The Hamiltonian matrix
The idea, then, is that to describe the quantum mechanical world we need to pick a set of base states $i$ and to write the physical laws by giving the matrix of coefficients $H_{ij}$. Then we have everything—we can answer any question about what will happen. So we have to learn what the rules are for finding the $H$’s to go with any physical situation—what corresponds to a magnetic field, or an electric field, and so on. And that’s the hardest part. For instance, for the new strange particles, we have no idea what $H_{ij}$’s to use. In other words, no one knows the complete $H_{ij}$ for the whole world. (Part of the difficulty is that one can hardly hope to discover the $H_{ij}$ when no one even knows what the base states are!) We do have excellent approximations for nonrelativistic phenomena and for some other special cases. In particular, we have the forms that are needed for the motions of electrons in atoms—to describe chemistry. But we don’t know the full true $H$ for the whole universe. The coefficients $H_{ij}$ are called the Hamiltonian matrix or, for short, just the Hamiltonian. (How Hamilton, who worked in the 1830s, got his name on a quantum mechanical matrix is a tale of history.) It would be much better called the energy matrix, for reasons that will become apparent as we work with it. So the problem is: Know your Hamiltonian! The Hamiltonian has one property that can be deduced right away, namely, that \begin{equation} \label{Eq:III:8:40} H_{ij}\cconj=H_{ji}. \end{equation}  This follows from the condition that the total probability that the system is in some state does not change. If you start with a particle—an object or the world—then you’ve still got it as time goes on. The total probability of finding it somewhere is \begin{equation*} \sum_i\abs{C_i(t)}^2, \end{equation*}  which must not vary with time. If this is to be true for any starting condition $\phi$, then Eq. (8.40) must also be true. As our first example, we take a situation in which the physical circumstances are not changing with time; we mean the external physical conditions, so that $H$ is independent of time. Nobody is turning magnets on and off. We also pick a system for which only one base state is required for the description; it is an approximation we could make for a hydrogen atom at rest, or something similar. Equation (8.39) then says \begin{equation} \label{Eq:III:8:41} i\hbar\,\ddt{C_1}{t}=H_{11}C_1. \end{equation}  Only one equation—that’s all! And if $H_{11}$ is constant, this differential equation is easily solved to give \begin{equation} \label{Eq:III:8:42} C_1=(\text{const})e^{-(i/\hbar)H_{11}t}. \end{equation}  This is the time dependence of a state with a definite energy $E=H_{11}$. You see why $H_{ij}$ ought to be called the energy matrix. It is the generalization of the energy for more complex situations. Next, to understand a little more about what the equations mean, we look at a system which has two base states. Then Eq. (8.39) reads \begin{equation} \begin{aligned} i\hbar\,\ddt{C_1}{t}&=H_{11}C_1+H_{12}C_2,\\[1ex] i\hbar\,\ddt{C_2}{t}&=H_{21}C_1+H_{22}C_2. \end{aligned} \label{Eq:III:8:43} \end{equation}  If the $H$’s are again independent of time, you can easily solve these equations. We leave you to try for fun, and we’ll come back and do them later. Yes, you can solve the quantum mechanics without knowing the $H$’s, so long as they are independent of time.