The solution of the two-state equations
We can now write our two-state equation in various forms, for example, either as \begin{equation*} i\hbar\,\ddt{C_i}{t}=\sum_jH_{ij}C_j \end{equation*} or \begin{equation} \label{Eq:III:11:30} i\hbar\,\ddt{\,\ket{\psi}}{t}=\Hop\,\ket{\psi}. \end{equation} They both mean the same thing. For a spin one-half particle in a magnetic field, the Hamiltonian $H$ is given by Eq. (11.8) or by Eq. (11.13).  If the field is in the $z$-direction, then—as we have seen several times by now—the solution is that the state $\ket{\psi}$, whatever it is, precesses around the $z$-axis (just as if you were to take the physical object and rotate it bodily around the $z$-axis) at an angular velocity equal to twice the magnetic field times $\mu/\hbar$. The same is true, of course, for a magnetic field along any other direction, because the physics is independent of the coordinate system. If we have a situation where the magnetic field varies from time to time in a complicated way, then we can analyze the situation in the following way. Suppose you start with the spin in the $+z$-direction and you have an $x$-magnetic field. The spin starts to turn. Then if the $x$-field is turned off, the spin stops turning. Now if a $z$-field is turned on, the spin precesses about $z$, and so on. So depending on how the fields vary in time, you can figure out what the final state is—along what axis it will point. Then you can refer that state back to the original $\ket{+}$ and $\ket{-}$ with respect to $z$ by using the projection formulas we had in Chapter 10 (or Chapter 6). If the state ends up with its spin in the direction $(\theta,\phi)$, it will have an up-amplitude $\cos\,(\theta/2)e^{-i\phi/2}$ and a down-amplitude $\sin\,(\theta/2)e^{+i\phi/2}$. That solves any problem. It is a word description of the solution of the differential equations.  The solution just described is sufficiently general to take care of any two-state system. Let’s take our example of the ammonia molecule—including the effects of an electric field. If we describe the system in terms of the states $\ketsl{\slI}$ and $\ketsl{\slII}$, the equations (9.38) and (9.39) look like this: \begin{equation} \begin{aligned} i\hbar\,\ddt{C_{\slI}}{t}&= +AC_{\slI}+\mu\Efield C_{\slII},\\[2ex] i\hbar\,\ddt{C_{\slII}}{t}&= -AC_{\slII}+\mu\Efield C_{\slI}. \end{aligned} \label{Eq:III:11:31} \end{equation} You say, “No, I remember there was an $E_0$ in there.” Well, we have shifted the origin of energy to make the $E_0$ zero. (You can always do that by changing both amplitudes by the same factor—$e^{iE_0t/\hbar}$—and get rid of any constant energy.) Now if corresponding equations always have the same solutions, then we really don’t have to do it twice. If we look at these equations and look at Eq. (11.1), then we can make the following identification. Let’s call $\ketsl{\slI}$ the state $\ket{+}$ and $\ketsl{\slII}$ the state $\ket{-}$. That does not mean that we are lining-up the ammonia in space, or that $\ket{+}$ and $\ket{-}$ has anything to do with the $z$-axis. It is purely artificial. We have an artificial space that we might call the “ammonia molecule representative space,” or something—a three-dimensional “diagram” in which being “up” corresponds to having the molecule in the state $\ketsl{\slI}$ and being “down” along this false $z$-axis represents having a molecule in the state $\ketsl{\slII}$. Then, the equations will be identified as follows. First of all, you see that the Hamiltonian can be written in terms of the sigma matrices as \begin{equation} \label{Eq:III:11:32} H=+A\sigma_z+\mu\Efield\sigma_x. \end{equation} Or, putting it another way, $\mu B_z$ in Eq. (11.1) corresponds to $-A$ in Eq. (11.32), and $\mu B_x$ corresponds to $-\mu\Efield$. In our “model” space, then, we have a constant $B$ field along the $z$-direction. If we have an electric field $\Efield$ which is changing with time, then we have a $B$ field along the $x$-direction which varies in proportion. So the behavior of an electron in a magnetic field with a constant component in the $z$-direction and an oscillating component in the $x$-direction is mathematically analogous and corresponds exactly to the behavior of an ammonia molecule in an oscillating electric field. Unfortunately, we do not have the time to go any further into the details of this correspondence, or to work out any of the technical details. We only wished to make the point that all systems of two states can be made analogous to a spin one-half object precessing in a magnetic field. 