The Schrödinger equation
So far we have just been worrying about how we can describe states which may involve an electron being anywhere at all in space. Now we have to worry about putting into our description the physics of what can happen in various circumstances. As before, we have to worry about how states can change with time. If we have a state $\ket{\psi}$ which goes over into another state $\ket{\psi'}$ sometime later, we can describe the situation for all times by making the wave function—which is just the amplitude $\braket{\FLPr}{\psi}$—a function of time as well as a function of the coordinate. A particle in a given situation can then be described by giving a time-varying wave function $\psi(\FLPr,t)=\psi(x,y,z,t)$. This time-varying wave function describes the evolution of successive states that occur as time develops. This so-called “coordinate representation”—which gives the projections of the state $\ket{\psi}$ into the base states $\ket{\FLPr}$ may not always be the most convenient one to use—but we will consider it first. In Chapter 8 we described how states varied in time in terms of the Hamiltonian $H_{ij}$. We saw that the time variation of the various amplitudes was given in terms of the matrix equation \begin{equation} \label{Eq:III:16:49} i\hbar\,\ddt{C_i}{t}=\sum_jH_{ij}C_j. \end{equation}  This equation says that the time variation of each amplitude $C_i$ is proportional to all of the other amplitudes $C_j$, with the coefficients $H_{ij}$. How would we expect Eq. (16.49) to look when we are using the continuum of base states $\ket{x}$? Let’s first remember that Eq. (16.49) can also be written as \begin{equation*} i\hbar\,\ddt{}{t}\,\braket{i}{\psi}= \sum_j\bracket{i}{\Hop}{j}\braket{j}{\psi}. \end{equation*}  Now it is clear what we should do. For the $x$-representation we would expect \begin{equation} \label{Eq:III:16:50} i\hbar\,\ddp{}{t}\,\braket{x}{\psi}= \int\bracket{x}{\Hop}{x'}\braket{x'}{\psi}\,dx'. \end{equation}  The sum over the base states $\ket{j}$, gets replaced by an integral over $x'$. Since $\bracket{x}{\Hop}{x'}$ should be some function of $x$ and $x'$, we can write it as $H(x,x')$—which corresponds to $H_{ij}$ in Eq. (16.49). Then Eq. (16.50) is the same as \begin{gather} \label{Eq:III:16:51} i\hbar\,\ddp{}{t}\,\psi(x)=\int \kern -.6ex H(x,x')\psi(x')\,dx' \end{gather} with \begin{gather*} H(x,x')\equiv\bracket{x}{\Hop}{x'}. \end{gather*}  According to Eq. (16.51), the rate of change of $\psi$ at $x$ would depend on the value of $\psi$ at all other points $x'$; the factor $H(x,x')$ is the amplitude per unit time that the electron will jump from $x'$ to $x$. It turns out in nature, however, that this amplitude is zero except for points $x'$ very close to $x$. This means—as we saw in the example of the chain of atoms at the beginning of the chapter, Eq. (16.12)—that the right-hand side of Eq. (16.51) can be expressed completely in terms of $\psi$ and the derivatives of $\psi$ with respect to $x$, all evaluated at the position $x$. For a particle moving freely in space with no forces, no disturbances, the correct law of physics is \begin{equation*} \int\kern -.6ex H(x,x')\psi(x')\,dx'=-\frac{\hbar^2}{2m}\, \frac{\partial^2}{\partial x^2}\,\psi(x). \end{equation*}  Where did we get that from? Nowhere. It’s not possible to derive it from anything you know. It came out of the mind of Schrödinger, invented in his struggle to find an understanding of the experimental observations of the real world. You can perhaps get some clue of why it should be that way by thinking of our derivation of Eq. (16.12) which came from looking at the propagation of an electron in a crystal. Of course, free particles are not very exciting. What happens if we put forces on the particle? Well, if the force of a particle can be described in terms of a scalar potential $V(x)$—which means we are thinking of electric forces but not magnetic forces—and if we stick to low energies so that we can ignore complexities which come from relativistic motions, then the Hamiltonian which fits the real world gives  \begin{equation} \label{Eq:III:16:52} \int\kern -.6ex H(x,x')\psi(x')\,dx'=-\frac{\hbar^2}{2m}\, \frac{\partial^2}{\partial x^2}\,\psi(x) +V(x)\psi(x). \end{equation} 
 \begin{equation} \begin{aligned} \int \kern -.6ex H&(x,x')\psi(x')\,dx'=\\ &-\frac{\hbar^2}{2m}\,\frac{\partial^2}{\partial x^2}\,\psi(x) +V(x)\psi(x).\notag \end{aligned} \label{Eq:III:16:52} \end{equation}  Again, you can get some clue as to the origin of this equation if you go back to the motion of an electron in a crystal, and see how the equations would have to be modified if the energy of the electron varied slowly from one atomic site to the other—as it might do if there were an electric field across the crystal. Then the term $E_0$ in Eq. (16.7) would vary slowly with position and would correspond to the new term we have added in (16.52). [You may be wondering why we went straight from Eq. (16.51) to Eq. (16.52) instead of just giving you the correct function for the amplitude $H(x,x')=\bracket{x}{\Hop}{x'}$. We did that because $H(x,x')$ can only be written in terms of strange algebraic functions, although the whole integral on the right-hand side of Eq. (16.51) comes out in terms of things you are used to. If you are really curious, $H(x,x')$ can be written in the following way: \begin{equation*} H(x,x')=-\frac{\hbar^2}{2m}\,\delta''(x-x') +V(x)\,\delta(x-x'), \end{equation*}  where $\delta''$ means the second derivative of the delta function. This rather strange function can be replaced by a somewhat more convenient algebraic differential operator, which is completely equivalent: \begin{equation*} H(x,x')=\biggl\{ -\frac{\hbar^2}{2m}\,\frac{\partial^2}{\partial x^2}+V(x)\biggr\} \delta(x-x'). \end{equation*}  We will not be using these forms, but will work directly with the form in Eq. (16.52).] If we now use the expression we have in (16.52) for the integral in (16.50) we get the following differential equation for $\psi(x)=\braket{x}{\psi}$: \begin{equation} \label{Eq:III:16:53} i\hbar\,\ddp{\psi}{t}=-\frac{\hbar^2}{2m}\, \frac{\partial^2}{\partial x^2}\,\psi(x)+V(x)\psi(x). \end{equation}  It is fairly obvious what we should use instead of Eq. (16.53) if we are interested in motion in three dimensions. The only changes are that $\partial^2/\partial x^2$ gets replaced by \begin{equation*} \nabla^2=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}, \end{equation*}  and $V(x)$ gets replaced by $V(x,y,z)$. The amplitude $\psi(x,y,z)$ for an electron moving in a potential $V(x,y,z)$ obeys the differential equation \begin{equation} \label{Eq:III:16:54} i\hbar\,\ddp{\psi}{t}=-\frac{\hbar^2}{2m}\, \nabla^2\psi+V\psi. \end{equation}  It is called the Schrödinger equation, and was the first quantum-mechanical equation ever known. It was written down by Schrödinger before any of the other quantum equations we have described in this book were discovered. Although we have approached the subject along a completely different route, the great historical moment marking the birth of the quantum mechanical description of matter occurred when Schrödinger first wrote down his equation in 1926. For many years the internal atomic structure of matter had been a great mystery. No one had been able to understand what held matter together, why there was chemical binding, and especially how it could be that atoms could be stable. Although Bohr had been able to give a description of the internal motion of an electron in a hydrogen atom which seemed to explain the observed spectrum of light emitted by this atom, the reason that electrons moved in this way remained a mystery. Schrödinger’s discovery of the proper equations of motion for electrons on an atomic scale provided a theory from which atomic phenomena could be calculated quantitatively, accurately, and in detail. In principle, Schrödinger’s equation is capable of explaining all atomic phenomena except those involving magnetism and relativity. It explains the energy levels of an atom, and all the facts of chemical binding. This is, however, true only in principle—the mathematics soon becomes too complicated to solve exactly any but the simplest problems. Only the hydrogen and helium atoms have been calculated to a high accuracy. However, with various approximations, some fairly sloppy, many of the facts of more complicated atoms and of the chemical binding of molecules can be understood. We have shown you some of these approximations in earlier chapters. The Schrödinger equation as we have written it does not take into account any magnetic effects. It is possible to take such effects into account in an approximate way by adding some more terms to the equation. However, as we have seen in Volume II, magnetism is essentially a relativistic effect, and so a correct description of the motion of an electron in an arbitrary electromagnetic field can only be discussed in a proper relativistic equation. The correct relativistic equation for the motion of an electron was discovered by Dirac a year after Schrödinger brought forth his equation, and takes on quite a different form. We will not be able to discuss it at all here. Before we go on to look at some of the consequences of the Schrödinger equation, we would like to show you what it looks like for a system with a large number of particles. We will not be making any use of the equation, but just want to show it to you to emphasize that the wave function $\psi$ is not simply an ordinary wave in space, but is a function of many variables. If there are many particles, the equation becomes  \begin{equation} \label{Eq:III:16:55} i\hbar\,\ddp{\psi(\FLPr_1,\FLPr_2,\FLPr_3,\dotsc)}{t}= \sum_i-\frac{\hbar^2}{2m_i}\biggl\{\! \frac{\partial^2\psi}{\partial x_i^2} +\frac{\partial^2\psi}{\partial y_i^2} +\frac{\partial^2\psi}{\partial z_i^2}\!\biggr\} +V(\FLPr_1,\FLPr_2,\dotsc)\psi. \end{equation} 
  \begin{gather} \label{Eq:III:16:55} i\hbar\,\ddp{\psi(\FLPr_1,\FLPr_2,\FLPr_3,\dotsc)}{t}=\\[1.2ex] \sum_i-\frac{\hbar^2}{2m_i}\biggl\{\! \frac{\partial^2\psi}{\partial x_i^2} +\frac{\partial^2\psi}{\partial y_i^2} +\frac{\partial^2\psi}{\partial z_i^2}\!\biggr\} +V(\FLPr_1,\FLPr_2,\dotsc)\psi.\notag \end{gather}    The potential function $V$ is what corresponds classically to the total potential energy of all the particles. If there are no external forces acting on the particles, the function $V$ is simply the electrostatic energy of interaction of all the particles. That is, if the $i$th particle carries the charge $Z_iq_e$, then the function $V$ is simply3 \begin{equation} \label{Eq:III:16:56} V(\FLPr_1,\FLPr_2,\FLPr_3,\dotsc)= \sum_{\substack{\text{all}\\\text{pairs}}} \frac{Z_iZ_j}{r_{ij}}\,e^2. \end{equation} 