The Polar Equation: Here we verify the claim leading to equations (7-27) while obtaining p...

The Polar Equation: Here we verify the claim leading to equations (7-27) while obtaining plots that agree with the functions Θℓ,m,(θ) given in Table 7.3. First, review Section 5.10 and the Guidelines at the beginning of the Chapter 5 Computational Exercises. If we expand the derivatives in polar equation (7-26) and also divide both sides by sin20, we obtain

The symmetry of the hydrogen atom argues that the probability density should be the same as we move away from the. xy-plane in either the positive or negative z-direction. This, in turn, demands that Θℓ,θ = (θ) be either an odd or even function about the xy-plane, where θ = п/2. Therefore, let us change variables, with β = п/2, So that Θℓ,θ = (β) must either be 0 (odd) or have 0 slope (even) at β = 0 (i.e., the xy-plane). With this replacement, cos θ = -sin β and sin θ = cos β, and the polar equation becomes

(a) By writing the second derivative in the "finite difference" form of Section 5.10 and the first derivative similarly as [Θ(β)- Θ(β- Δβ)]/Δβ, show that the differential equation can be rewritten as

(b) Just as in the one-dimensional linear cases, letting β = Δβ and choosing values for Θ(Δβ) and Θ(0) would allow calculation of Θ(2Δβ), then Θ(3Δβ), then Θ(4Δβ), and so on. For Δβ, use 0.001. Now, choose the simplest values of C and mℓ(, allowed by equations (7-27), decide whether the corresponding function Θℓ,m(θ) from Table 7.3 is an odd or even function of β, choose Θ(0) and Θ(Δβ) accordingly (again, see the Chapter 5 guidelines), then plot 0 at all positive multiples of Δβ out to β = 1.57 (i.e., -п/2). Afterward, vary C and mf by small amounts and describe the effect on 0. Repeat the process for two other allowed values of C, and all allowed values of mℓ, for each C, and verify that all these sets give physically acceptable solutions. Note that the symmetries make it unnecessary to plot negative values of β. For each set of C and nip, discuss how your 0 relates to the functions in Table 7.3. (Note: Plots for larger C may seem to have a slight divergence near the end. This isn't "real." Rather, it is due to the fact that our limited precision fails to accurately "cancel" the tan β and sec β, which grow very rapidly near п/2.)