In molecular and solid-state applications, one often uses a basis of orbitals aligned with the cartesian axes rather than the basis used throughout this chapter. For example, the orbitals

are a basis for the hydrogen states with n = 2 and ℓ = 1.
(a) Show that each of these orbitals can be written as a linear combination of the orbitals Ψ_nℓm with n = 2, ℓ = 1and m = 1,0,1.
(b) Show that the states Ψ_2pi are eigenstates of the corresponding component of angular momentum: L̂_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.

Figure 4.9

2. (r,, ) = 2p, (r,, ) = 2p. (r,6,) = 1 e V32 1 V32 1 X 323 -1/2a -/2a