The hydrogen wave functions
Let’s review what we have discovered. The states which satisfy Schrödinger’s equation for an electron in a Coulomb field are characterized by three quantum numbers $n$, $l$, $m$, all integers. The angular distribution of the electron amplitude can have only certain forms which we call $Y_{l,m}$. They are labeled by $l$, the quantum number of total angular momentum, and $m$, the “magnetic” quantum number, which can range from $-l$ to $+l$. For each angular configuration, various possible radial distributions $F_{n,l}(r)$ of the electron amplitude are possible; they are labeled by the principal quantum number $n$—which can range from $l+1$ to $\infty$. The energy of the state depends only on $n$, and increases with increasing $n$.  The lowest energy, or ground, state is an $s$-state. It has $l=0$, $n=1$, and $m=0$. It is a “nondegenerate” state—there is only one with this energy, and its wave function is spherically symmetric. The amplitude to find the electron is a maximum at the center, and falls off monotonically with increasing distance from the center. We can visualize the electron amplitude as a blob as shown in Fig. 19–6(a).  There are other $s$-states with higher energies, for $n=2$, $3$, $4$, … For each energy there is only one version ($m=0$), and they are all spherically symmetric. These states have amplitudes which alternate in sign one or more times with increasing $r$. There are $n-1$ spherical nodal surfaces—the places where $\psi$ goes through zero. The $2s$-state ($l=0$, $n=2$), for example, will look as sketched in Fig. 19–6(b). (The dark areas indicate regions where the amplitude is large, and the plus and minus signs indicate the relative phases of the amplitude.) The energy levels of the $s$-states are shown in the first column of Fig. 19–7.  Then there are the $p$-states—with $l=1$. For each $n$, which must be $2$ or greater, there are three states of the same energy, one each for $m=+1$, $m=0$, and $m=-1$. The energy levels are as shown in Fig. 19–7. The angular dependences of these states are given in Table 19–1. For instance, for $m=0$, if the amplitude is positive for $\theta$ near zero, it will be negative for $\theta$ near $180^\circ$. There is a nodal plane coincident with the $xy$-plane. For $n>2$ there are also spherical nodes. The $n=2$, $m=0$ amplitude is sketched in Fig. 19–6(c), and the $n=3$, $m=0$ wave function is sketched in Fig. 19–6(d).  You might think that since $m$ represents a kind of “orientation” in space, there should be similar distributions with the peaks of amplitude along the $x$-axis or along the $y$-axis. Are these perhaps the $m=+1$ and $m=-1$ states? No. But since we have three states with equal energies, any linear combinations of the three will also be stationary states of the same energy. It turns out that the “$x$”-state—which corresponds to the “$z$”-state, or $m=0$ state, of Fig. 19–6(c)—is a linear combination of the $m=+1$ and $m=-1$ states. The corresponding “$y$”-state is another combination. Specifically, we mean that \begin{align*} \text{“$z$”}&=\ket{1,0},\\[1ex] \text{“$x$”}&=-\frac{\ket{1,+1}-\ket{1,-1}}{\sqrt{2}},\\[1ex] \text{“$y$”}&=-\frac{\ket{1,+1}+\ket{1,-1}}{i\sqrt{2}}. \end{align*} These states all look the same when referred to their particular axes.  The $d$-states ($l=2$) have five possible values of $m$ for each energy, the lowest energy has $n=3$. The levels go as shown in Fig. 19–7. The angular dependences get more complicated. For instance the $m=0$ states have two conical nodes, so the wave function reverses phase from $+$, to $-$, to $+$ as you go around from the north pole to the south pole. The rough form of the amplitude is sketched in (e) and (f) of Fig. 19–6 for the $m=0$ states with $n=3$ and $n=4$. Again, the larger $n$’s have spherical nodes.  We will not try to describe any more of the possible states. You will find the hydrogen wave functions described in more detail in many books. Two good references are L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics, McGraw-Hill (1935); and R. B. Leighton, Principles of Modern Physics, McGraw-Hill (1959). You will find in them graphs of some of the functions and pictorial representations of many states.  We would like to mention one particular feature of the wave functions for higher $l$: for $l>0$ the amplitudes are zero at the center. That is not surprising, since it’s hard for an electron to have angular momentum when its radius arm is very small. For this reason, the higher the $l$, the more the amplitudes are “pushed away” from the center. If you look at the way the radial functions $F_{n,l}(r)$ vary for small $r$, you find from (19.53) that \begin{equation*} F_{n,l}(r)\approx r^l. \end{equation*} Such a dependence on $r$ means that for larger $l$’s you have to go farther from $r=0$ before you get an appreciable amplitude. This behavior is, incidentally, determined by the centrifugal force term in the radial equation, so the same thing will apply for any potential that varies slower than $1/r^2$ for small $r$—which most atomic potentials do. 