$l=0$ orbitals

The $l=0$ orbitals are called $s$ ($s$ for ``sharp'') orbitals. When $l=0$, $m=0$ as well, and the wave functions are of the form

$$\psi_{n00}(r,\theta,\phi) = R_{n0}(r)Y_{00}(\theta,\phi) = \left({1 \over 4\pi}\right)^{1/2}R_{n0}(r)$$

There is no dependence on $\theta$ or $\phi$ because $Y_{00}$ is a constant. Thus, all of these orbitals are spherically symmetric.

Figure: $n=1$, $n=2$, and $n=3$ $s$-orbitals.

Note several things about these orbitals. First, the density of points dies off exponentially as $r$ increases, consistent with the exponential dependence of the functions $R_{n0}(r)$. We show this in more detail in the figure below:

Figure: Details of the hydrogen 1$s$ orbital.

In particular, (d) shows a scatter plot of points selected randomly from $\psi_{100}(r)$. The plot shows that we go further from the origin, the points become less dense due to the exponential decay of the wave function.

As $n$ increases, the exponentials decay more slowly as $e^{-r/a_0}$ for $n=1$, $e^{-r/2a_0}$ for $n=2$ and $e^{-r/3a_0}$ for $n=3$. Note, also, that the wave functions are peaked at $r=0$, which would suggest that the amplitude is maximal to find the electron right on top of the nucleus! In fact, we need to be careful about this interpretation, since the radial probability density $p_{n0}(r)$ contains an extra $r^2$ factor from the volume element:

$$p_{n0}(r) = r^2 R_{n0}^2(r)$$

which goes to 0 as $r\rightarrow 0$. The figure below shows the probability densities of various radial wave functions. For now, let us focus on the first row, which are the $l=0$ probabilities upt to $n=4$:

Figure: Radial probabilities up to $n=4$.

We also see that the wave functions have radial nodes. The number of nodes for $R_{n0}(r)$ is $n-1$.