The laws for combining amplitudes
When Schrödinger first discovered the correct laws of quantum mechanics, he wrote an equation which described the amplitude to find a particle in various places. This equation was very similar to the equations that were already known to classical physicists—equations that they had used in describing the motion of air in a sound wave, the transmission of light, and so on. So most of the time at the beginning of quantum mechanics was spent in solving this equation. But at the same time an understanding was being developed, particularly by Born and Dirac, of the basically new physical ideas behind quantum mechanics. As quantum mechanics developed further, it turned out that there were a large number of things which were not directly encompassed in the Schrödinger equation—such as the spin of the electron, and various relativistic phenomena. Traditionally, all courses in quantum mechanics have begun in the same way, retracing the path followed in the historical development of the subject. One first learns a great deal about classical mechanics so that he will be able to understand how to solve the Schrödinger equation. Then he spends a long time working out various solutions. Only after a detailed study of this equation does he get to the “advanced” subject of the electron’s spin. We had also originally considered that the right way to conclude these lectures on physics was to show how to solve the equations of classical physics in complicated situations—such as the description of sound waves in enclosed regions, modes of electromagnetic radiation in cylindrical cavities, and so on. That was the original plan for this course. However, we have decided to abandon that plan and to give instead an introduction to the quantum mechanics. We have come to the conclusion that what are usually called the advanced parts of quantum mechanics are, in fact, quite simple. The mathematics that is involved is particularly simple, involving simple algebraic operations and no differential equations or at most only very simple ones. The only problem is that we must jump the gap of no longer being able to describe the behavior in detail of particles in space. So this is what we are going to try to do: to tell you about what conventionally would be called the “advanced” parts of quantum mechanics. But they are, we assure you, by all odds the simplest parts—in a deep sense of the word—as well as the most basic parts. This is frankly a pedagogical experiment; it has never been done before, as far as we know. In this subject we have, of course, the difficulty that the quantum mechanical behavior of things is quite strange. Nobody has an everyday experience to lean on to get a rough, intuitive idea of what will happen. So there are two ways of presenting the subject: We could either describe what can happen in a rather rough physical way, telling you more or less what happens without giving the precise laws of everything; or we could, on the other hand, give the precise laws in their abstract form. But, then because of the abstractions, you wouldn’t know what they were all about, physically. The latter method is unsatisfactory because it is completely abstract, and the first way leaves an uncomfortable feeling because one doesn’t know exactly what is true and what is false. We are not sure how to overcome this difficulty. You will notice, in fact, that Chapters 1 and 2 showed this problem. The first chapter was relatively precise; but the second chapter was a rough description of the characteristics of different phenomena. Here, we will try to find a happy medium between the two extremes. We will begin in this chapter by dealing with some general quantum mechanical ideas. Some of the statements will be quite precise, others only partially precise. It will be hard to tell you as we go along which is which, but by the time you have finished the rest of the book, you will understand in looking back which parts hold up and which parts were only explained roughly. The chapters which follow this one will not be so imprecise. In fact, one of the reasons we have tried carefully to be precise in the succeeding chapters is so that we can show you one of the most beautiful things about quantum mechanics—how much can be deduced from so little. We begin by discussing again the superposition of probability amplitudes. As an example we will refer to the experiment described in Chapter 1, and shown again here in Fig. 3–1. There is a source $s$ of particles, say electrons; then there is a wall with two slits in it; after the wall, there is a detector located at some position $x$. We ask for the probability that a particle will be found at $x$. Our first general principle in quantum mechanics is that the probability that a particle will arrive at $x$, when let out at the source $s$, can be represented quantitatively by the absolute square of a complex number called a probability amplitude—in this case, the “amplitude that a particle from $s$ will arrive at $x$.” We will use such amplitudes so frequently that we will use a shorthand notation—invented by Dirac and generally used in quantum mechanics—to represent this idea. We write the probability amplitude this way: \begin{equation} \label{Eq:III:3:1} \braket{\text{Particle arrives at $x$}}{\text{particle leaves $s$}}. \end{equation}  In other words, the two brackets $\langle\:\rangle$ are a sign equivalent to “the amplitude that”; the expression at the right of the vertical line always gives the starting condition, and the one at the left, the final condition. Sometimes it will also be convenient to abbreviate still more and describe the initial and final conditions by single letters. For example, we may on occasion write the amplitude (3.1) as \begin{equation} \label{Eq:III:3:2} \braket{x}{s}. \end{equation}  We want to emphasize that such an amplitude is, of course, just a single number—a complex number. We have already seen in the discussion of Chapter 1 that when there are two ways for the particle to reach the detector, the resulting probability is not the sum of the two probabilities, but must be written as the absolute square of the sum of two amplitudes. We had that the probability that an electron arrives at the detector when both paths are open is \begin{equation} \label{Eq:III:3:3} P_{12}=\abs{\phi_1+\phi_2}^2. \end{equation}  We wish now to put this result in terms of our new notation. First, however, we want to state our second general principle of quantum mechanics: When a particle can reach a given state by two possible routes, the total amplitude for the process is the sum of the amplitudes for the two routes considered separately. In our new notation we write that \begin{equation} \label{Eq:III:3:4} \braket{x}{s}_{\text{both holes open}}= \braket{x}{s}_{\text{through $1$}}+ \braket{x}{s}_{\text{through $2$}}. \end{equation}  Incidentally, we are going to suppose that the holes $1$ and $2$ are small enough that when we say an electron goes through the hole, we don’t have to discuss which part of the hole. We could, of course, split each hole into pieces with a certain amplitude that the electron goes to the top of the hole and the bottom of the hole and so on. We will suppose that the hole is small enough so that we don’t have to worry about this detail. That is part of the roughness involved; the matter can be made more precise, but we don’t want to do so at this stage. Now we want to write out in more detail what we can say about the amplitude for the process in which the electron reaches the detector at $x$ by way of hole $1$. We can do that by using our third general principle: When a particle goes by some particular route the amplitude for that route can be written as the product of the amplitude to go part way with the amplitude to go the rest of the way. For the setup of Fig. 3–1 the amplitude to go from $s$ to $x$ by way of hole $1$ is equal to the amplitude to go from $s$ to $1$, multiplied by the amplitude to go from $1$ to $x$. \begin{equation} \label{Eq:III:3:5} \braket{x}{s}_{\text{via $1$}}=\braket{x}{1}\braket{1}{s}. \end{equation}  Again this result is not completely precise. We should also include a factor for the amplitude that the electron will get through the hole at $1$; but in the present case it is a simple hole, and we will take this factor to be unity. You will note that Eq. (3.5) appears to be written in reverse order. It is to be read from right to left: The electron goes from $s$ to $1$ and then from $1$ to $x$. In summary, if events occur in succession—that is, if you can analyze one of the routes of the particle by saying it does this, then it does this, then it does that—the resultant amplitude for that route is calculated by multiplying in succession the amplitude for each of the successive events. Using this law we can rewrite Eq. (3.4) as \begin{equation*} \braket{x}{s}_{\text{both}}=\braket{x}{1}\braket{1}{s}+ \braket{x}{2}\braket{2}{s}. \end{equation*}  Now we wish to show that just using these principles we can calculate a much more complicated problem like the one shown in Fig. 3–2. Here we have two walls, one with two holes, $1$ and $2$, and another which has three holes, $a$, $b$, and $c$. Behind the second wall there is a detector at $x$, and we want to know the amplitude for a particle to arrive there. Well, one way you can find this is by calculating the superposition, or interference, of the waves that go through; but you can also do it by saying that there are six possible routes and superposing an amplitude for each. The electron can go through hole $1$, then through hole $a$, and then to $x$; or it could go through hole $1$, then through hole $b$, and then to $x$; and so on. According to our second principle, the amplitudes for alternative routes add, so we should be able to write the amplitude from $s$ to $x$ as a sum of six separate amplitudes. On the other hand, using the third principle, each of these separate amplitudes can be written as a product of three amplitudes. For example, one of them is the amplitude for $s$ to $1$, times the amplitude for $1$ to $a$, times the amplitude for $a$ to $x$. Using our shorthand notation, we can write the complete amplitude to go from $s$ to $x$ as  \begin{equation*} \braket{x}{s}= \braket{x}{a}\braket{a}{1}\braket{1}{s}+ \braket{x}{b}\braket{b}{1}\braket{1}{s}+ \dotsb+ \braket{x}{c}\braket{c}{2}\braket{2}{s}. \end{equation*} 
  \begin{align*} \braket{x}{s}&= \braket{x}{a}\braket{a}{1}\braket{1}{s}+ \braket{x}{b}\braket{b}{1}\braket{1}{s}+\\[.75ex] &\dotsb+\braket{x}{c}\braket{c}{2}\braket{2}{s}. \end{align*}   We can save writing by using the summation notation \begin{equation} \label{Eq:III:3:6} \braket{x}{s}=\sum_{\substack{i=1,2\\\alpha=a,b,c}} \braket{x}{\alpha}\braket{\alpha}{i}\braket{i}{s}. \end{equation}  In order to make any calculations using these methods, it is, naturally, necessary to know the amplitude to get from one place to another. We will give a rough idea of a typical amplitude. It leaves out certain things like the polarization of light or the spin of the electron, but aside from such features it is quite accurate. We give it so that you can solve problems involving various combinations of slits. Suppose a particle with a definite energy is going in empty space from a location $\FLPr_1$ to a location $\FLPr_2$. In other words, it is a free particle with no forces on it. Except for a numerical factor in front, the amplitude to go from $\FLPr_1$ to $\FLPr_2$ is \begin{equation} \label{Eq:III:3:7} \braket{\FLPr_2}{\FLPr_1}=\frac{e^{ipr_{12}/\hbar}}{r_{12}}, \end{equation}  where $r_{12}=\abs{\FLPr_2-\FLPr_1}$, and $p$ is the momentum which is related to the energy $E$ by the relativistic equation \begin{equation*} p^2c^2=E^2-(m_0c^2)^2, \end{equation*}  or the nonrelativistic equation \begin{equation*} \frac{p^2}{2m}=\text{Kinetic energy}. \end{equation*}  Equation (3.7) says in effect that the particle has wavelike properties, the amplitude propagating as a wave with a wave number equal to the momentum divided by $\hbar$. In the most general case, the amplitude and the corresponding probability will also involve the time. For most of these initial discussions we will suppose that the source always emits the particles with a given energy so we will not need to worry about the time. But we could, in the general case, be interested in some other questions. Suppose that a particle is liberated at a certain place $P$ at a certain time, and you would like to know the amplitude for it to arrive at some location, say $\FLPr$, at some later time. This could be represented symbolically as the amplitude $\braket{\FLPr,t=t_1}{P,t=0}$. Clearly, this will depend upon both $\FLPr$ and $t$. You will get different results if you put the detector in different places and measure at different times. This function of $\FLPr$ and $t$, in general, satisfies a differential equation which is a wave equation. For example, in a nonrelativistic case it is the Schrödinger equation. One has then a wave equation analogous to the equation for electromagnetic waves or waves of sound in a gas. However, it must be emphasized that the wave function that satisfies the equation is not like a real wave in space; one cannot picture any kind of reality to this wave as one does for a sound wave. Although one may be tempted to think in terms of “particle waves” when dealing with one particle, it is not a good idea, for if there are, say, two particles, the amplitude to find one at $\FLPr_1$ and the other at $\FLPr_2$ is not a simple wave in three-dimensional space, but depends on the six space variables $\FLPr_1$ and $\FLPr_2$. If we are, for example, dealing with two (or more) particles, we will need the following additional principle: Provided that the two particles do not interact, the amplitude that one particle will do one thing and the other one something else is the product of the two amplitudes that the two particles would do the two things separately. For example, if $\braket{a}{s_1}$ is the amplitude for particle $1$ to go from $s_1$ to $a$, and $\braket{b}{s_2}$ is the amplitude for particle $2$ to go from $s_2$ to $b$, the amplitude that both things will happen together is \begin{equation*} \braket{a}{s_1}\braket{b}{s_2}. \end{equation*}  There is one more point to emphasize. Suppose that we didn’t know where the particles in Fig. 3–2 come from before arriving at holes $1$ and $2$ of the first wall. We can still make a prediction of what will happen beyond the wall (for example, the amplitude to arrive at $x$) provided that we are given two numbers: the amplitude to have arrived at $1$ and the amplitude to have arrived at $2$. In other words, because of the fact that the amplitude for successive events multiplies, as shown in Eq. (3.6), all you need to know to continue the analysis is two numbers—in this particular case $\braket{1}{s}$ and $\braket{2}{s}$. These two complex numbers are enough to predict all the future. That is what really makes quantum mechanics easy. It turns out that in later chapters we are going to do just such a thing when we specify a starting condition in terms of two (or a few) numbers. Of course, these numbers depend upon where the source is located and possibly other details about the apparatus, but given the two numbers, we do not need to know any more about such details.