States with an angular dependence
In the states described by the $\psi_n(r)$ we have found that the probability amplitude for finding the electron is spherically symmetric—depending only on $r$, the distance for the proton. Such states have zero orbital angular momentum. We should now inquire about states which may have some angular dependences.  We could, if we wished, just investigate the strictly mathematical problem of finding the functions of $r$, $\theta$, and $\phi$ which satisfy the differential equation (19.7)—putting in the additional physical conditions that the only acceptable functions are ones which go to zero for large $r$. You will find this done in many books. We are going to take a short cut by using the knowledge we already have about how amplitudes depend on angles in space.  The hydrogen atom in any particular state is a particle with a certain “spin” $j$—the quantum number of the total angular momentum. Part of this spin comes from the electron’s intrinsic spin, and part from the electron’s motion. Since each of these two components acts independently (to an excellent approximation) we will again ignore the spin part and think only about the “orbital” angular momentum. This orbital motion behaves, however, just like a spin. For example, if the orbital quantum number is $l$, the $z$-component of angular momentum can be $l$, $l-1$, $l-2$, …, $-l$. (We are, as usual, measuring in units of $\hbar$.)  Also, all the rotation matrices and other properties we have worked out still apply. (From now on we will really ignore the electron’s spin; when we speak of “angular momentum” we will mean only the orbital part.)  Since the potential $V$ in which the electron moves depends only on $r$ and not on $\theta$ or $\phi$, the Hamiltonian is symmetric under all rotations. It follows that the angular momentum and all its components are conserved. (This is true for motion in any “central field”—one which depends only on $r$—so is not a special feature of the Coulomb $e^2/r$ potential.)  Now let’s think of some possible state of the electron; its internal angular structure will be characterized by the quantum number $l$. Depending on the “orientation” of the total angular momentum with respect to the $z$-axis, the $z$-component of angular momentum will be $m$, which is one of the $2l+1$ possibilities between $+l$ and $-l$. Let’s say $m=1$. With what amplitude will the electron be found on the $z$-axis at some distance $r$? Zero. An electron on the $z$-axis cannot have any orbital angular momentum around that axis. Alright, suppose $m$ is zero, then there can be some nonzero amplitude to find the electron at each distance from the proton. We’ll call this amplitude $F_l(r)$. It is the amplitude to find the electron at the distance $r$ up along the $z$-axis, when the atom is in the state $\ket{l,0}$, by which we mean orbital spin $l$ and $z$-component $m=0$.  If we know $F_l(r)$ everything is known. For any state $\ket{l,m}$, we know the amplitude $\psi_{l,m}(\FLPr)$ to find the electron anywhere in the atom. How? Watch. Suppose we have the atom in the state $\ket{l,m}$, what is the amplitude to find the electron at the angle $\theta,\phi$ and the distance $r$ from the origin? Put a new $z$-axis, say $z'$, at that angle (see Fig. 19–3), and ask what is the amplitude that the electron will be at the distance $r$ along the new axis $z'$? We know that it cannot be found along $z'$ unless its $z'$-component of angular momentum, say $m'$, is zero. When $m'$ is zero, however, the amplitude to find the electron along $z'$ is $F_l(r)$. Therefore, the result is the product of two factors. The first is the amplitude that an atom in the state $\ket{l,m}$ along the $z$-axis will be in the state $\ket{l,m'=0}$ with respect to the $z'$-axis. Multiply that amplitude by $F_l(r)$ and you have the amplitude $\psi_{l,m}(\FLPr)$ to find the electron at $(r,\theta,\phi)$ with respect to the original axes.  Let’s write it out. We have worked out earlier the transformation matrices for rotations. To go from the frame $x,y,z$ to the frame $x',y',z'$ of Fig. 19–3, we can rotate first around the $z$-axis by the angle $\phi$, and then rotate about the new $y$-axis ($y'$) by the angle $\theta$. This combined rotation is the product \begin{equation*} R_y(\theta)R_z(\phi). \end{equation*} The amplitude to find the state $l,m'=0$ after the rotation is \begin{equation} \label{Eq:III:19:31} \bracket{l,0}{R_y(\theta)R_z(\phi)}{l,m}. \end{equation} Our result, then, is \begin{equation} \label{Eq:III:19:32} \psi_{l,m}(\FLPr)=\bracket{l,0}{R_y(\theta)R_z(\phi)}{l,m}F_l(r). \end{equation}  The orbital motion can have only integral values of $l$. (If the electron can be found anywhere at $r\neq0$, there is some amplitude to have $m=0$ in that direction. And $m=0$ states exist only for integral spins.) The rotation matrices for $l=1$ are given in Table 17–2. For larger $l$ you can use the general formulas we worked out in Chapter 18. The matrices for $R_z(\phi)$ and $R_y(\theta)$ appear separately, but you know how to combine them. For the general case you would start with the state $\ket{l,m}$ and operate with $R_z(\phi)$ to get the new state $R_z(\phi)\,\ket{l,m}$ (which is just $e^{im\phi}\,\ket{l,m}$). Then you operate on this state with $R_y(\theta)$ to get the state $R_y(\theta)R_z(\phi)\,\ket{l,m}$. Multiplying by $\bra{l,0}$ gives the matrix element (19.31).  The matrix elements of the rotation operation are functions of $\theta$ and $\phi$. The particular functions which appear in (19.31) also show up in many kinds of problems which involve waves in spherical geometries and so has been given a special name. Not everyone uses the same convention; but one of the most common ones is \begin{equation} \label{Eq:III:19:33} \bracket{l,0}{R_y(\theta)R_z(\phi)}{l,m}\equiv a\,Y_{l,m}(\theta,\phi). \end{equation}  The functions $Y_{l,m}(\theta,\phi)$ are called the spherical harmonics, and $a$ is just a numerical factor which depends on the definition chosen for $Y_{l,m}$. For the usual definition, \begin{equation} \label{Eq:III:19:34} a=\sqrt{\frac{4\pi}{2l+1}}. \end{equation} With this notation, the hydrogen wave functions can be written \begin{equation} \label{Eq:III:19:35} \psi_{l,m}(\FLPr)=a\,Y_{l,m}(\theta,\phi)F_l(r). \end{equation}  The angle functions $Y_{l,m}(\theta,\phi)$ are important not only in many quantum-mechanical problems, but also in many areas of classical physics in which the $\nabla^2$ operator appears, such as electromagnetism. As another example of their use in quantum mechanics, consider the disintegration of an excited state of Ne$^{20}$ (such as we discussed in the last chapter) which decays by emitting an $\alpha$-particle and going into O$^{16}$: \begin{equation*} \text{Ne}^{20*}\to\text{O}^{16}+\text{He}^4. \end{equation*} Suppose that the excited state has some spin $l$ (necessarily an integer) and that the $z$-component of angular momentum is $m$. We might now ask the following: given $l$ and $m$, what is the amplitude that we will find the $\alpha$-particle going off in a direction which makes the angle $\theta$ with respect to the $z$-axis and the angle $\phi$ with respect to the $xz$-plane—as shown in Fig. 19–4.  To solve this problem we make, first, the following observation. A decay in which the $\alpha$-particle goes straight up along $z$ must come from a state with $m=0$. This is so because both O$^{16}$ and the $\alpha$-particle have spin zero, and because their motion cannot have any angular momentum about the $z$-axis. Let’s call this amplitude $a$ (per unit solid angle). Then, to find the amplitude for a decay at the arbitrary angle of Fig. 19–4, all we need to know is what amplitude the given initial state has zero angular momentum about the decay direction. The amplitude for the decay at $\theta$ and $\phi$ is then $a$ times the amplitude that a state $\ket{l,m}$ with respect to the $z$-axis will be in the state $\ket{l,0}$ with respect to $z'$—the decay direction. This latter amplitude is just what we have written in (19.31). The probability to see the $\alpha$-particle at $\theta,\phi$ is \begin{equation*} P(\theta,\phi)=a^2\abs{\bracket{l,0}{R_y(\theta)R_z(\phi)}{l,m}}^2. \end{equation*}  As an example, consider an initial state with $l=1$ and various values of $m$. From Table 17–2 we know the necessary amplitudes. They are \begin{equation} \begin{aligned} \bracket{1,0}{R_y(\theta)R_z(\phi)}{1,+1}&= -\frac{1}{\sqrt{2}}\sin\theta e^{i\phi},\\[1ex] \bracket{1,0}{R_y(\theta)R_z(\phi)}{1,0}&=\cos\theta,\\[1ex] \bracket{1,0}{R_y(\theta)R_z(\phi)}{1,-1}&= \frac{1}{\sqrt{2}}\sin\theta e^{-i\phi}. \end{aligned} \label{Eq:III:19:36} \end{equation} These are the three possible angular distribution amplitudes—depending on the $m$-value of the initial nucleus.  Amplitudes such as the ones in (19.36) appear so often and are sufficiently important that they are given several names. If the angular distribution amplitude is proportional to any one of the three functions or any linear combination of them, we say, “The system has an orbital angular momentum of one.” Or we may say, “The Ne$^{20*}$ emits a $p$-wave $\alpha$-particle.” Or we say, “The $\alpha$-particle is emitted in an $l=1$ state.” Because there are so many ways of saying the same thing it is useful to have a dictionary. If you are going to understand what other physicists are talking about, you will just have to memorize the language. In Table 19–1 we give a dictionary of orbital angular momentum.  ($l=j=$ an integer) If the orbital angular momentum is zero, then there is no change when you rotate the coordinate system and there is no variation with angle—the “dependence” on angle is as a constant, say $1$. This is also called an “$s$-state”, and there is only one such state—as far as angular dependence is concerned. If the orbital angular momentum is $1$, then the amplitude of the angular variation may be any one of the three functions given—depending on the value of $m$—or it may be a linear combination. These are called “$p$-states,” and there are three of them. If the orbital angular momentum is $2$ then there are the five functions shown. Any linear combination is called an “$l=2$,” or a “$d$-wave” amplitude. Now you can immediately guess what the next letter is—what should come after $s$, $p$, $d$? Well, of course, $f$, $g$, $h$, and so on down the alphabet!  The letters don’t mean anything. (They did once mean something—they meant “sharp” lines, “principal” lines, “diffuse” lines and “fundamental” lines of the optical spectra of atoms. But those were in the days when people did not know where the lines came from. After $f$ there were no special names, so we now just continue with $g$, $h$, and so on.)  The angular functions in the table go by several names—and are sometimes defined with slightly different conventions about the numerical factors that appear out in front. Sometimes they are called “spherical harmonics,” and written as $Y_{l,m}(\theta,\phi)$. Sometimes they are written $P_l^m(\cos\theta)e^{im\phi}$, and if $m=0$, simply as $P_l(\cos\theta)$. The functions $P_l(\cos\theta)$ are called the “Legendre polynomials” in $\cos\theta$, and the functions $P_l^m(\cos\theta)$ are called the “associated Legendre functions.” You will find tables of these functions in many books.  Notice, incidentally, that all the functions for a given $l$ have the property that they have the same parity—for odd $l$ they change sign under an inversion and for even $l$ they don’t change. So we can write that the parity of a state of orbital angular momentum $l$ is $(-1)^l$.  As we have seen, these angular distributions may refer to a nuclear disintegration or some other process, or to the distribution of the amplitude to find an electron at some place in the hydrogen atom. For instance, if an electron is in a $p$-state ($l=1$) the amplitude to find it can depend on the angle in many possible ways—but all are linear combinations of the three functions for $l=1$ in Table 19–1. Let’s take the case $\cos\theta$. That’s interesting. That means that the amplitude is positive, say, in the upper part ($\theta < \pi/2$), is negative in the lower part ($\theta > \pi/2$), and is zero when $\theta$ is $90^\circ$. Squaring this amplitude we see that the probability of finding the electron varies with $\theta$ as shown in Fig. 19–5—and is independent of $\phi$. This angular distribution is responsible for the fact that in molecular binding the attraction of an electron in an $l=1$ state for another atom depends on direction—it is the origin of the directed valences of chemical attraction. 