Amplitudes on a line
We are now going to discuss how the probability amplitudes of quantum mechanics vary in space. In some of the earlier chapters you may have had a rather uncomfortable feeling that some things were being left out. For example, when we were talking about the ammonia molecule, we chose to describe it in terms of two base states. For one base state we picked the situation in which the nitrogen atom was “above” the plane of the three hydrogen atoms, and for the other base state we picked the condition in which the nitrogen atom was “below” the plane of the three hydrogen atoms. Why did we pick just these two states?  Why is it not possible that the nitrogen atom could be at $2$ angstroms above the plane of the three hydrogen atoms, or at $3$ angstroms, or at $4$ angstroms above the plane? Certainly, there are many positions that the nitrogen atom could occupy. Again when we talked about the hydrogen molecular ion, in which there is one electron shared by two protons, we imagined two base states: one for the electron in the neighborhood of proton number one, and the other for the electron in the neighborhood of proton number two. Clearly we were leaving out many details. The electron is not exactly at proton number two but is only in the neighborhood. It could be somewhere above the proton, somewhere below the proton, somewhere to the left of the proton, or somewhere to the right of the proton. We intentionally avoided discussing these details. We said that we were interested in only certain features of the problem, so we were imagining that when the electron was in the vicinity of proton number one, it would take up a certain rather definite condition. In that condition the probability to find the electron would have some rather definite distribution around the proton, but we were not interested in the details. We can also put it another way. In our discussion of a hydrogen molecular ion we chose an approximate description when we described the situation in terms of two base states. In reality there are lots and lots of these states. An electron can take up a condition around a proton in its lowest, or ground, state, but there are also many excited states. For each excited state the distribution of the electron around the proton is different. We ignored these excited states, saying that we were interested in only the conditions of low energy. But it is just these other excited states which give the possibility of various distributions of the electron around the proton. If we want to describe in detail the hydrogen molecular ion, we have to take into account also these other possible base states. We could do this in several ways, and one way is to consider in greater detail states in which the location of the electron in space is more carefully described. We are now ready to consider a more elaborate procedure which will allow us to talk in detail about the position of the electron, by giving a probability amplitude to find the electron anywhere and everywhere in a given situation. This more complete theory provides the underpinning for the approximations we have been making in our earlier discussions. In a sense, our early equations can be derived as a kind of approximation to the more complete theory. You may be wondering why we did not begin with the more complete theory and make the approximations as we went along. We have felt that it would be much easier for you to gain an understanding of the basic machinery of quantum mechanics by beginning with the two-state approximations and working gradually up to the more complete theory than to approach the subject the other way around. For this reason our approach to the subject appears to be in the reverse order to the one you will find in many books.  As we go into the subject of this chapter you will notice that we are breaking a rule we have always followed in the past. Whenever we have taken up any subject we have always tried to give a more or less complete description of the physics—showing you as much as we could about where the ideas led to. We have tried to describe the general consequences of a theory as well as describing some specific detail so that you could see where the theory would lead. We are now going to break that rule; we are going to describe how one can talk about probability amplitudes in space and show you the differential equations which they satisfy. We will not have time to go on and discuss many of the obvious implications which come out of the theory. Indeed we will not even be able to get far enough to relate this theory to some of the approximate formulations we have used earlier—for example, to the hydrogen molecule or to the ammonia molecule. For once, we must leave our business unfinished and open-ended. We are approaching the end of our course, and we must satisfy ourselves with trying to give you an introduction to the general ideas and with indicating the connections between what we have been describing and some of the other ways of approaching the subject of quantum mechanics. We hope to give you enough of an idea that you can go off by yourself and by reading books learn about many of the implications of the equations we are going to describe. We must, after all, leave something for the future. Let’s review once more what we have found out about how an electron can move along a line of atoms. When an electron has an amplitude to jump from one atom to the next, there are definite energy states in which the probability amplitude for finding the electron is distributed along the lattice in the form of a traveling wave. For long wavelengths—for small values of the wave number $k$—the energy of the state is proportional to the square of the wave number. For a crystal lattice with the spacing $b$, in which the amplitude per unit time for the electron to jump from one atom to the next is $iA/\hbar$, the energy of the state is related to $k$ (for small $kb$) by \begin{equation} \label{Eq:III:16:1} E=Ak^2b^2 \end{equation}  (see Section 13–2). We also saw that groups of such waves with similar energies would make up a wave packet which would behave like a classical particle with a mass $m_{\text{eff}}$ given by: \begin{equation} \label{Eq:III:16:2} m_{\text{eff}}=\frac{\hbar^2}{2Ab^2}. \end{equation}  Since waves of probability amplitude in a crystal behave like a particle, one might well expect that the general quantum mechanical description of a particle would show the same kind of wave behavior we observed for the lattice. Suppose we were to think of a lattice on a line and imagine that the lattice spacing $b$ were to be made smaller and smaller. In the limit we would be thinking of a case in which the electron could be anywhere along the line. We would have gone over to a continuous distribution of probability amplitudes. We would have the amplitude to find an electron anywhere along the line. This would be one way to describe the motion of an electron in a vacuum. In other words, if we imagine that space can be labeled by an infinity of points all very close together and we can work out the equations that relate the amplitudes at one point to the amplitudes at neighboring points, we will have the quantum mechanical laws of motion of an electron in space. Let’s begin by recalling some of the general principles of quantum mechanics. Suppose we have a particle which can exist in various conditions in a quantum mechanical system. Any particular condition an electron can be found in, we call a “state,” which we label with a state vector, say $\ket{\phi}$. Some other condition would be labeled with another state vector, say $\ket{\psi}$. We then introduce the idea of base states. We say that there is a set of states $\ket{1}$, $\ket{2}$, $\ket{3}$, $\ket{4}$, and so on, which have the following properties. First, all of these states are quite distinct—we say they are orthogonal. By this we mean that for any two of the base states $\ket{i}$ and $\ket{j}$ the amplitude $\braket{i}{j}$ that an electron known to be in the state $\ket{i}$ is also in the state $\ket{j}$ is equal to zero—unless, of course, $\ket{i}$ and $\ket{j}$ stand for the same state. We represent this symbolically by \begin{equation} \label{Eq:III:16:3} \braket{i}{j}=\delta_{ij}. \end{equation}  You will remember that $\delta_{ij}=0$ if $i$ and $j$ are different, and $\delta_{ij}=1$ if $i$ and $j$ are the same number. Second, the base states $\ket{i}$ must be a complete set, so that any state at all can be described in terms of them. That is, any state $\ket{\phi}$ at all can be described completely by giving all of the amplitudes $\braket{i}{\phi}$ that a particle in the state $\ket{\phi}$ will also be found in the state $\ket{i}$. In fact, the state vector $\ket{\phi}$ is equal to the sum of the base states each multiplied by a coefficient which is the amplitude that the state $\ket{\phi}$ is also in the state $\ket{i}$: \begin{equation} \label{Eq:III:16:4} \ket{\phi}=\sum_i\ket{i}\braket{i}{\phi}. \end{equation}  Finally, if we consider any two states $\ket{\phi}$ and $\ket{\psi}$, the amplitude that the state $\ket{\psi}$ will also be in the state $\ket{\phi}$ can be found by first projecting the state $\ket{\psi}$ into the base states and then projecting from each base state into the state $\ket{\phi}$. We write that in the following way: \begin{equation} \label{Eq:III:16:5} \braket{\phi}{\psi}=\sum_i\braket{\phi}{i}\braket{i}{\psi}. \end{equation}  The summation is, of course, to be carried out over the whole set of base states $\ket{i}$. In Chapter 13 when we were working out what happens with an electron placed on a linear array of atoms, we chose a set of base states in which the electron was localized at one or other of the atoms in the line. The base state $\ket{n}$ represented the condition in which the electron was localized at atom number “$n$.” (There is, of course, no significance to the fact that we called our base states $\ket{n}$ instead of $\ket{i}$.) A little later, we found it convenient to label the base states by the coordinate $x_n$ of the atom rather than by the number of the atom in the array. The state $\ket{x_n}$ is just another way of writing the state $\ket{n}$. Then, following the general rules, any state at all, say $\ket{\psi}$ is described by giving the amplitudes that an electron in the state $\ket{\psi}$ is also in one of the states $\ket{x_n}$. For convenience we have chosen to let the symbol $C_n$ stand for these amplitudes, \begin{equation} \label{Eq:III:16:6} C_n=\braket{x_n}{\psi}. \end{equation}  Since the base states are associated with a location along the line, we can think of the amplitude $C_n$ as a function of the coordinate $x$ and write it as $C(x_n)$. The amplitudes $C(x_n)$ will, in general, vary with time and are, therefore, also functions of $t$. We will not generally bother to show explicitly this dependence. In Chapter 13 we then proposed that the amplitudes $C(x_n)$ should vary with time in a way described by the Hamiltonian equation (Eq. 13.3). In our new notation this equation is \begin{equation} \label{Eq:III:16:7} i\hbar\ddp{C(x_n)}{t}\!=\!E_0C(x_n)\!-\! AC(x_n\!+\!b)\!-\!AC(x_n\!-\!b). \end{equation}  The last two terms on the right-hand side represent the process in which an electron at atom $(n+1)$ or at atom $(n-1)$ can feed into atom $n$. We found that Eq. (16.7) has solutions corresponding to definite energy states, which we wrote as \begin{equation} \label{Eq:III:16:8} C(x_n)=e^{-iEt/\hbar}e^{ikx_n}. \end{equation}  For the low-energy states the wavelengths are large ($k$ is small), and the energy is related to $k$ by \begin{equation} \label{Eq:III:16:9} E=(E_0-2A)+Ak^2b^2, \end{equation}  or, choosing our zero of energy so that $(E_0-2A)=0$, the energy is given by Eq. (16.1). Let’s see what might happen if we were to let the lattice spacing $b$ go to zero, keeping the wave number $k$ fixed. If that is all that were to happen the last term in Eq. (16.9) would just go to zero and there would be no physics. But suppose $A$ and $b$ are varied together so that as $b$ goes to zero the product $Ab^2$ is kept constant1—using Eq. (16.2) we will write $Ab^2$ as the constant $\hbar^2/2m_{\text{eff}}$. Under these circumstances, Eq. (16.9) would be unchanged, but what would happen to the differential equation (16.7)? First we will rewrite Eq. (16.7) as  \begin{equation} \label{Eq:III:16:10} i\hbar\,\ddp{C(x_n)}{t}=(E_0-2A)C(x_n)+A[2C(x_n)- C(x_n+b)-C(x_n-b)]. \end{equation} 
 \begin{align} \label{Eq:III:16:10} i\hbar\ddp{C(x_n)}{t}&=(E_0-2A)C(x_n)\,+\\[.5ex] &A[2C(x_n)\!-\!C(x_n\!+b)\!-\!C(x_n\!-b)].\notag \end{align}  For our choice of $E_0$, the first term drops out. Next, we can think of a continuous function $C(x)$ that goes smoothly through the proper values $C(x_n)$ at each $x_n$. As the spacing $b$ goes to zero, the points $x_n$ get closer and closer together, and (if we keep the variation of $C(x)$ fairly smooth) the quantity in the brackets is just proportional to the second derivative of $C(x)$. We can write—as you can see by making a Taylor expansion of each term—the equality \begin{equation} \label{Eq:III:16:11} 2C(x)-C(x+b)-C(x-b)\approx-b^2\, \frac{\partial^2C(x)}{\partial x^2}. \end{equation}  In the limit, then, as $b$ goes to zero, keeping $b^2A$ equal to $\hbar^2/2m_{\text{eff}}$, Eq. (16.7) goes over into \begin{equation} \label{Eq:III:16:12} i\hbar\,\ddp{C(x)}{t}=-\frac{\hbar^2}{2m_{\text{eff}}}\, \frac{\partial^2C(x)}{\partial x^2}. \end{equation}  We have an equation which says that the time rate of change of $C(x)$—the amplitude to find the electron at $x$—depends on the amplitude to find the electron at nearby points in a way which is proportional to the second derivative of the amplitude with respect to position. The correct quantum mechanical equation for the motion of an electron in free space was first discovered by Schrödinger. For motion along a line it has exactly the form of Eq. (16.12) if we replace $m_{\text{eff}}$ by $m$, the free-space mass of the electron. For motion along a line in free space the Schrödinger equation is \begin{equation} \label{Eq:III:16:13} i\hbar\,\ddp{C(x)}{t}=-\frac{\hbar^2}{2m}\, \frac{\partial^2C(x)}{\partial x^2}. \end{equation}  We do not intend to have you think we have derived the Schrödinger equation but only wish to show you one way of thinking about it. When Schrödinger first wrote it down, he gave a kind of derivation based on some heuristic arguments and some brilliant intuitive guesses. Some of the arguments he used were even false, but that does not matter; the only important thing is that the ultimate equation gives a correct description of nature. The purpose of our discussion is then simply to show you that the correct fundamental quantum mechanical equation (16.13) has the same form you get for the limiting case of an electron moving along a line of atoms. This means that we can think of the differential equation in (16.13) as describing the diffusion of a probability amplitude from one point to the next along the line. That is, if an electron has a certain amplitude to be at one point, it will, a little time later, have some amplitude to be at neighboring points. In fact, the equation looks something like the diffusion equations which we have used in Volume I. But there is one main difference: the imaginary coefficient in front of the time derivative makes the behavior completely different from the ordinary diffusion such as you would have for a gas spreading out along a thin tube. Ordinary diffusion gives rise to real exponential solutions, whereas the solutions of Eq. (16.13) are complex waves. 