How states change with time
We have already talked about how we can represent a situation in which we put something through an apparatus. Now one convenient, delightful “apparatus” to consider is merely a wait of a few minutes; that is, you prepare a state $\phi$, and then before you analyze it, you just let it sit. Perhaps you let it sit in some particular electric or magnetic field—it depends on the physical circumstances in the world. At any rate, whatever the conditions are, you let the object sit from time $t_1$ to time $t_2$. Suppose that it is let out of your first apparatus in the condition $\phi$ at $t_1$. And then it goes through an “apparatus,” but the “apparatus” consists of just delay until $t_2$. During the delay, various things could be going on—external forces applied or other shenanigans—so that something is happening. At the end of the delay, the amplitude to find the thing in some state $\chi$ is no longer exactly the same as it would have been without the delay. Since “waiting” is just a special case of an “apparatus,” we can describe what happens by giving an amplitude with the same form as Eq. (8.17). Because the operation of “waiting” is especially important, we’ll call it $U$ instead of $A$, and to specify the starting and finishing times $t_1$ and $t_2$, we’ll write $U(t_2,t_1)$. The amplitude we want is \begin{equation} \label{Eq:III:8:27} \bracket{\chi}{U(t_2,t_1)}{\phi}. \end{equation}  Like any other such amplitude, it can be represented in some base system or other by writing it \begin{equation} \label{Eq:III:8:28} \sum_{ij}\braket{\chi}{i}\bracket{i}{U(t_2,t_1)}{j} \braket{j}{\phi}. \end{equation}  Then $U$ is completely described by giving the whole set of amplitudes—the matrix \begin{equation} \label{Eq:III:8:29} \bracket{i}{U(t_2,t_1)}{j}. \end{equation}  We can point out, incidentally, that the matrix $\bracket{i}{U(t_2,t_1)}{j}$ gives much more detail than may be needed. The high-class theoretical physicist working in high-energy physics considers problems of the following general nature (because it’s the way experiments are usually done). He starts with a couple of particles, like a proton and a proton, coming together from infinity. (In the lab, usually one particle is standing still, and the other comes from an accelerator that is practically at infinity on atomic level.) The things go crash and out come, say, two K-mesons, six $\pi$-mesons, and two neutrons in certain directions with certain momenta. What’s the amplitude for this to happen? The mathematics looks like this: The $\phi$-state specifies the spins and momenta of the incoming particles. The $\chi$ would be the question about what comes out. For instance, with what amplitude do you get the six mesons going in such-and-such directions, and the two neutrons going off in these directions, with their spins so-and-so. In other words, $\chi$ would be specified by giving all the momenta, and spins, and so on of the final products. Then the job of the theorist is to calculate the amplitude (8.27). However, he is really only interested in the special case that $t_1$ is $-\infty$ and $t_2$ is $+\infty$. (There is no experimental evidence on the details of the process, only on what comes in and what goes out.) The limiting case of $U(t_2,t_1)$ as $t_1\to-\infty$ and $t_2\to+\infty$ is called $S$, and what he wants is \begin{equation*} \bracket{\chi}{S}{\phi}. \end{equation*}  Or, using the form (8.28), he would calculate the matrix \begin{equation*} \bracket{i}{S}{j}, \end{equation*}  which is called the $S$-matrix. So if you see a theoretical physicist pacing the floor and saying, “All I have to do is calculate the $S$-matrix,” you will know what he is worried about. How to analyze—how to specify the laws for—the $S$-matrix is an interesting question. In relativistic quantum mechanics for high energies, it is done one way, but in nonrelativistic quantum mechanics it can be done another way, which is very convenient. (This other way can also be done in the relativistic case, but then it is not so convenient.) It is to work out the $U$-matrix for a small interval of time—in other words for $t_2$ and $t_1$ close together. If we can find a sequence of such $U$’s for successive intervals of time we can watch how things go as a function of time. You can appreciate immediately that this way is not so good for relativity, because you don’t want to have to specify how everything looks “simultaneously” everywhere. But we won’t worry about that—we’re just going to worry about nonrelativistic mechanics. Suppose we think of the matrix $U$ for a delay from $t_1$ until $t_3$ which is greater than $t_2$. In other words, let’s take three successive times: $t_1$ less than $t_2$ less than $t_3$. Then we claim that the matrix that goes between $t_1$ and $t_3$ is the product in succession of what happens when you delay from $t_1$ until $t_2$ and then from $t_2$ until $t_3$. It’s just like the situation when we had two apparatuses $B$ and $A$ in series. We can then write, following the notation of Section 5–6, \begin{equation} \label{Eq:III:8:30} U(t_3,t_1)=U(t_3,t_2)\cdot U(t_2,t_1). \end{equation}  In other words, we can analyze any time interval if we can analyze a sequence of short time intervals in between. We just multiply together all the pieces; that’s the way that quantum mechanics is analyzed nonrelativistically. Our problem, then, is to understand the matrix $U(t_2,t_1)$ for an infinitesimal time interval—for $t_2=t_1+\Delta t$. We ask ourselves this: If we have a state $\phi$ now, what does the state look like an infinitesimal time $\Delta t$ later? Let’s see how we write that out. Call the state at the time $t$, $\ket{\psi(t)}$ (we show the time dependence of $\psi$ to be perfectly clear that we mean the condition at the time $t$). Now we ask the question: What is the condition after the small interval of time $\Delta t$ later? The answer is \begin{equation} \label{Eq:III:8:31} \ket{\psi(t+\Delta t)}=U(t+\Delta t,t)\,\ket{\psi(t)}. \end{equation}  This means the same as we meant by (8.25), namely, that the amplitude to find $\chi$ at the time $t+\Delta t$, is \begin{equation} \label{Eq:III:8:32} \braket{\chi}{\psi(t+\Delta t)}=\bracket{\chi}{U(t+\Delta t,t)}{\psi(t)}. \end{equation}  Since we’re not yet too good at these abstract things, let’s project our amplitudes into a definite representation. If we multiply both sides of Eq. (8.31) by $\bra{i}$, we get \begin{equation} \label{Eq:III:8:33} \braket{i}{\psi(t+\Delta t)}=\bracket{i}{U(t+\Delta t,t)}{\psi(t)}. \end{equation}  We can also resolve the $\ket{\psi(t)}$ into base states and write \begin{equation} \label{Eq:III:8:34} \braket{i}{\psi(t+\Delta t)}=\sum_j \bracket{i}{U(t+\Delta t,t)}{j}\braket{j}{\psi(t)}. \end{equation}  We can understand Eq. (8.34) in the following way. If we let $C_i(t)=\braket{i}{\psi(t)}$ stand for the amplitude to be in the base state $i$ at the time $t$, then we can think of this amplitude (just a number, remember!)  varying with time. Each $C_i$ becomes a function of $t$. And we also have some information on how the amplitudes $C_i$ vary with time. Each amplitude at $(t+\Delta t)$ is proportional to all of the other amplitudes at $t$ multiplied by a set of coefficients. Let’s call the $U$-matrix $U_{ij}$, by which we mean \begin{equation*} U_{ij}=\bracket{i}{U}{j}. \end{equation*}  Then we can write Eq. (8.34) as \begin{equation} \label{Eq:III:8:35} C_i(t+\Delta t)=\sum_jU_{ij}(t+\Delta t,t)C_j(t). \end{equation}  This, then, is how the dynamics of quantum mechanics is going to look. We don’t know much about the $U_{ij}$ yet, except for one thing. We know that if $\Delta t$ goes to zero, nothing can happen—we should get just the original state. So, $U_{ii}\to1$ and $U_{ij}\to0$, if $i\neq j$. In other words, $U_{ij}\to\delta_{ij}$ for $\Delta t\to0$. Also, we can suppose that for small $\Delta t$, each of the coefficients $U_{ij}$ should differ from $\delta_{ij}$ by amounts proportional to $\Delta t$; so we can write \begin{equation} \label{Eq:III:8:36} U_{ij}=\delta_{ij}+K_{ij}\,\Delta t. \end{equation}  However, it is usual to take the factor $(-i/\hbar)$2 out of the coefficients $K_{ij}$, for historical and other reasons; we prefer to write \begin{equation} \label{Eq:III:8:37} U_{ij}(t+\Delta t,t)=\delta_{ij}-\frac{i}{\hbar}\,H_{ij}(t)\,\Delta t. \end{equation}  It is, of course, the same as Eq. (8.36) and, if you wish, just defines the coefficients $H_{ij}(t)$. The terms $H_{ij}$ are just the derivatives with respect to $t_2$ of the coefficients $U_{ij}(t_2,t_1)$, evaluated at $t_2=t_1=t$. Using this form for $U$ in Eq. (8.35), we have \begin{equation} \label{Eq:III:8:38} C_i(t+\Delta t)=\sum_j\biggl[ \delta_{ij}-\frac{i}{\hbar}\,H_{ij}(t)\,\Delta t \biggr]C_j(t). \end{equation}  Taking the sum over the $\delta_{ij}$ term, we get just $C_i(t)$, which we can put on the other side of the equation. Then dividing by $\Delta t$, we have what we recognize as a derivative \begin{equation} \frac{C_i(t+\Delta t)-C_i(t)}{\Delta t}= -\frac{i}{\hbar}\sum_jH_{ij}(t)C_j(t)\notag \end{equation}  or \begin{equation} \label{Eq:III:8:39} i\hbar\,\ddt{C_i(t)}{t}=\sum_jH_{ij}(t)C_j(t). \end{equation}  You remember that $C_i(t)$ is the amplitude $\braket{i}{\psi}$ to find the state $\psi$ in one of the base states $i$ (at the time $t$). So Eq. (8.39) tells us how each of the coefficients $\braket{i}{\psi}$ varies with time. But that is the same as saying that Eq. (8.39) tells us how the state $\psi$ varies with time, since we are describing $\psi$ in terms of the amplitudes $\braket{i}{\psi}$. The variation of $\psi$ in time is described in terms of the matrix $H_{ij}$, which has to include, of course, the things we are doing to the system to cause it to change. If we know the $H_{ij}$—which contains the physics of the situation and can, in general, depend on the time—we have a complete description of the behavior in time of the system. Equation (8.39) is then the quantum mechanical law for the dynamics of the world. (We should say that we will always take a set of base states which are fixed and do not vary with time. There are people who use base states that also vary. However, that’s like using a rotating coordinate system in mechanics, and we don’t want to get involved in such complications.)