$l=1$ orbitals

The $l=1$ orbitals are known as $p$ (``$p$'' for ``principal'') orbitals. The orbitals take the form

$$\psi_{n1m}(r,\theta,\phi) = R_{n1}(r)Y_{1m}(\theta,\phi)$$

where

$$Y_{1\pm 1}(\theta,\phi) = \mp\left({3 \over 8\pi}\right)^{1/2} \sin\theta e^{\pm i\phi}$$

$$Y_{10}(\theta,\phi) = \left({3 \over 4\pi}\right)^{1/2} \cos\theta$$

Thus, these orbitals are not spherically symmetric.

The $m=0$ orbital is known as the $p_z$ orbital because of the $\cos\theta$ dependence and lack of $\phi$ dependence. This resembles the spherical coordinate transformation for $z$, $z=r\cos\theta$. The figure below shows the basic shape of a $p$ orbital for $n=2$ and $n=3$:

Figure: $s$, $p$ and $d$ orbital plots.

The figure below shows the $2p$ orbitals in more detail, including scatter plots of the wave functions:

Figure: Details of the $2p$ orbitals.

At $n=3$, the nodal structure of the radial part of the orbital can be seen superimposed on the $p$-like structure, as can be seen in Fig. 6) and in the top part of the figure below:

Figure: Details of the $3p$ and $3d$ orbitals.

The nodal plane in the $p$ orbital at $\theta = \pi/2$ arises because $\cos(\pi/2)=0$ for all $\phi$, meaning that the entire $xy$ plane is a nodal plane.

The orbitals $\psi_{n11}(r,\theta,\phi)$ and $\psi_{n1-1}(r,\theta,\phi)$ are not real because of the $\exp(\pm i\phi)$ dependence of $Y_{1\pm 1}(\theta,\phi)$. Thus, these orbitals are not entire convenient to work with. Fortunately, because $\psi_{n11}$ and $\psi_{n1-1}$ are solutions of the Schrödinger equation with the same energy $E_n$, we can take any combination of these two functions we wish, and we still have a solution of the Schrödinger equation with the same energy. Thus, it is useful to take two combinations that give us two real orbitals. Consider, for example:

$$\tilde{\psi}_{p_x} = {1 \over \sqrt{2}} \left[\psi_{n1-1}(r,\theta,\phi)-\psi_{n11}(r,\theta,\phi)\right]$$

$$\tilde{\psi}_{p_y} = {i \over \sqrt{2}} \left[\psi_{n1-1}(r,\theta,\phi)+\psi_{n11}(r,\theta,\phi)\right]$$

which corresponds to defining new spherical harmonics:

$$Y_{p_x}(\theta,\phi) = {1 \over \sqrt{2}} \left[Y_{1-1}(\theta,\phi) - Y_{11}(\theta,\phi)\right] = \left({3 \over 4\pi}\right)^{3/2}\sin\theta\cos\phi$$

$$Y_{p_y}(\theta,\phi) = {i \over \sqrt{2}} \left[Y_{1-1}(\theta,\phi) + Y_{11}(\theta,\phi)\right] = \left({3 \over 4\pi}\right)^{3/2}\sin\theta\sin\phi$$

Again, the notation $p_x$ and $p_y$ is used because of the similarity to the spherical coordinate transformations $x = r\sin\theta\cos\phi$ and $y=r\sin\theta\sin\phi$. These orbitals have the same shape as the $p_z$ orbital but are rotated to be oriented along the $x$-axis for the $p_x$ orbital and along the $y$-axis for the $p_y$ orbital.

The nodal structure of the radial probability distribution functions is shown in the second row of Fig. 5. Note that these have one less node at each $n$ than those for $l=0$. That is, the number of nodes is $n-2$, or $n-l-1$. Overall, however, the total number of nodes remains the same, i.e. $n-1$. What has happened is that a radial node is ``exchange'' for an angular node.