Quantized magnetic states
In the last chapter we described how in quantum mechanics the angular momentum of a thing does not have an arbitrary direction, but its component along a given axis can take on only certain equally spaced, discrete values. It is a shocking and peculiar thing. You may think that perhaps we should not go into such things until your minds are more advanced and ready to accept this kind of an idea. Actually, your minds will never become more advanced—in the sense of being able to accept such a thing easily. There isn’t any descriptive way of making it intelligible that isn’t so subtle and advanced in its own form that it is more complicated than the thing you were trying to explain. The behavior of matter on a small scale—as we have remarked many times—is different from anything that you are used to and is very strange indeed. As we proceed with classical physics, it is a good idea to try to get a growing acquaintance with the behavior of things on a small scale, at first as a kind of experience without any deep understanding. Understanding of these matters comes very slowly, if at all. Of course, one does get better able to know what is going to happen in a quantum-mechanical situation—if that is what understanding means—but one never gets a comfortable feeling that these quantum-mechanical rules are “natural.” Of course they are, but they are not natural to our own experience at an ordinary level. We should explain that the attitude that we are going to take with regard to this rule about angular momentum is quite different from many of the other things we have talked about. We are not going to try to “explain” it, but we must at least tell you what happens; it would be dishonest to describe the magnetic properties of materials without mentioning the fact that the classical description of magnetism—of angular momentum and magnetic moments—is incorrect.  One of the most shocking and disturbing features about quantum mechanics is that if you take the angular momentum along any particular axis you find that it is always an integer or half-integer times $\hbar$. This is so no matter which axis you take. The subtleties involved in that curious fact—that you can take any other axis and find that the component for it is also locked to the same set of values—we will leave to a later chapter, when you will experience the delight of seeing how this apparent paradox is ultimately resolved.  We will now just accept the fact that for every atomic system there is a number $j$, called the spin of the system—which must be an integer or a half-integer—and that the component of the angular momentum along any particular axis will always have one of the following values between $+j\hbar$ and $-j\hbar$: \begin{equation} \label{Eq:II:35:1} J_z=\text{one of}\, \left\{ \begin{array}{@{}l@{}} \phantom{-}j\\ \phantom{-}j-1\\ \phantom{-}j-2\\ \phantom{-j}\:\:\vdots\\ -j+2\\ -j+1\\ -j \end{array} \right\} \cdot\hbar. \end{equation}  We have also mentioned that every simple atomic system has a magnetic moment which has the same direction as the angular momentum. This is true not only for atoms and nuclei but also for the fundamental particles. Each fundamental particle has its own characteristic value of $j$ and its magnetic moment. (For some particles, both are zero.) What we mean by “the magnetic moment” in this statement is that the energy of the system in a magnetic field, say in the $z$-direction, can be written as $-\mu_zB$ for small magnetic fields. We must have the condition that the field should not be too great, otherwise it could disturb the internal motions of the system and the energy would not be a measure of the magnetic moment that was there before the field was turned on. But if the field is sufficiently weak, the field changes the energy by the amount \begin{equation} \label{Eq:II:35:2} \Delta U=-\mu_zB, \end{equation} with the understanding that in this equation we are to replace $\mu_z$ by \begin{equation} \label{Eq:II:35:3} \mu_z=g\biggl(\frac{q}{2m}\biggr)J_z, \end{equation} where $J_z$ has one of the values in Eq. (35.1).  Suppose we take a system with a spin $j=3/2$. Without a magnetic field, the system has four different possible states corresponding to the different values of $J_z$, all of which have exactly the same energy. But the moment we turn on the magnetic field, there is an additional energy of interaction which separates these states into four slightly different energy levels. The energies of these levels are given by a certain energy proportional to $B$, multiplied by $\hbar$ times $3/2$, $1/2$, $-1/2$, and $-3/2$—the values of $J_z$. The splitting of the energy levels for atomic systems with spins of $1/2$, $1$, and $3/2$ are shown in the diagrams of Fig. 35-1. (Remember that for any arrangement of electrons the magnetic moment is always directed opposite to the angular momentum.)  You will notice from the diagrams that the “center of gravity” of the energy levels is the same with and without a magnetic field. Also notice that the spacings from one level to the next are always equal for a given particle in a given magnetic field. We are going to write the energy spacing, for a given magnetic field $B$, as $\hbar\omega_p$—which is just a definition of $\omega_p$. Using Eqs. (35.2) and (35.3), we have \begin{equation*} \hbar\omega_p=g\,\frac{q}{2m}\,\hbar B \end{equation*} or \begin{equation} \label{Eq:II:35:4} \phantom{\hbar}\omega_p=g\,\frac{q}{2m}\,B. \end{equation}  The quantity $g(q/2m)$ is just the ratio of the magnetic moment to the angular momentum—it is a property of the particle. Equation (35.4) is the same formula that we got in Chapter 34 for the angular velocity of precession in a magnetic field, for a gyroscope whose angular momentum is $\FLPJ$ and whose magnetic moment is $\FLPmu$. 