Filtering atoms with a Stern-Gerlach apparatus
In this chapter we really begin the quantum mechanics proper—in the sense that we are going to describe a quantum mechanical phenomenon in a completely quantum mechanical way. We will make no apologies and no attempt to find connections to classical mechanics. We want to talk about something new in a new language. The particular situation which we are going to describe is the behavior of the so-called quantization of the angular momentum, for a particle of spin one. But we won’t use words like “angular momentum” or other concepts of classical mechanics until later. We have chosen this particular example because it is relatively simple, although not the simplest possible example. It is sufficiently complicated that it can stand as a prototype which can be generalized for the description of all quantum mechanical phenomena. Thus, although we are dealing with a particular example, all the laws which we mention are immediately generalizable, and we will give the generalizations so that you will see the general characteristics of a quantum mechanical description. We begin with the phenomenon of the splitting of a beam of atoms into three separate beams in a Stern-Gerlach experiment. You remember that if we have an inhomogeneous magnetic field made by a magnet with a pointed pole tip and we send a beam through the apparatus, the beam of particles may be split into a number of beams—the number depending on the particular kind of atom and its state. We are going to take the case of an atom which gives three beams, and we are going to call that a particle of spin one. You can do for yourself the case of five beams, seven beams, two beams, etc.—you just copy everything down and where we have three terms, you will have five terms, seven terms, and so on. Imagine the apparatus drawn schematically in Fig. 5–1. A beam of atoms (or particles of any kind) is collimated by some slits and passes through a nonuniform field. Let’s say that the beam moves in the $y$-direction and that the magnetic field and its gradient are both in the $z$-direction. Then, looking from the side, we will see the beam split vertically into three beams, as shown in the figure. Now at the output end of the magnet we could put small counters which count the rate of arrival of particles in any one of the three beams. Or we can block off two of the beams and let the third one go on. Suppose we block off the lower two beams and let the top-most beam go on and enter a second Stern-Gerlach apparatus of the same kind, as shown in Fig. 5–2. What happens? There are not three beams in the second apparatus; there is only the top beam.1 This is what you would expect if you think of the second apparatus as simply an extension of the first. Those atoms which are being pushed upward continue to be pushed upward in the second magnet. You can see then that the first apparatus has produced a beam of “purified” objects—atoms that get bent upward in the particular inhomogeneous field. The atoms, as they enter the original Stern-Gerlach apparatus, are of three “varieties,” and the three kinds take different trajectories. By filtering out all but one of the varieties, we can make a beam whose future behavior in the same kind of apparatus is determined and predictable. We will call this a filtered beam, or a polarized beam, or a beam in which the atoms all are known to be in a definite state. For the rest of our discussion, it will be more convenient if we consider a somewhat modified apparatus of the Stern-Gerlach type. The apparatus looks more complicated at first, but it will make all the arguments simpler. Anyway, since they are only “thought experiments,” it doesn’t cost anything to complicate the equipment. (Incidentally, no one has ever done all of the experiments we will describe in just this way, but we know what would happen from the laws of quantum mechanics, which are, of course, based on other similar experiments. These other experiments are harder to understand at the beginning, so we want to describe some idealized—but possible—experiments.)  Figure 5–3(a) shows a drawing of the “modified Stern-Gerlach apparatus” we would like to use. It consists of a sequence of three high-gradient magnets. The first one (on the left) is just the usual Stern-Gerlach magnet and splits the incoming beam of spin-one particles into three separate beams. The second magnet has the same cross section as the first, but is twice as long and the polarity of its magnetic field is opposite the field in magnet $1$. The second magnet pushes in the opposite direction on the atomic magnets and bends their paths back toward the axis, as shown in the trajectories drawn in the lower part of the figure. The third magnet is just like the first, and brings the three beams back together again, so that leaves the exit hole along the axis. Finally, we would like to imagine that in front of the hole at $A$ there is some mechanism which can get the atoms started from rest and that after the exit hole at $B$ there is a decelerating mechanism that brings the atoms back to rest at $B$. That is not essential, but it will mean that in our analysis we won’t have to worry about including any effects of the motion as the atoms come out, and can concentrate on those matters having only to do with the spin. The whole purpose of the “improved” apparatus is just to bring all the particles to the same place, and with zero speed. Now if we want to do an experiment like the one in Fig. 5–2, we can first make a filtered beam by putting a plate in the middle of the apparatus that blocks two of the beams, as shown in Fig. 5–4. If we now put the polarized atoms through a second identical apparatus, all the atoms will take the upper path, as can be verified by putting similar plates in the way of the various beams of the second $S$ filter and seeing whether particles get through. Suppose we call the first apparatus by the name $S$. (We are going to consider all sorts of combinations, and we will need labels to keep things straight.) We will say that the atoms which take the top path in $S$ are in the “plus state with respect to $S$”; the ones which take the middle path are in the “zero state with respect to $S$”; and the ones which take the lowest path are in the “minus state with respect to $S$.” (In the more usual language we would say that the $z$-component of the angular momentum was $+1\hbar$, $0$, and $-1\hbar$, but we are not using that language now.) Now in Fig. 5–4 the second apparatus is oriented just like the first, so the filtered atoms will all go on the upper path. Or if we had blocked off the upper and lower beams in the first apparatus and let only the zero state through, all the filtered atoms would go through the middle path of the second apparatus. And if we had blocked off all but the lowest beam in the first, there would be only a low beam in the second. We can say that in each case our first apparatus has produced a filtered beam in a pure state with respect to $S$ ($+$, $0$, or $-$), and we can test which state is present by putting the atoms through a second, identical apparatus. We can make our second apparatus so that it transmits only atoms of a particular state—by putting masks inside it as we did for the first one—and then we can test the state of the incoming beam just by seeing whether anything comes out the far end. For instance, if we block off the two lower paths in the second apparatus, $100$ percent of the atoms will still come through; but if we block off the upper path, nothing will get through. To make this kind of discussion easier, we are going to invent a shorthand symbol to represent one of our improved Stern-Gerlach apparatuses. We will let the symbol \begin{equation} \label{Eq:III:5:1} \SG{S} \end{equation}  stand for one complete apparatus. (This is not a symbol you will ever find used in quantum mechanics; we’ve just invented it for this chapter. It is simply meant to be a shorthand picture of the apparatus of Fig. 5–3.) Since we are going to want to use several apparatuses at once, and with various orientations, we will identify each with a letter underneath. So the symbol in (5.1) stands for the apparatus $S$. When we block off one or more of the beams inside, we will show that by some vertical bars indicating which beam is blocked, like this: \begin{equation} \label{Eq:III:5:2} \SGP{S}\!\!\!\cdotp \end{equation}  The various possible combinations we will be using are shown in Fig. 5–5. If we have two filters in succession (as in Fig. 5–4), we will put the two symbols next to each other, like this: \begin{equation} \label{Eq:III:5:3} \SGP{S} \!\! \SG{S} \!\!\!\cdotp \end{equation}  For this setup, everything that comes through the first also gets through the second. In fact, even if we block off the “zero” and “minus” channels of the second apparatus, so that we have  \begin{equation} \label{Eq:III:5:4} \SGP{S} \!\! \SGP{S} \!\!\!, \end{equation}  we still get $100$ percent transmission through the second apparatus. On the other hand, if we have  \begin{equation} \label{Eq:III:5:5} \SGP{S} \!\! \SGM{S} \!\!\!, \end{equation}  nothing at all comes out of the far end. Similarly,  \begin{equation} \label{Eq:III:5:6} \SGM{S} \!\! \SGP{S} \end{equation}  would give nothing out. On the other hand,  \begin{equation} \label{Eq:III:5:7} \begin{array}{cc} \SGM{S} \!\! \SGM{S} \end{array} \end{equation}  would be just equivalent to  \begin{equation*} \SGM{S} \end{equation*}  by itself. Now we want to describe these experiments quantum mechanically. We will say that an atom is in the $(+S)$ state if it has gone through the apparatus of Fig. 5–5(b), that it is in a $(\OS)$ state if it has gone through (c), and in a $(-S)$ state if it has gone through (d).2 Then we let $\braket{b}{a}$ be the amplitude that an atom which is in state $a$ will get through an apparatus into the $b$ state. We can say: $\braket{b}{a}$ is the amplitude for an atom in the state $a$ to get into the state $b$. The experiment (5.4) gives us that \begin{equation*} \braket{+S}{+S}=1, \end{equation*}  whereas (5.5) gives us \begin{equation*} \braket{-S}{+S}=0. \end{equation*}  Similarly, the result of (5.6) is \begin{equation*} \braket{+S}{-S}=0, \end{equation*}  and of (5.7) is \begin{equation*} \braket{-S}{-S}=1. \end{equation*}  As long as we deal only with “pure” states—that is, we have only one channel open—there are nine such amplitudes, and we can write them in a table: \begin{equation} \label{Eq:III:5:8} \text{to}\, \begin{array}{c} \begin{array}{c}     \text{from} \end{array}  \\[-.5ex] \begin{array}{c|ccc} &+S & 0S & -S  \\    \hline +S&1 & 0 & 0\\ 0S&0 & 1 & 0\\ -S&0 & 0 & 1 \end{array} \end{array} \end{equation}  This array of nine numbers—called a matrix—summarizes the phenomena we’ve been describing.