The magnetic energy of atoms
Now we want to talk again about the magnetic moment. We have said that in quantum mechanics the magnetic moment of a particular atomic system can be written in terms of the angular momentum by Eq. (34.6); \begin{equation} \label{Eq:II:34:27} \FLPmu=-g\biggl(\frac{q_e}{2m}\biggr)\FLPJ, \end{equation} where $-q_e$ and $m$ are the charge and mass of the electron.  An atomic magnet placed in an external magnetic field will have an extra magnetic energy which depends on the component of its magnetic moment along the field direction. We know that \begin{equation} \label{Eq:II:34:28} U_{\text{mag}}=-\FLPmu\cdot\FLPB. \end{equation} Choosing our $z$-axis along the direction of $\FLPB$, \begin{equation} \label{Eq:II:34:29} U_{\text{mag}}=-\mu_zB. \end{equation} Using Eq. (34.27), we have that \begin{equation*} U_{\text{mag}}=g\biggl(\frac{q_e}{2m}\biggr)J_zB. \end{equation*} Quantum mechanics says that $J_z$ can have only certain values: $j\hbar$, $(j-1)\hbar$, …, $-j\hbar$. Therefore, the magnetic energy of an atomic system is not arbitrary; it can have only certain values. Its maximum value, for instance, is \begin{equation*} g\biggl(\frac{q_e}{2m}\biggr)\hbar jB. \end{equation*} The quantity $q_e\hbar/2m$ is usually given the name “the Bohr magneton” and written $\mu_B$: \begin{equation*} \mu_B=\frac{q_e\hbar}{2m}. \end{equation*} The possible values of the magnetic energy are \begin{equation*} U_{\text{mag}}=g\mu_BB\,\frac{J_z}{\hbar}, \end{equation*} where $J_z/\hbar$ takes on the possible values $j$, $(j-1)$, $(j-2)$, …, $(-j+1)$, $-j$.  In other words, the energy of an atomic system is changed when it is put in a magnetic field by an amount that is proportional to the field, and proportional to $J_z$. We say that the energy of an atomic system is “split into $2j+1$ levels” by a magnetic field. For instance, an atom whose energy is $U_0$ outside a magnetic field and whose $j$ is $3/2$, will have four possible energies when placed in a field. We can show these energies by an energy-level diagram like that drawn in Fig. 34-5. Any particular atom can have only one of the four possible energies in any given field $B$. That is what quantum mechanics says about the behavior of an atomic system in a magnetic field.  The simplest “atomic” system is a single electron. The spin of an electron is $1/2$, so there are two possible states: $J_z=\hbar/2$ and $J_z=-\hbar/2$. For an electron at rest (no orbital motion), the spin magnetic moment has a $g$-value of $2$, so the magnetic energy can be either $\pm\mu_BB$. The possible energies in a magnetic field are shown in Fig. 34-6. Speaking loosely we say that the electron either has its spin “up” (along the field) or “down” (opposite the field).  For systems with higher spins, there are more states. We can think that the spin is “up” or “down” or cocked at some “angle” in between, depending on the value of $J_z$.  We will use these quantum mechanical results to discuss the magnetic properties of materials in the next chapter. 