3.4 Specular reflection and Bessel theory

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), Fig. 3.3–3.5 and §§3.3.2–3.4.1. Rewritten in original wording. Continues 3.3b Series labs. Labs: 3.4b Specular and Bessel labs.

Epilogue on series: correlated error (Fig. 3.3)

The alternating sum for $e^{-x}$ is a simple, highly correlated process. Its error-versus-$N$ curves dip (algorithmic error falling) and then rise sharply — but that rise is not the $\sqrt{N}\,\epsilon_m$ random walk from §3.2. Cancellation of large opposite terms feeds round-off coherently. The $\sqrt{N}$ cartoon returns when you get to integration (Chapter 5).

Slide $x$ upward: the dip moves right and the floor gets worse — the same stack of curves as Figure 3.3.

3.4.1 Specular reflection in a circle

For a perfect mirror, angle of incidence equals angle of reflection. Put the origin at the center of a circular mirror and track the ray by an angle $\theta$. Walk the bounce advance, the closing condition, and $\phi/\pi$ rational — pick the matching equation at each step.

Theorem (closed specular orbit). A ray with step $2\phi$ on a circle closes if and only if $\phi/\pi$ is rational (up to the $2\pi$ identification of the hit point).

Finite precision changes the story: if you round $\phi$ and $\theta$ to four decimals each bounce (round(x, 4)), a tiny angle error accumulates with the step count — the same long-calculation pathology as in §3.1–3.2.

Quiz

Specular orbits

1 / 3

Each bounce advances θ by…

Pick an answer to see if you are right.

3.4.2 Spherical Bessel functions

Spherical Bessel $j_\ell(x)$ and Neumann $n_\ell(x)$ are the regular / irregular solutions of

$$x^{2} f'' + 2x f' + \bigl[x^{2}-\ell(\ell+1)\bigr] f = 0.$$

They appear in partial-wave expansions, e.g.

$$e^{i\mathbf{k}\cdot\mathbf{r}} = \sum_{\ell=0}^{\infty} i^{\ell}(2\ell+1)\, j_\ell(kr)\, P_\ell(\cos\theta).$$

Recurrence: up looks easy, down is stable

Walk the analytic seeds $j_0,j_1,n_0,n_1$, the three-term recurrence (up and down), and Miller’s rescale — pick the matching equation at each step.

Upward from $j_0,j_1$: you eventually subtract two large numbers to make a tiny $j_\ell$ — subtractive cancellation. Because $n_\ell$ obeys the same recurrence, the computer’s value is effectively $j_\ell^{(c)}=j_\ell+\epsilon\,n_\ell$. When $|n_\ell|\gg|j_\ell|$, even a speck of Neumann swamps the answer.

Miller’s device: start at large $L$ with arbitrary seeds, recur downward (small + small → larger), then renormalize so $j_0$ matches $\sin x/x$. Relative $\ell$-dependence stabilizes; absolute scale comes from $j_0$.

Quiz

Bessel recurrence

1 / 3

Upward recurrence fails mainly because of…

Pick an answer to see if you are right.

Sample reference values (Maple-style, for labs):

$x$	$j_3(x)$	$j_5(x)$	$j_8(x)$
0.1	$\sim 9.52\times 10^{-6}$	$\sim 9.62\times 10^{-10}$	$\sim 2.90\times 10^{-16}$
1	$\sim 9.01\times 10^{-3}$	$\sim 9.26\times 10^{-5}$	$\sim 2.83\times 10^{-8}$
10	$\sim -3.95\times 10^{-2}$	$\sim -5.55\times 10^{-2}$	$\sim 1.26\times 10^{-1}$

Next: 3.4b Specular and Bessel labs.