Scattering

Manchester, 1909. Geiger and Marsden are doing the most boring experiment imaginable: firing alpha particles at a gold foil a few thousand atoms thick and counting tiny flashes on a zinc-sulfide screen, hour after hour, in a dark room. The prevailing picture of the atom — charge spread out like pudding — says every alpha should sail through with at most a fraction of a degree of deflection. Thousands do exactly that. Then the count comes in: about one alpha in eight thousand comes back at more than 90°. Rutherford later said it was the most incredible event of his life — a heavy shell rebounding from tissue paper.

Think about what that one-in-eight-thousand means before any formalism. A rebound needs something in the atom that is both very heavy and very concentrated — spread the same charge over the whole atom and no single encounter is violent enough to turn an alpha around. From the fraction of large-angle counts, Rutherford could even compute how concentrated: the positive charge sits in a region about 10⁻¹⁴ m across — ten thousand times smaller than the atom's own 10⁻¹⁰ m, so the nucleus holds a trillionth of the atom's volume and the rest is empty. The nucleus was discovered not by seeing it but by counting ricochets at angles — and that is the template. Beam in, count at angle θ, infer the target. Hofstadter measured the proton's size that way in the 1950s; SLAC found quarks inside it that way in 1969; every collider since is the same experiment with better beams.

So the question this chapter must answer: a beam of particles with a definite momentum comes in, a detector sits at angle θ — what does quantum mechanics predict for the count rate? Chapters 9–13 solved bound states, where the spectrum was the payoff. Here nothing is bound; the energy is whatever we dial the beam to. The payoff must be something else, and finding out what — a phase, it will turn out, one number per angular momentum channel — is the story of this chapter.

First the bookkeeping that connects counters to theory, because every scattering paper you will ever read speaks this dialect. Send a beam of flux Φ (particles per area per second) at a target. A detector subtending solid angle dΩ at angle (θ, φ) clicks dN times per second. The ratio

has units of area and is the whole experimental deliverable: the differential cross-section. It is the effective bullseye area the target presents for deflecting particles into that particular direction. Integrate over all directions and you get the total cross-section σ — the target's total effective area for scattering anywhere. Nuclear physicists measure σ in barns, 10⁻²⁸ m², as in “can't hit the broad side of.” The name is a real joke made by real people during the Manhattan Project, and it stuck.

Now the quantum side. Set up the stationary problem: a particle of energy E = ħ²k²/2m in a potential V(r) that dies off at large r. Far from the target, what can the wavefunction look like? There is an incoming plane wave. And there must be an outgoing disturbance produced by the target — a wave radiating from the origin, which at large r means $e^{ikr}/r$ (the 1/r keeps the outgoing probability flux finite, exactly like light from a lamp). The most general thing the target can do is modulate that outgoing wave with a direction-dependent strength. So we propose:

and the function f(θ) — the scattering amplitude, with units of length — is where all the physics lives. To connect it to (14.1), compute the probability current (5.7) for each piece: the plane wave carries flux ħk/m; the scattered wave carries radial flux (ħk/m)|f(θ)|²/r² — and a detector of area dA at distance r subtends dΩ = dA/r², so the r² cancels and the count rate per solid angle is flux × |f|². Divide as (14.1) instructs:

That is the dictionary, complete. The detector measures the squared modulus of an amplitude — the same Born-rule structure as everything since Chapter 1, now stated for a continuum. And notice what the dictionary quietly implies: since only |f|² is measured at each angle, f can hide a phase from any single detector — but not from an interference. The forward direction is where the scattered wave overlaps the surviving beam, and their interference must account for every particle removed from that beam. Worked through (exercise 2), that bookkeeping becomes the optical theorem:

— the total cross-section is fixed by the imaginary part of the amplitude at exactly θ = 0. Scattering anywhere must cast a shadow forward, and a shadow is interference. Keep that in your pocket for the hard sphere in 14.5.

We need f(θ) from V(r). The honest equation is an integral equation (the scattered wave at r depends on the wave everywhere inside the potential, which depends on the scattered wave…), and Chapter 12 taught us exactly what to do with a problem like that: expand in powers of the perturbation and keep the first term. Inside the target, replace the true wave by the incident wave — pretend the potential is too weak to distort what it scatters. Each volume element then radiates a wavelet with strength V(r), and summing wavelets from all elements — each carried to the detector by the free outgoing Green's function $e^{ik|\vec{r}-\vec{r}{}^{\prime }|}/|\vec{r}-\vec{r}{}^{\prime }|$, the response that turns the scattering equation into an integral (the outgoing wave a single point source produces) — gives the first Born approximation:

Stop and look at what that says, because it is one of the great punchlines of the theory: the scattering amplitude is the Fourier transform of the potential, evaluated at the momentum the collision transfers. A detector swept through angle is a Fourier analyzer pointed at the target. Sharp features in V of size d live at q ≈ 1/d; to see them you need 2k sin(θ/2) to reach 1/d — which is precisely why seeing small things requires large momentum, and why “microscope” and “accelerator” are the same instrument at different budgets.

For a central potential the angular integral in (14.5) is elementary and leaves a one-dimensional transform,

and one worked example pays for the whole section. Take the Yukawa potential $V=-g{e}^{-\mu r}/r$ — a Coulomb force wearing an exponential blanket of range 1/μ (this is what the Coulomb field of a nucleus looks like through its screen of atomic electrons, and, in Yukawa's original reading, what a force carried by a particle of mass μħ/c looks like). The integral is doable in your head via $e^{-\mu r}\sin (qr)$:

Remove the screen and out falls the Rutherford formula — the very 1/sin⁴(θ/2) law Rutherford derived in 1911 with classical hyperbolic orbits, the law Geiger and Marsden's flashes obeyed. That a planetary-orbit calculation and a first-order wave calculation give the identical answer is a famous accident special to 1/r (even the exact quantum Coulomb amplitude only differs by a phase). But it explains something historically important: Rutherford could not have known he was doing the Born approximation fifteen years early — 1911 to 1926 — yet his formula fit, so the nucleus was believed. Physics got lucky exactly once, in the one potential where luck was guaranteed.

When can you trust Born? The derivation replaced the true wave by the incident wave inside the target, so the condition is that the potential distorts the wave only a little where it matters — weak V or fast beam. We will not settle for the hand-wave: the lab in 14.6 measures the quality of the approximation against an exact calculation, and — better — checks that its error scales linearly with V₀, which is the signature of a truncated first-order series and not of some other sin.

Born is a weak-potential story. For strong potentials — the hard spheres, the deep wells, the resonances where cross-sections blow up a hundredfold — we need machinery that is exact, and Chapter 9 already built it. A central potential conserves angular momentum, so a beam is best chopped into angular-momentum channels: the plane wave decomposes as

a superposition of spherical waves, one per l, with the spherical Bessel function j_l as radial profile. Two facts about the pieces, both doing heavy lifting. First, at large r each free channel is a standing wave, j_l(kr) → sin(kr − lπ/2)/kr: an incoming and an outgoing spherical wave in perfect balance. Second — this is the classical picture hiding inside — a particle with impact parameter b carries angular momentum ħkb, so the l-th channel represents impact parameters near b ≈ l/k, and a potential of range a can only touch channels with

A slow beam (ka ≪ 1) interacts through one channel only, l = 0. Hold that thought.

Now put the potential in. It conserves each l separately, and it cannot destroy probability — whatever flows in per channel must flow out. Staring at the standing wave sin(kr − lπ/2), there is exactly one thing a potential is allowed to do to it: slide it.

One real number per channel, the phase shift δ_l — that is the entire effect of any central potential on a beam, the continuum's answer to “what replaces the energy spectrum.” The sign even carries intuition: an attractive well pulls the wave in and drags the pattern outward-in (δ > 0); a repulsive core pushes it out (δ < 0). Subtract the free wave from the shifted wave to isolate the outgoing part, match coefficients against (14.2), and the amplitude and cross-section assemble themselves:

Check it against the optical theorem (14.4): at θ = 0 every P_l is 1, and $\mathrm{Im}{e}^{i\delta }\sin \delta ={\sin }^2\delta$ — the two formulas in (14.11) are the same statement. Unitarity is built in, and it puts a ceiling on each channel: sin²δ_l ≤ 1 means channel l can contribute at most 4π(2l+1)/k², no matter how monstrous the potential. When a channel hits that ceiling — δ_l = π/2 — the channel is said to be at resonance. Section 14.5 is about what makes it happen.

The explorer is running the real calculation — a Numerov integration through the potential, one channel at a time, the same algorithm the lab in 14.6 runs. It is deliberately not the same computation: to re-solve on every slider drag the browser takes a step twice as coarse and stops the integration well short of where the lab stops it, so the widget is an intuition instrument and the lab's scoreboard is the measurement. Read the trends here; quote the precision from there.

With that said, (14.9) is the first thing to see in it: drop k and watch the channels die until l = 0 stands alone and the angular pattern goes spherical. This is why low-energy nuclear physics got away with s-waves for decades, and why the low-energy limit of any short-range potential is a single number (the scattering length a_s = −lim δ₀/k, exercise 4). Then park k low and deepen the well — the explorer's slider sets the interior value V(r < a), so a well of depth V₀ sits at V(r < a) = −V₀ — until you reach −π²/8 ≈ −1.23. Something dramatic happens near there, and it is the subject of the next section.

Three exactly solvable stories, in rising order of drama. They are the referees of the lab, so we state them carefully.

The hard sphere. Inside r = a the wave must vanish; outside it is free. In each channel u_l is a combination of the two free solutions that has a node at a, which pins the phase immediately: tan δ_l = j_l(ka)/n_l(ka). For a slow beam only l = 0 survives, δ₀ = −ka exactly, and (14.11) gives σ → 4π/k² · sin²(ka) → 4πa². Four times the area the marble blocks! A slow wave has wavelength much larger than the sphere; it does not graze edges, it wraps the whole obstacle and feels its full surface. And at high energy, where you would bet on geometry, σ settles to 2πa² — the extra πa² is the shadow, the forward-peaked diffraction required by the optical theorem to cancel the beam behind the sphere. A sharp shadow is not the absence of waves; it is a wave achievement.

The square well. Radius a and depth V₀, which from here on means exactly one thing: $V(r)=-V_0$ inside r = a and zero outside, with $V_0>0$. (The lab, the scoreboard and this section all quote that positive depth; the explorer's slider, which must also be able to raise a barrier, quotes the signed interior value $V(r<a)=-V_0$ instead.) Inside, the l = 0 wave is sin(Kr) with K = √(k² + 2mV₀/ħ²); outside, sin(kr + δ₀). Matching the logarithmic derivative ${\psi }^{\prime }/\psi$ at the edge — the scale-free slope-to-value ratio that must run continuously across a boundary, since the unknown normalization cancels in the ratio — gives the classic

which is worth reading, not just using: the −ka is a trivial geometric offset, and the interesting term compares the wave's slope at the edge to what a free wave would have. Now the drama. As the well deepens, tan(Ka) passes through infinity at Ka = π/2 — exactly the depth (V₀ = π²ħ²/8ma²) at which the well captures its first bound state (Chapter 6's condition). Near that depth, at low k, δ₀ swings through π/2, sin²δ₀ hits its unitarity ceiling, and σ₀ = 4π sin²δ₀/k² — with k small — becomes enormous: a zero-energy resonance. A state that barely binds (or barely fails to bind) makes the target look orders of magnitude bigger than it is. Nature ran the experiment: the neutron–proton system has one barely-bound state (the deuteron) and one barely-unbound (the spin-singlet), so thermal neutrons see protons ~100× geometric size — the accident that makes ordinary water a good moderator and nuclear reactors feasible.

Levinson's census. Push the well deeper still and the drama repeats at each new capture — and something cleaner emerges. Track δ₀(k) continuously from high energy (where it must vanish) down to threshold. Each bound state the well holds contributes exactly π to δ₀(0):

Sit with the strangeness of that. A scattering experiment — beams in, beams out, positive energies only — counts the negative-energy states it never populates. The deep reason is that bound states and scattering states are one analytic family (the S-matrix $e^{2i\delta (k)}$ has a pole at every bound state, exercise 6), and the phase shift, being a winding, keeps integer count of the poles. The lab verifies the census numerically at four well depths — the integer read off the tracked phase against the exact square-well count, tabulated in 14.6.

One vocabulary item before the lab, because you will meet it from nuclear physics to particle listings: near any energy E_R where a channel's phase crosses π/2 steeply — a well depth that almost traps a state behind a centrifugal or potential barrier — expanding cot δ about E_R turns (14.11) into

the Breit–Wigner peak: a Lorentzian bump of width Γ, the continuum's memory of an almost-bound state that lives for a time ħ/Γ before leaking out. Every “particle” discovered as a bump in a cross-section — the Δ, the J/ψ, the Higgs — is this equation wearing different units.

The lab (Rust-QP/ch14-scatter) builds one honest instrument: integrate u_l outward through any central potential with Numerov (Chapter 10's workhorse), then read the phase shift by matching to free solutions at two radii — the ratio of samples kills the unknown normalization, and no fitting is involved anywhere:

1/// integrate u'' = [l(l+1)/r^2 + 2V(r) - k^2] u outward and return
2/// tan(delta_l) from Riccati-Bessel matching at two radii.
3///
4/// Starting from the ORIGIN we must never evaluate f(0) — the
5/// centrifugal term is 0/0 or infinite there, and the resulting NaN
6/// poisons everything downstream. (Worse: f64::max IGNORES NaN, so the
7/// first version of this lab had referees "passing" on NaN — the Born
8/// ratio check, a division, was the only one honest enough to fail.)
9/// Cure: seed with the exact limiting behavior u ~ r^{l+1} at r = H and
10/// 2H, and start the recurrence at r = 2H.
11fn phase_shift<V: Fn(f64) -> f64>(l: usize, k: f64, v: &V, r_start: f64, r_max: f64) -> f64 {
12    let f = |r: f64| (l * (l + 1)) as f64 / (r * r) + 2.0 * v(r) - k * k;
13    let (mut u_prev, mut u_curr, start_r);
14    if r_start > 0.0 {
15        // hard-wall start: u(r_start) = 0, any small value one step in
16        u_prev = 0.0;
17        u_curr = 1e-6;
18        start_r = r_start + H;
19    } else {
20        // origin start: power-law seeds, first f evaluation at r = H
21        u_prev = H.powi(l as i32 + 1);
22        u_curr = (2.0 * H).powi(l as i32 + 1);
23        start_r = 2.0 * H;
24    }
25    // …
26    let (u2, r2_used) = (u_curr, r);
27    // asymptotically u ~ S_l(kr) cos d - C_l(kr) sin d, with the
28    // Riccati-Bessel functions S = kr j_l, C = kr n_l; the ratio of u at
29    // two radii then pins tan(delta) with no normalization needed:
30    let (j1v, n1v) = sph_jn(l, k * r1_used);
31    let (j2v, n2v) = sph_jn(l, k * r2_used);
32    let s1 = k * r1_used * j1v[l];
33    let c1 = k * r1_used * n1v[l];
34    let s2 = k * r2_used * j2v[l];
35    let c2 = k * r2_used * n2v[l];
36    let rho = u2 / u1;
37    (s2 - rho * s1) / (c2 - rho * c1)
38}

Then it referees the instrument against every exact result this chapter derived. This lab earned its keep twice before passing, and both war stories are worth the page they cost. First: the hard-sphere referee failed at |δ − exact| = 1.556 — not noise, wrongness — and bisecting the failure landed on the spherical Bessel routine: Miller's downward recurrence for j_l is only self-purifying where j_l is the minimal solution, l ≳ x, so its start order must exceed the argument, not just the top order requested. Second: the first version never evaluated the centrifugal term at r = 0 — or so we thought. It did, minted a NaN, and four referees passed anyway, because IEEE's max ignores NaN and a NaN error looks like no error. Only the Born scaling check — a division — refused to launder it. The fix is in the code above; the lesson (assert finiteness first, then smallness) is now stamped into every referee this book will write from here on.

1// j_l: Miller's downward recurrence, normalized by j_0. The start
2    // order must exceed BOTH l_max and x: below x the recurrence is
3    // oscillatory and never purifies toward j (our first version started
4    // at l_max + 15 regardless of x, and j_1(58) came out 57x too large
5    // — caught by the hard-sphere referee).
6    let start = l_max + 20 + x.ceil() as usize;
7    // …
8    // -- 2. square well: delta_0 numeric vs exact --------------------------
9    // A sharp edge sitting ON a grid point is ambiguous at the one-cell
10    // scale: the sampled well is effectively ~H/2 narrower, and since
11    // d(delta)/da = O(1) that costs ~5e-4 in the phase — hundreds of times
12    // our tolerance (measured before this fix: 3e-4..1.3e-3). Same lesson as
13    // the Chapter 6 barriers: place the discontinuity MIDWAY between
14    // samples. Grid points sit at integer multiples of H, so an edge at
15    // a + H/2 is represented symmetrically; we use that radius in the
16    // analytic formula too, so both sides describe the same well.
17    let a_edge = a + 0.5 * H;

One referee in the list is worth singling out, because it audits a number this chapter has been quoting since 14.5. Every resonance curve below is labelled by its distance from the critical depth π²/8 — so the lab is made to find that depth rather than assume it. It bisects on the tracked zero-energy phase, stopping where δ₀(0) = π/2 (exercise 6's prediction for a well poised exactly at capture), and compares the depth it landed on with Chapter 6's bound-state condition — a formula the Numerov sweep never sees. Two independent roads to one number:

1// -- 3b. WHERE is the capture threshold? --------------------------------
2    // The resonance curves above are drawn at depths quoted as fractions of
3    // pi^2/8 — so that number had better be earned, not assumed. Bisect the
4    // depth at which the tracked zero-energy phase equals pi/2 (exercise 6's
5    // prediction for a well exactly at critical depth) and compare with
6    // Chapter 6's bound-state condition V0 = pi^2 hbar^2 / 8 m a^2. The
7    // Numerov sweep never sees that formula; the bracket [0.5, 2.0] comes
8    // from the Levinson table above, which measured no bound state at the
9    // first depth and one at the second.
10    let mut v_lo = 0.5;
11    let mut v_hi = 2.0;
12    for _ in 0..17 {
13        let v_mid = 0.5 * (v_lo + v_hi);
14        if delta0_tracked_over_pi(v_mid, a_edge, a + 40.0) < 0.5 {
15            v_lo = v_mid;
16        } else {
17            v_hi = v_mid;
18        }
19    }
20    let v_crit_measured = 0.5 * (v_lo + v_hi);
21    // the analytic threshold for the SAMPLED well, whose edge sits at a_edge
22    let v_crit_exact = PI * PI / (8.0 * a_edge * a_edge);
23    let v_crit_err = (v_crit_measured - v_crit_exact).abs();

The full slate, then — the panel below counts it for you — with each referee reporting a deviation from something exact, each tolerance set just above the accuracy the run actually achieves, and each printed with its own PASS/FAIL before the program will admit to succeeding. Two of them deserve their asterisks stated out loud. The Levinson check is an integer comparison with room to spare, which sounds lax until you notice that nothing short of a correct phase, tracked continuously through every mod-π jump from k = 8 down to threshold, lands anywhere near an integer. And the Born check is deliberately restricted: it grades the approximation inside the forward cone θ ≤ 60°, where it is supposed to work, because in the far backward tail — where the cross-section has fallen by orders of magnitude and the second-order terms Born threw away are all that is left — it fails badly. The panel prints both numbers, and the failure is the interesting one.

1/// One line of the scoreboard. `value` is always a measured DEVIATION from
2/// an exact result, so `pass` is uniformly "finite, then small".
3#[derive(Serialize)]
4struct RefereeRow {
5    name: String,
6    value: f64,
7    tol: f64,
8    pass: bool,
9}
10
11fn referee_row(name: &str, value: f64, tol: f64) -> RefereeRow {
12    RefereeRow { name: name.to_string(), value, tol, pass: value.is_finite() && value < tol }
13}
14// …
15    // Seven referees, every one a DEVIATION from something exact, and every
16    // tolerance sitting just above the accuracy actually achieved.
17    let referees = vec![
18        // Numerov + two-radius matching vs tan(delta_l) = j_l(ka)/n_l(ka),
19        // l = 0..5 over four beam energies:
20        referee_row("hard_sphere_worst", hs_worst, 1e-8),
21        // the wave answer for a slow beam: sigma -> 4 pi a^2, not pi a^2:
22        referee_row("sigma_low_k_vs_4pia2", (sigma_ratio - 1.0).abs(), 1e-4),
23        // Numerov vs delta_0 = atan[(k/K) tan(Ka)] - ka, three depths:
24        referee_row("well_delta0_worst", well_worst, 1e-6),
25        // the census: tracked delta_0(0)/pi vs the exact bound-state count:
26        referee_row("levinson_worst", lev_worst, 0.02),
27        // where the well first binds: bisected threshold vs pi^2/8 a^2:
28        referee_row("v_crit_vs_pi2_over_8", v_crit_err, 5e-4),
29        // Born inside its cone (theta <= 60 deg), NOT over the whole sweep:
30        referee_row("born_forward_worst", forward_errs[0], 0.01),
31        // and the fingerprint of a first-order theory: halve V0, halve the
32        // error. This one is a RATIO -- the only referee that can catch a
33        // NaN by refusing to divide, which is how the NaN below was found:
34        referee_row("born_scaling_vs_2", (born_ratio - 2.0).abs(), 0.25),
35    ];
36
37    // FINITENESS FIRST, then smallness. f64::max IGNORES NaN, so a "worst
38    // error" of NaN sails straight through every < tolerance test below.
39    // Four referees once "passed" on a NaN here. Never again.
40    for r in &referees {
41        assert!(r.value.is_finite(), "{} is not finite: {}", r.name, r.value);
42    }
43    // …
44    let all_passed = data.referees.iter().all(|r| r.pass);
45    assert!(all_passed, "at least one Chapter 14 referee failed");
46    println!("ALL REFEREES PASSED.");

1. (F) Carry out the current bookkeeping behind (14.3): compute $\vec{j}=\frac{\hslash }{m}\mathrm{Im}({\psi }^{\ast }\nabla \psi )$ for the scattered piece of (14.2), keep the leading term in 1/r, and show the interference terms between plane and scattered waves average to zero at any fixed θ ≠ 0 (rapidly oscillating in r) — which is why the two fluxes can be counted separately away from the forward direction.
2. (F) Prove the optical theorem from partial waves: evaluate Im f(0) from (14.11) and match it to σ_tot. Then explain in words why a purely real f — however large — cannot describe any scattering at all without violating conservation of particles.
3. (F) Do the Yukawa integral in (14.7) by writing sin(qr) as complex exponentials, and verify the Rutherford limit. Then restore the screen: for gold foil (Z = 79, screening at the Bohr-ish radius of the atom), estimate below which scattering angle Geiger and Marsden would have seen the pudding-model — i.e. screened — deflections instead of Rutherford's law.
4. (C) In the partial-wave explorer, set the interior potential to V(r < a) = −1.0 — a well of depth V₀ = 1.0 — and measure the scattering length: record δ₀ at k = 0.15, 0.20, 0.25 and check a_s = −δ₀/k is already constant. Then repeat at depth V₀ = 1.20 and watch a_s grow as the threshold approaches. (The diverging scattering length near a threshold bound state is exactly the knob cold-atom experiments turn — there via magnetic field — to make interactions arbitrarily strong: the Feshbach resonance.)
5. (F) Derive tan δ_l = j_l(ka)/n_l(ka) for the hard sphere from the node condition, and get the two limits honestly: σ → 4πa² as ka → 0, and — summing sin² over the ~ka active channels with the asymptotic forms — σ → 2πa² as ka → ∞.
6. (F, hard) The S-matrix and its poles. For the l = 0 square well, write the exterior solution as incoming plus outgoing wave, $u\propto e^{-ikr}-S(k)e^{+ikr}$ with $S(k)=e^{2i{\delta }_0(k)}$. Continue analytically to imaginary momentum k = iκ, κ > 0, and show S has a pole precisely where the well's bound-state quantization condition holds. Conclude that (14.13) is a winding-number statement, and predict what the lab's continuous δ₀ tracking would report for a well at exactly the critical depth π²/8. (The Levinson table's two shallowest rows bracket the answer, ½, from below and above — and the lab's threshold referee bisects on precisely that prediction to recover π²/8 from the numerics. Check its scoreboard line against your reasoning.)
7. (F, hard) Derive Breit–Wigner (14.14): assume δ_l crosses π/2 at E_R and expand cot δ_l(E) ≈ 2(E_R − E)/Γ to first order. Show the peak σ saturates unitarity, that the time delay of a scattered wave packet, 2ħ dδ/dE, equals 2ħ/Γ at the peak — the resonance lives that long — and that a background phase δ_bg turns the symmetric bump into an asymmetric Fano line shape.
8. (P, hard) Give the lab an l = 1 shape resonance: a well of depth V₀ plus its own centrifugal barrier can trap a quasi-bound p-state at positive energy. Sweep k finely for a = 1, V₀ ≈ 6, locate the energy where δ₁ crosses π/2, and fit σ₁(E) to (14.14) to extract E_R and Γ. Referee the fit two ways: the fitted E_R must match the δ₁ = π/2 crossing, and the width must match the slope via Γ = 2 (dδ₁/dE)⁻¹ at E_R. Then show the resonance narrows as V₀ deepens (the barrier gets relatively taller) — you are watching a decay rate obey a tunneling law.