In molecular and solid-state applications, one often uses a basis of orbitals aligned with the cartesian axes rather than the basis \psi_{n \ell m} used throughout this chapter. For example, the orbitals
\psi_{2 p_{x}}(r, \theta, \phi)=\frac{1}{\sqrt{32 \pi a^{3}}} \frac{x}{a} e^{-r / 2 a} ,
\psi_{2 p_{y}}(r, \theta, \phi)=\frac{1}{\sqrt{32 \pi a^{3}}} \frac{y}{a} e^{-r / 2 a} ,
\psi_{2 p_{z}}(r, \theta, \phi)=\frac{1}{\sqrt{32 \pi a^{3}}} \frac{z}{a} e^{-r / 2 a} .
are a basis for the hydrogen states with n=2 \text { and } \ell=1 .
(a) Show that each of these orbitals can be written as a linear combination of the orbitals \psi_{n \ell m} \text { with } n=2, \ell=1 \text {, and } m=-1,0,1 .
(b) Show that the states \psi_{2} p_{i} are eigenstates of the corresponding component of angular momentum: \hat{L}_{i} What is the eigenvalue in each case.
(c) Make contour plots (as in Figure 4.9) for the three orbitals. In Mathematica use ContourPlot3D.
