ are all at right angles, we have the relation \begin{equation} \label{Eq:III:8:3} \FLPe_i\cdot\FLPe_j=\delta_{ij}. \end{equation}  This corresponds to the relations (5.25) among the base states $i$, \begin{equation} \label{Eq:III:8:4} \braket{i}{j}=\delta_{ij}. \end{equation}  You see now why one says that the base states $i$ are all “orthogonal.” There is one minor difference between Eq. (8.1) and the dot product. We have that \begin{equation} \label{Eq:III:8:5} \braket{\phi}{\chi}=\braket{\chi}{\phi}\cconj. \end{equation} But in vector algebra \begin{equation*} \FLPA\cdot\FLPB=\FLPB\cdot\FLPA.\notag \end{equation*}  With the complex numbers of quantum mechanics we have to keep straight the order of the terms, whereas in the dot product, the order doesn’t matter.  Now consider the following vector equation: \begin{equation} \label{Eq:III:8:6} \FLPA=\sum_i\FLPe_i(\FLPe_i\cdot\FLPA). \end{equation}  It’s a little unusual, but correct. It means the same thing as \begin{equation} \label{Eq:III:8:7} \FLPA=\sum_iA_i\FLPe_i=A_x\FLPe_x+A_y\FLPe_y+A_z\FLPe_z. \end{equation}  Notice, though, that Eq. (8.6) involves a quantity which is different from a dot product. A dot product is just a number, whereas Eq. (8.6) is a vector equation. One of the great tricks of vector analysis was to abstract away from the equations the idea of a vector itself. One might be similarly inclined to abstract a thing that is the analog of a “vector” from the quantum mechanical formula Eq. (8.1)—and one can indeed. We remove the $\bra{\chi}$ from both sides of Eq. (8.1) and write the following equation (don’t get frightened—it’s just a notation and in a few minutes you will find out what the symbols mean): \begin{equation} \label{Eq:III:8:8} \ket{\phi}=\sum_i\ket{i}\braket{i}{\phi}. \end{equation}  One thinks of the bracket $\braket{\chi}{\phi}$ as being divided into two pieces. The second piece $\ket{\phi}$ is often called a ket, and the first piece $\bra{\chi}$ is called a bra (put together, they make a “bra-ket”—a notation proposed by Dirac); the half-symbols $\ket{\phi}$ and $\bra{\chi}$ are also called state vectors. In any case, they are not numbers, and, in general, we want the results of our calculations to come out as numbers; so such “unfinished” quantities are only part-way steps in our calculations. It happens that until now we have written all our results in terms of numbers. How have we managed to avoid vectors? It is amusing to note that even in ordinary vector algebra we could make all equations involve only numbers. For instance, instead of a vector equation like \begin{equation*} \FLPF=m\FLPa, \end{equation*}  we could always have written \begin{equation*} \FLPC\cdot\FLPF=\FLPC\cdot(m\FLPa). \end{equation*}  We have then an equation between dot products that is true for any vector $\FLPC$. But if it is true for any $\FLPC$, it hardly makes sense at all to keep writing the $\FLPC$! Now look at Eq. (8.1). It is an equation that is true for any $\chi$. So to save writing, we should just leave out the $\chi$ and write Eq. (8.8) instead. It has the same information provided we understand that it should always be “finished” by “multiplying on the left by”—which simply means reinserting—some $\bra{\chi}$ on both sides. So Eq. (8.8) means exactly the same thing as Eq. (8.1)—no more, no less. When you want numbers, you put in the $\bra{\chi}$ you want. Maybe you have already wondered about the $\phi$ in Eq. (8.8). Since the equation is true for any $\phi$, why do we keep it? Indeed, Dirac suggests that the $\phi$ also can just as well be abstracted away, so that we have only \begin{equation} \label{Eq:III:8:9} |=\sum_i\ket{i}\bra{i}. \end{equation}  And this is the great law of quantum mechanics! (There is no analog in vector analysis.) It says that if you put in any two states $\chi$ and $\phi$ on the left and right of both sides, you get back Eq. (8.1). It is not really very useful, but it’s a nice reminder that the equation is true for any two states.