Chapter 22: Scattering Theory: Quantum Collisions

"The S-matrix is the observable; the potential is the theory." --- Werner Heisenberg

"In a very real sense, scattering experiments are the only experiments we can do. We throw things at other things and watch what comes out."

Every piece of experimental knowledge about nuclei, hadrons, atoms, and elementary particles has been extracted from the same basic procedure: you hurl a projectile at a target and carefully measure what happens. Rutherford discovered the nucleus this way in 1911, bombarding gold foil with alpha particles and watching them bounce back "as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you." Nearly everything we know about the subatomic world --- the quark structure of the proton, the existence of the Higgs boson, the range of the nuclear force --- ultimately rests on scattering data.

This chapter develops the quantum theory of scattering from first principles. We begin with the physical setup: an incident beam encounters a potential, and we ask what fraction of particles emerge at each angle. We then build two complementary mathematical frameworks --- the Born approximation (perturbative, best for weak potentials or high energies) and partial wave analysis (exact in principle, best for low energies or short-range potentials). The two approaches converge on a remarkable result, the optical theorem, which connects the total amount of scattering to the forward scattering amplitude alone. We close by applying the full machinery to the hydrogen-like Coulomb potential, recovering Rutherford's classical formula from quantum mechanics and discovering where quantum corrections appear.

The mathematics in this chapter draws on spherical harmonics and the angular momentum machinery of Chapter 5, the Dirac notation and Green's function ideas of Chapter 8, and the perturbative mindset of Chapter 17. If any of these feel rusty, a brief review of the relevant sections is worthwhile before proceeding.

🏃 Fast Track: If your primary interest is applying scattering theory rather than deriving it, you may skim the derivations in Sections 22.3--22.4 and 22.5--22.6 on a first pass, focusing on the boxed results and the physical interpretation following each derivation. Sections 22.1--22.2 (setup and cross sections) and Sections 22.7--22.8 (resonances and the optical theorem) are essential for everyone.

22.1 The Scattering Problem

The Physical Setup

Consider a beam of particles, each with well-defined momentum $\hbar\mathbf{k}$, incident on a localized potential $V(\mathbf{r})$ centered at the origin. "Localized" means that $V(\mathbf{r}) \to 0$ faster than $1/r$ as $r \to \infty$ (we will handle the Coulomb potential, which violates this condition, as a special case in Section 22.10). Far from the scatterer, the particles are essentially free.

The key assumption is that the incident beam can be represented by a plane wave:

$$|\psi_{\text{inc}}\rangle \longrightarrow \psi_{\text{inc}}(\mathbf{r}) = Ae^{i\mathbf{k}\cdot\mathbf{r}}$$

where we choose $\mathbf{k} = k\hat{z}$ (beam along the $z$-axis) without loss of generality.

After the interaction, at large distances from the scatterer, the total wavefunction must take the form:

$$\boxed{\psi(\mathbf{r}) \xrightarrow{r \to \infty} A\left[e^{ikz} + f(\theta, \phi)\frac{e^{ikr}}{r}\right]}$$

This is the scattering boundary condition, and it encodes the entire physics of the problem.

💡 Key Insight: The first term is the undisturbed incident plane wave, still propagating in the $+z$ direction. The second term is the scattered wave --- an outgoing spherical wave (the $e^{ikr}/r$ factor guarantees it is outgoing and falls off as required by flux conservation in 3D). All the nontrivial physics is contained in the scattering amplitude $f(\theta, \phi)$, which tells us how much scattering occurs in each direction.

Why an Outgoing Spherical Wave?

In three dimensions, the free-particle Green's function for the Helmholtz equation $(\nabla^2 + k^2)\psi = 0$ has the asymptotic form $e^{\pm ikr}/r$. We select the $+$ sign because the scattered wave must represent particles moving away from the scatterer (an outgoing wave). The $1/r$ falloff is necessary so that the probability flux through a sphere of radius $r$ remains constant --- since the surface area grows as $r^2$ and the flux density goes as $|\psi|^2 \propto 1/r^2$, the integrated flux is $r$-independent.

The Scattering Amplitude $f(\theta, \phi)$

The scattering amplitude $f(\theta, \phi)$ has dimensions of length. It depends on: - The incident energy $E = \hbar^2 k^2 / 2m$ - The scattering angles $(\theta, \phi)$ - The potential $V(\mathbf{r})$

For a central potential (spherically symmetric $V = V(r)$), the scattering amplitude is independent of $\phi$:

$$f(\theta, \phi) \to f(\theta)$$

This is because the combination of an incident beam along $\hat{z}$ and a spherically symmetric potential has azimuthal symmetry about $\hat{z}$. We will focus on central potentials for most of this chapter.

⚠️ Common Misconception: Students sometimes think the scattering amplitude is a probability amplitude in the Born-rule sense. It is not --- it is not normalized, and $|f|^2$ does not give a probability. Rather, $|f|^2$ gives a differential cross section, which has dimensions of area and represents an effective target size. The precise connection is the subject of the next section.

The Wave Packet Perspective

A plane wave $e^{ikz}$ is not normalizable and extends over all space, so how can it represent a localized beam? The resolution is that the incident beam is more accurately described as a wave packet --- a superposition of plane waves peaked around $\mathbf{k}_0 = k_0\hat{z}$:

$$\psi_{\text{inc}}(\mathbf{r}, t) = \int \tilde{A}(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r} - \omega_k t)}\frac{d^3\mathbf{k}}{(2\pi)^3}$$

where $\tilde{A}(\mathbf{k})$ is sharply peaked around $\mathbf{k}_0$ with a spread $\Delta k \ll k_0$. If the packet is wide compared to the range of the potential ($\Delta x \gg a$) but narrow compared to the distance to the detector, then the scattering of each Fourier component can be treated independently, and the stationary-state cross section $|f(\theta)|^2$ gives the correct angular distribution. The mathematical proof of this equivalence is due to Goldberger and Watson (1964) and requires careful control of the large-$r$ limit.

The upshot is: the plane-wave boundary condition is a mathematical convenience that yields the correct physical cross section, provided the wave packet is much wider than the scatterer and has a narrow energy spread. These conditions are overwhelmingly satisfied in any real scattering experiment.

🔵 Historical Note: The plane-wave formulation of scattering theory was developed independently by Born (1926), Faxen and Holtsmark (1927), and Mott (1928). The wave-packet justification, which places the formalism on rigorous mathematical footing, was completed much later by Dollard (1964) for short-range potentials and by Dollard (1966) for the Coulomb case, which requires a modified wave-packet analysis due to the long-range nature of the potential.

🔄 Check Your Understanding (Spaced Review --- Ch 5): Why is the choice of spherical coordinates natural when $V = V(r)$? What property of the Laplacian makes the angular and radial parts separable?

22.2 Cross Sections: Differential and Total

The Differential Cross Section

The experimentally measurable quantity in any scattering experiment is the differential cross section $d\sigma/d\Omega$, defined as:

$$\frac{d\sigma}{d\Omega} \equiv \frac{\text{number of particles scattered into } d\Omega \text{ per unit time}}{\text{incident flux}}$$

where $d\Omega = \sin\theta\,d\theta\,d\phi$ is an element of solid angle.

To connect this to the scattering amplitude, we compute the probability current density. For the incident plane wave $\psi_{\text{inc}} = Ae^{ikz}$:

$$\mathbf{j}_{\text{inc}} = \frac{\hbar}{2mi}\left(\psi^*\nabla\psi - \psi\nabla\psi^*\right) = |A|^2 \frac{\hbar k}{m}\hat{z} = |A|^2 v\hat{z}$$

where $v = \hbar k / m$ is the particle velocity.

For the scattered wave $\psi_{\text{sc}} = Af(\theta)e^{ikr}/r$, the radial component of the current at large $r$ is:

$$j_{\text{sc}} = |A|^2\frac{\hbar k}{m}\frac{|f(\theta)|^2}{r^2}$$

The number of particles scattered per unit time into solid angle $d\Omega$ is $j_{\text{sc}} \cdot r^2\,d\Omega$, so:

$$\frac{d\sigma}{d\Omega} = \frac{j_{\text{sc}} \cdot r^2}{j_{\text{inc}}} = |f(\theta)|^2$$

This is the central result:

$$\boxed{\frac{d\sigma}{d\Omega} = |f(\theta)|^2}$$

The differential cross section has dimensions of area (length squared), consistent with $f$ having dimensions of length.

📊 By the Numbers: Cross sections in nuclear and particle physics are often quoted in barns (b): $1\;\text{barn} = 10^{-24}\;\text{cm}^2 = 10^{-28}\;\text{m}^2$. Typical nuclear cross sections are of order $1\;\text{barn}$, while the proton-proton total cross section at LHC energies ($\sqrt{s} = 13\;\text{TeV}$) is about $110\;\text{mb}$ (millibarns). The word "barn" was coined during the Manhattan Project --- to a nuclear physicist, a nucleus is "as big as a barn."

The Total Cross Section

The total cross section is obtained by integrating over all solid angles:

$$\sigma_{\text{tot}} = \int \frac{d\sigma}{d\Omega}\,d\Omega = \int_0^{2\pi} d\phi \int_0^{\pi} |f(\theta)|^2\sin\theta\,d\theta$$

For a central potential ($f$ independent of $\phi$):

$$\boxed{\sigma_{\text{tot}} = 2\pi\int_0^{\pi} |f(\theta)|^2\sin\theta\,d\theta}$$

💡 Key Insight: The cross section is an effective area. If the scatterer were a classical hard sphere of radius $R$, the total cross section would be $\sigma = \pi R^2$ (the geometric cross section). Quantum mechanics can give cross sections much larger (resonances) or much smaller (destructive interference) than the geometric value.

Scattering into a Detector

In practice, a detector subtends a finite solid angle $\Delta\Omega$ at angles $(\theta_0, \phi_0)$. The rate at which the detector registers particles is:

$$\dot{N}_{\text{det}} = j_{\text{inc}} \cdot n_{\text{target}} \cdot \frac{d\sigma}{d\Omega}\bigg|_{(\theta_0, \phi_0)} \cdot \Delta\Omega$$

where $n_{\text{target}}$ is the areal density (particles per unit area) of the target. This formula connects the abstract scattering amplitude to actual count rates in the laboratory.

Example: Isotropic Scattering

If the scattering amplitude is a constant $f(\theta) = -a$ (independent of angle), the scattering is isotropic:

$$\frac{d\sigma}{d\Omega} = a^2, \qquad \sigma_{\text{tot}} = 4\pi a^2$$

This occurs for pure $s$-wave scattering at low energy. The parameter $a$ is called the scattering length and is one of the most important quantities in low-energy scattering. For neutron-proton scattering in the spin-singlet state, $a_s = -23.7\;\text{fm}$, giving $\sigma = 4\pi(23.7)^2 \approx 7060\;\text{fm}^2 \approx 70.6\;\text{barns}$. This is an enormous cross section --- far larger than the geometric size of the proton ($\sigma_{\text{geom}} \sim \pi r_p^2 \sim 3\;\text{fm}^2$). The large scattering length indicates that the neutron-proton interaction nearly supports a bound state in the singlet channel. When a potential almost has a bound state at zero energy, the scattering length diverges --- a phenomenon called a zero-energy resonance.

✅ Checkpoint: Verify that $d\sigma/d\Omega$ has dimensions of area, and $\sigma_{\text{tot}}$ has dimensions of area. If a scattering experiment measures $d\sigma/d\Omega = 10\;\text{mb/sr}$ and the angular distribution is isotropic, what is $\sigma_{\text{tot}}$ in millibarns?

22.3 The Born Approximation

Integral Form of the Schrodinger Equation

The time-independent Schrodinger equation for the scattering problem is:

$$(\nabla^2 + k^2)\psi(\mathbf{r}) = \frac{2m}{\hbar^2}V(\mathbf{r})\psi(\mathbf{r})$$

where $k^2 = 2mE/\hbar^2$. We can rewrite this as an inhomogeneous Helmholtz equation and solve it using the Green's function:

$$\psi(\mathbf{r}) = \psi_{\text{inc}}(\mathbf{r}) + \int G_0^{(+)}(\mathbf{r}, \mathbf{r}')V(\mathbf{r}')\psi(\mathbf{r}')\,d^3\mathbf{r}'$$

The outgoing-wave Green's function (retarded Green's function) satisfies $(\nabla^2 + k^2)G_0^{(+)} = \delta^3(\mathbf{r} - \mathbf{r}')$ and is given by:

$$G_0^{(+)}(\mathbf{r}, \mathbf{r}') = -\frac{1}{4\pi}\frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{|\mathbf{r} - \mathbf{r}'|}$$

This is the Lippmann-Schwinger equation:

$$\boxed{|\psi\rangle = |\psi_{\text{inc}}\rangle + \hat{G}_0^{(+)}\hat{V}|\psi\rangle}$$

where $\hat{G}_0^{(+)} = (E - \hat{H}_0 + i\epsilon)^{-1}$ in operator form. The $+i\epsilon$ prescription selects outgoing waves (causality).

🔗 Connection: The Lippmann-Schwinger equation is the scattering analogue of the integral equation approach you encountered in perturbation theory (Chapter 17). There, we expanded in powers of the perturbation. Here, we will do the same --- the result is the Born series.

The Green's Function: Where Does It Come From?

The retarded Green's function $G_0^{(+)}(\mathbf{r}, \mathbf{r}')$ can be derived by Fourier transform. We need the solution to:

$$(\nabla^2 + k^2)G_0^{(+)}(\mathbf{r}, \mathbf{r}') = \delta^3(\mathbf{r} - \mathbf{r}')$$

Fourier transforming both sides and working in the variable $\mathbf{s} = \mathbf{r} - \mathbf{r}'$:

$$G_0^{(+)}(\mathbf{s}) = \int\frac{e^{i\mathbf{p}\cdot\mathbf{s}}}{-p^2 + k^2 + i\epsilon}\frac{d^3\mathbf{p}}{(2\pi)^3}$$

The $+i\epsilon$ prescription pushes the poles of the integrand off the real $p$-axis in a way that selects outgoing waves. Evaluating the angular integrals and then the radial integral by contour integration (closing in the upper half-plane for $s > 0$), one obtains:

$$G_0^{(+)}(\mathbf{r}, \mathbf{r}') = -\frac{1}{4\pi}\frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{|\mathbf{r} - \mathbf{r}'|}$$

Had we chosen $-i\epsilon$ instead, we would get the advanced Green's function with $e^{-ik|\mathbf{r} - \mathbf{r}'|}$ --- an incoming spherical wave, which is unphysical for scattering (it would correspond to particles converging toward the scatterer from infinity). The choice of $+i\epsilon$ is therefore dictated by causality: effects propagate outward from the interaction region.

💡 Key Insight: The $+i\epsilon$ prescription in quantum scattering is deeply related to the Feynman $i\epsilon$ prescription in quantum field theory. In both cases, the correct boundary condition (outgoing waves, forward propagation in time) is selected by an infinitesimal imaginary part in the energy denominator. This seemingly technical detail encodes the fundamental arrow of time in quantum physics.

The Far-Field Approximation

For a detector far from the scatterer ($r \gg r'$, where $r'$ ranges over the support of $V$), we can expand:

$$|\mathbf{r} - \mathbf{r}'| \approx r - \hat{r}\cdot\mathbf{r}' + \mathcal{O}(r'/r)$$

This gives:

$$\frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{|\mathbf{r} - \mathbf{r}'|} \approx \frac{e^{ikr}}{r}e^{-i\mathbf{k}'\cdot\mathbf{r}'}$$

where $\mathbf{k}' = k\hat{r}$ is the wavevector in the direction of the detector. The Lippmann-Schwinger equation then yields:

$$\psi(\mathbf{r}) \xrightarrow{r \to \infty} e^{ikz} - \frac{1}{4\pi}\frac{2m}{\hbar^2}\frac{e^{ikr}}{r}\int e^{-i\mathbf{k}'\cdot\mathbf{r}'}V(\mathbf{r}')\psi(\mathbf{r}')\,d^3\mathbf{r}'$$

Comparing with the scattering boundary condition $\psi \sim e^{ikz} + f(\theta)e^{ikr}/r$, we identify:

$$f(\theta) = -\frac{m}{2\pi\hbar^2}\int e^{-i\mathbf{k}'\cdot\mathbf{r}'}V(\mathbf{r}')\psi(\mathbf{r}')\,d^3\mathbf{r}'$$

This is an exact expression, but it is not yet useful because the unknown $\psi$ appears inside the integral.

First Born Approximation

The Born approximation replaces the exact wavefunction $\psi(\mathbf{r}')$ inside the integral with the incident plane wave $e^{i\mathbf{k}\cdot\mathbf{r}'}$. This is the first-order term in the iterative solution of the Lippmann-Schwinger equation:

$$\boxed{f_{\text{Born}}(\theta) = -\frac{m}{2\pi\hbar^2}\int e^{i(\mathbf{k} - \mathbf{k}')\cdot\mathbf{r}'}V(\mathbf{r}')\,d^3\mathbf{r}' = -\frac{m}{2\pi\hbar^2}\tilde{V}(\mathbf{q})}$$

where $\mathbf{q} = \mathbf{k} - \mathbf{k}'$ is the momentum transfer and $\tilde{V}(\mathbf{q})$ is the three-dimensional Fourier transform of the potential.

💡 Key Insight: In the Born approximation, the scattering amplitude is simply (proportional to) the Fourier transform of the potential evaluated at the momentum transfer. This is a profoundly important result: scattering experiments effectively measure the Fourier transform of the interaction potential. This is exactly how the structure of the proton was discovered --- deep inelastic scattering experiments at SLAC in the late 1960s revealed point-like constituents (quarks) by measuring the scattering cross section as a function of momentum transfer.

Computing the Momentum Transfer

Since $|\mathbf{k}| = |\mathbf{k}'| = k$ (elastic scattering), the magnitude of the momentum transfer is:

$$q = |\mathbf{q}| = 2k\sin(\theta/2)$$

This ranges from $q = 0$ (forward scattering, $\theta = 0$) to $q = 2k$ (backscattering, $\theta = \pi$).

Example: Yukawa Potential

For the Yukawa (screened Coulomb) potential:

$$V(r) = V_0\frac{e^{-\mu r}}{r}$$

where $\mu^{-1}$ is the range of the potential, the Fourier transform is a standard integral:

$$\tilde{V}(\mathbf{q}) = \frac{4\pi V_0}{q^2 + \mu^2}$$

The Born approximation gives:

$$f_{\text{Born}}(\theta) = -\frac{2mV_0}{\hbar^2}\frac{1}{q^2 + \mu^2} = -\frac{2mV_0}{\hbar^2}\frac{1}{4k^2\sin^2(\theta/2) + \mu^2}$$

and the differential cross section:

$$\frac{d\sigma}{d\Omega}\bigg|_{\text{Born}} = \left(\frac{2mV_0}{\hbar^2}\right)^2 \frac{1}{\left(4k^2\sin^2(\theta/2) + \mu^2\right)^2}$$

✅ Checkpoint: Verify that this expression has dimensions of length squared. What happens as $\mu \to 0$ (unscreened Coulomb limit)? You should recover a result resembling Rutherford's formula.

Example: The Gaussian Potential

For the Gaussian potential $V(r) = V_0 e^{-r^2/a^2}$, the Fourier transform is also a Gaussian:

$$\tilde{V}(\mathbf{q}) = V_0(\pi a^2)^{3/2}e^{-q^2 a^2/4}$$

The Born cross section is:

$$\frac{d\sigma}{d\Omega}\bigg|_{\text{Born}} = \left(\frac{m V_0}{\hbar^2}\right)^2\left(\frac{a^2}{2}\right)^3 \pi^3 \exp\left[-2k^2 a^2\sin^2(\theta/2)\right]$$

This is a Gaussian in $\sin^2(\theta/2)$, strongly peaked in the forward direction. The width of the forward peak is $\Delta\theta \sim 1/(ka)$: at high energies ($ka \gg 1$), the scattering becomes sharply forward-peaked, while at low energies ($ka \ll 1$), it becomes nearly isotropic.

💡 Key Insight: The forward peaking of the Born cross section at high energy is a general feature, not specific to the Gaussian. It follows from the Fourier relationship: a potential of spatial extent $a$ produces a scattering pattern with angular width $\Delta\theta \sim 1/(ka) = \lambda/(2\pi a)$. This is nothing but the diffraction limit, familiar from optics. A larger scatterer produces a narrower diffraction pattern, just as a wider slit produces a narrower central maximum.

🧪 Experiment: This Fourier relationship between the potential and the cross section is exploited in electron scattering experiments to determine the charge distribution of nuclei. The elastic electron-nucleus cross section, measured as a function of momentum transfer $q$, is essentially the Fourier transform of the nuclear charge density $\rho(r)$. The oscillations in $d\sigma/d\Omega$ as a function of angle (seen, for example, in Hofstadter's Nobel Prize-winning experiments at Stanford in the 1950s) are diffraction minima that reveal the nuclear radius with remarkable precision.

22.4 Born Series and Validity

The Born Series

The Lippmann-Schwinger equation can be iterated systematically. Define:

$$|\psi^{(0)}\rangle = |\psi_{\text{inc}}\rangle$$ $$|\psi^{(n)}\rangle = |\psi_{\text{inc}}\rangle + \hat{G}_0^{(+)}\hat{V}|\psi^{(n-1)}\rangle$$

The first Born approximation uses $|\psi^{(0)}\rangle$ in the integral. The second Born approximation uses $|\psi^{(1)}\rangle$, and so on. This generates the Born series:

$$f = f^{(1)} + f^{(2)} + f^{(3)} + \cdots$$

where:

$$f^{(1)} \propto \langle\mathbf{k}'|\hat{V}|\mathbf{k}\rangle$$ $$f^{(2)} \propto \langle\mathbf{k}'|\hat{V}\hat{G}_0^{(+)}\hat{V}|\mathbf{k}\rangle$$ $$f^{(3)} \propto \langle\mathbf{k}'|\hat{V}\hat{G}_0^{(+)}\hat{V}\hat{G}_0^{(+)}\hat{V}|\mathbf{k}\rangle$$

Each term describes scattering with one additional interaction with the potential. The first Born approximation accounts for single scattering; the second includes double scattering, and so on. This expansion converges when the potential is "weak" in an appropriate sense.

Validity of the First Born Approximation

The first Born approximation is reliable when the scattered wave is much weaker than the incident wave inside the scattering region. A sufficient condition is:

$$\frac{m}{2\pi\hbar^2}\int \frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{|\mathbf{r} - \mathbf{r}'|}|V(\mathbf{r}')|\,d^3\mathbf{r}' \ll 1$$

For a potential of range $a$ and depth $V_0$, this becomes (order of magnitude):

$$\boxed{\frac{2mV_0 a^2}{\hbar^2} \ll 1 \quad \text{(low energy, } ka \ll 1\text{)}}$$

$$\boxed{\frac{2mV_0 a}{\hbar^2 k} \ll 1 \quad \text{(high energy, } ka \gg 1\text{)}}$$

The first condition says the potential is too weak to support a bound state. The second says the kinetic energy dominates the potential energy. Notice that the Born approximation improves at high energy --- fast particles spend less time in the scattering region and are less perturbed.

⚠️ Common Misconception: The Born approximation is not restricted to weak potentials. It applies whenever the scattered wave is a small correction to the incident wave. For high-energy scattering off a strong potential, the Born approximation can be excellent even though the potential is not "weak" in absolute terms. What matters is the ratio of the potential energy to the kinetic energy.

🔵 Historical Note: Max Born introduced this approximation in 1926 in one of the founding papers of quantum scattering theory. It was this work on scattering --- not the Born rule for probabilities --- that Born himself considered his most important contribution. The Nobel committee, when they finally awarded Born the prize in 1954 (28 years late, many physicists believed), cited the Born rule. Born's response was characteristically dry: he felt the scattering work was more worthy.

22.5 Partial Wave Expansion

Motivation

The Born approximation is a perturbative method. When the potential is strong, or when we want exact results, we need a different approach. Partial wave analysis exploits the angular momentum structure of the problem and provides an exact expansion of the scattering amplitude.

The key idea is that for a central potential, angular momentum is conserved. Each partial wave --- characterized by angular momentum quantum number $l$ --- scatters independently. The total scattering amplitude is a sum over these independently scattered partial waves.

Expanding the Plane Wave

The incident plane wave $e^{ikz}$ can be expanded in Legendre polynomials (the Rayleigh expansion):

$$e^{ikz} = e^{ikr\cos\theta} = \sum_{l=0}^{\infty}(2l+1)i^l j_l(kr)P_l(\cos\theta)$$

where $j_l(kr)$ are the spherical Bessel functions and $P_l(\cos\theta)$ are Legendre polynomials. This is the fundamental expansion: a plane wave contains all angular momentum quantum numbers $l = 0, 1, 2, \ldots$

The spherical Bessel functions have the asymptotic behavior for $kr \gg l$:

$$j_l(kr) \xrightarrow{r \to \infty} \frac{\sin(kr - l\pi/2)}{kr} = \frac{1}{2ikr}\left[e^{i(kr - l\pi/2)} - e^{-i(kr - l\pi/2)}\right]$$

So each partial wave of the incident beam, at large $r$, consists of an incoming spherical wave and an outgoing spherical wave in equal measure.

The Partial Wave Scattering Amplitude

For a central potential, the full wavefunction is:

$$\psi(r, \theta) = \sum_{l=0}^{\infty}(2l+1)i^l R_l(r)P_l(\cos\theta)$$

where $R_l(r)$ satisfies the radial Schrodinger equation with the effective potential $V(r) + \hbar^2 l(l+1)/(2mr^2)$.

At large $r$ (outside the range of $V$), $R_l(r)$ must be a linear combination of the free-particle solutions. The scattering boundary condition requires that only the outgoing wave be modified:

$$R_l(r) \xrightarrow{r \to \infty} \frac{1}{2ikr}\left[S_l e^{i(kr - l\pi/2)} - e^{-i(kr - l\pi/2)}\right]$$

where $S_l$ is the S-matrix element for partial wave $l$.

Comparing the asymptotic form of the full wavefunction with the scattering boundary condition $\psi \sim e^{ikz} + f(\theta)e^{ikr}/r$, we extract the partial wave expansion of the scattering amplitude:

$$\boxed{f(\theta) = \frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)(S_l - 1)P_l(\cos\theta)}$$

Since the S-matrix is unitary for elastic scattering, $|S_l| = 1$, and we can write $S_l = e^{2i\delta_l}$ where $\delta_l$ is the phase shift for the $l$-th partial wave. Then:

$$\boxed{f(\theta) = \frac{1}{k}\sum_{l=0}^{\infty}(2l+1)e^{i\delta_l}\sin\delta_l \; P_l(\cos\theta)}$$

This is the partial wave expansion of the scattering amplitude. It is an exact result for any central potential.

🔗 Connection: The Legendre polynomials $P_l(\cos\theta)$ are the $m = 0$ spherical harmonics: $P_l(\cos\theta) = \sqrt{4\pi/(2l+1)}Y_l^0(\theta, \phi)$. Only $m = 0$ appears because the azimuthal symmetry of the problem (beam along $z$, spherical potential) selects $L_z = 0$.

Why Partial Waves Are Powerful

The partial wave expansion converts a three-dimensional scattering problem into a sequence of one-dimensional radial problems, one for each $l$. Several features make this decomposition extraordinarily useful:

1. Convergence at low energy. A classical particle with angular momentum $L = l\hbar$ and linear momentum $p = \hbar k$ has impact parameter $b = L/p = l/k$. For the particle to "hit" a potential of range $a$, we need $b \lesssim a$, i.e., $l \lesssim ka$. Partial waves with $l \gg ka$ miss the scatterer entirely and contribute negligible phase shifts. At low energies ($ka \ll 1$), only $l = 0$ (and perhaps $l = 1$) matters.

2. Each partial wave is independent. For a central potential, there is no coupling between different $l$ values. This is a consequence of angular momentum conservation: $[\hat{H}, \hat{L}^2] = 0$ and $[\hat{H}, \hat{L}_z] = 0$. Each partial wave scatters independently, which is why we can write $\sigma_{\text{tot}}$ as a simple sum over $l$.

3. Resonances are localized in $l$. A resonance typically appears in a single partial wave (the one whose centrifugal barrier supports a quasi-bound state at the right energy). The partial wave decomposition isolates the resonance cleanly.

4. The unitarity limit provides absolute bounds. No matter how strong the potential, a single partial wave cannot scatter more than $\sigma_l^{\max} = 4\pi(2l+1)/k^2$. This is a consequence of probability conservation and provides a powerful constraint on the scattering amplitude.

22.6 Phase Shifts

Physical Meaning

What does a phase shift mean? Compare the asymptotic form of the $l$-th partial wave with and without the potential:

• No potential: $R_l^{(0)}(r) \sim \sin(kr - l\pi/2)/(kr)$
• With potential: $R_l(r) \sim \sin(kr - l\pi/2 + \delta_l)/(kr)$

The scattering potential advances (or retards) the phase of each outgoing partial wave by $\delta_l$ relative to the free-particle case. That is the only effect of a localized potential at large distances: it shifts phases.

💡 Key Insight: An attractive potential ($V < 0$) speeds up the particle inside the scattering region (the local kinetic energy is larger), so the wavefunction accumulates extra phase: $\delta_l > 0$. A repulsive potential ($V > 0$) slows the particle, and $\delta_l < 0$. The sign of the phase shift tells you the sign of the potential.

Partial Wave Cross Sections

Using the orthogonality of Legendre polynomials $\int_{-1}^{1}P_l P_{l'}\,d(\cos\theta) = 2\delta_{ll'}/(2l+1)$, the total cross section becomes:

$$\boxed{\sigma_{\text{tot}} = \frac{4\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)\sin^2\delta_l}$$

Each partial wave contributes independently:

$$\sigma_l = \frac{4\pi}{k^2}(2l+1)\sin^2\delta_l$$

The maximum contribution from a single partial wave occurs when $\delta_l = \pi/2 + n\pi$ (the unitarity limit):

$$\sigma_l^{\max} = \frac{4\pi}{k^2}(2l+1)$$

Computing Phase Shifts

To find $\delta_l$ for a given potential, solve the radial Schrodinger equation:

$$-\frac{\hbar^2}{2m}\left[\frac{d^2 u_l}{dr^2} - \frac{l(l+1)}{r^2}u_l\right] + V(r)u_l = Eu_l$$

with $u_l(r) = rR_l(r)$, subject to $u_l(0) = 0$. Integrate outward to a radius $r_0$ well beyond the range of the potential, and match to the free-particle solution:

$$u_l(r_0) = A_l\left[\cos\delta_l \cdot \hat{j}_l(kr_0) - \sin\delta_l \cdot \hat{n}_l(kr_0)\right]$$

where $\hat{j}_l$ and $\hat{n}_l$ are the Riccati-Bessel functions ($\hat{j}_l(x) = xj_l(x)$, $\hat{n}_l(x) = -xn_l(x)$). The phase shift is extracted by:

$$\tan\delta_l = \frac{k r_0 j_l'(kr_0)/j_l(kr_0) - u_l'(r_0)/u_l(r_0)}{k r_0 n_l'(kr_0)/n_l(kr_0) - u_l'(r_0)/u_l(r_0)}$$

where primes denote derivatives with respect to the argument.

Example: Hard Sphere Scattering

For a hard sphere of radius $a$ ($V = \infty$ for $r < a$, $V = 0$ for $r > a$), the boundary condition is $\psi(a) = 0$, giving:

$$\tan\delta_l = \frac{j_l(ka)}{n_l(ka)}$$

Low-energy limit ($ka \ll 1$): Using the small-argument forms of the Bessel functions:

$$j_l(x) \approx \frac{x^l}{(2l+1)!!}, \quad n_l(x) \approx -\frac{(2l-1)!!}{x^{l+1}}$$

we find $\tan\delta_l \sim -(ka)^{2l+1}/(2l+1)!!\cdot(2l-1)!!$, which decreases rapidly with $l$. At low energies, only s-wave ($l = 0$) scattering matters.

For $l = 0$: $\tan\delta_0 = -\tan(ka)$, so $\delta_0 = -ka$. The $s$-wave cross section is:

$$\sigma_0 = \frac{4\pi}{k^2}\sin^2(ka) \xrightarrow{ka \ll 1} 4\pi a^2$$

📊 By the Numbers: The quantum hard-sphere cross section $4\pi a^2$ at low energy is four times the classical geometric cross section $\pi a^2$. This factor of 4 is a purely quantum effect --- it arises from the interference between the incident and scattered waves. At high energies ($ka \gg 1$), the total cross section approaches $2\pi a^2$ (twice the classical value), which is another wave phenomenon related to diffraction.

Example: Finite Square Well Phase Shifts

For the finite spherical square well $V(r) = -V_0$ for $r < a$, $V = 0$ for $r > a$, the $l = 0$ interior solution is:

$$u_0(r) = A\sin(Kr), \quad K = \sqrt{k^2 + \frac{2mV_0}{\hbar^2}}$$

where $K$ is the wavenumber inside the well (larger than $k$ because the kinetic energy is larger). The exterior solution is $u_0(r) = B\sin(kr + \delta_0)$. Matching the logarithmic derivative $u'/u$ at $r = a$:

$$K\cot(Ka) = k\cot(ka + \delta_0)$$

In the low-energy limit ($ka \ll 1$):

$$\tan\delta_0 \approx ka\left[\frac{\tan(Ka)}{Ka} - 1\right]$$

This result reveals the scattering length: $a_s = -\lim_{k \to 0}(\delta_0/k) = a[1 - \tan(K_0 a)/(K_0 a)]$ where $K_0 = \sqrt{2mV_0/\hbar^2}$.

Notice what happens when $K_0 a = (n + 1/2)\pi$: the tangent function diverges, and $a_s \to \pm\infty$. This is the zero-energy resonance --- the potential is on the verge of binding a new state. The scattering cross section $\sigma = 4\pi a_s^2$ diverges, even though the energy approaches zero. This phenomenon is crucial in ultracold atomic physics, where magnetic fields are used to tune the scattering length through a Feshbach resonance, enabling control over interaction strengths in quantum gases.

The Low-Energy Dominance of $s$-Waves

The rapid decrease of phase shifts with $l$ at low energies is not an accident --- it reflects the centrifugal barrier. For a potential of range $a$, the centrifugal barrier height is approximately:

$$V_{\text{barrier}} \sim \frac{\hbar^2 l(l+1)}{2ma^2}$$

When the incident energy $E = \hbar^2 k^2/(2m)$ is much less than this barrier height ($E \ll V_{\text{barrier}}$, i.e., $ka \ll \sqrt{l(l+1)}$), the particle cannot penetrate to the region where the potential acts, and $\delta_l$ is exponentially small. More precisely:

$$\delta_l \propto (ka)^{2l+1} \quad \text{for } ka \ll 1$$

This power-law suppression means that at sufficiently low energy, only the $s$-wave phase shift survives, and the scattering becomes isotropic (independent of angle). This is a universal feature of short-range scattering.

🔄 Check Your Understanding (Spaced Review --- Ch 17): In perturbation theory, we expanded the energy in powers of the perturbation $\hat{V}$. In the Born series, we expand the scattering amplitude in powers of $\hat{V}$. How is the partial wave method fundamentally different from both of these perturbative approaches?

22.7 Resonances and the Breit-Wigner Formula

What is a Scattering Resonance?

A scattering resonance occurs when the cross section for a particular partial wave passes through its maximum value (the unitarity limit). This happens when $\delta_l$ passes through $\pi/2$ (or an odd multiple thereof).

Physically, a resonance occurs when the incident energy is close to the energy of a quasi-bound state --- a state that would be truly bound if the potential barrier were infinitely high, but that can decay by tunneling through the barrier. The particle is temporarily trapped in the potential well, orbits many times, and then escapes. The long dwell time produces a large cross section.

🧪 Experiment: Neutron-nucleus scattering provides spectacular examples of resonances. When a slow neutron encounters a nucleus, the cross section can spike by factors of $10^3$ or more at specific energies corresponding to quasi-bound states of the compound nucleus. These resonances are so sharp that they were used to measure nuclear energy levels with extraordinary precision, a technique called neutron resonance spectroscopy.

The Breit-Wigner Formula

Near a resonance energy $E_r$ where $\delta_l(E_r) = \pi/2$, we can expand the phase shift:

$$\delta_l(E) \approx \frac{\pi}{2} + (E - E_r)\frac{d\delta_l}{dE}\bigg|_{E_r} + \cdots$$

Define the width $\Gamma$ through:

$$\frac{d\delta_l}{dE}\bigg|_{E_r} = -\frac{2}{\Gamma}$$

Then:

$$\cot\delta_l \approx -\frac{2(E - E_r)}{\Gamma}$$

which gives $\sin^2\delta_l = \Gamma^2/4/[(E - E_r)^2 + \Gamma^2/4]$.

The $l$-th partial wave cross section near the resonance is:

$$\boxed{\sigma_l(E) = \frac{4\pi}{k^2}(2l+1)\frac{\Gamma^2/4}{(E - E_r)^2 + \Gamma^2/4}}$$

This is the Breit-Wigner formula. It describes a Lorentzian peak centered at $E_r$ with full width at half maximum $\Gamma$.

Physical Interpretation of the Width

The width $\Gamma$ is directly related to the lifetime $\tau$ of the quasi-bound state through the energy-time uncertainty relation:

$$\Gamma = \frac{\hbar}{\tau}$$

A narrow resonance ($\Gamma$ small) corresponds to a long-lived quasi-bound state; a broad resonance ($\Gamma$ large) corresponds to a short-lived state that decays quickly.

💡 Key Insight: Every "unstable particle" in the particle physics zoo --- the $\Delta$ baryon, the $Z$ boson, the Higgs boson --- is actually a scattering resonance. When we say the $Z$ boson has a mass of 91.2 GeV and a width of 2.5 GeV, we are giving the Breit-Wigner parameters of a resonance in the $e^+e^- \to \text{hadrons}$ cross section.

Resonance Conditions

For a resonance to occur at angular momentum $l$, there must be a quasi-bound state behind a potential barrier. The effective potential for angular momentum $l$ is:

$$V_{\text{eff}}(r) = V(r) + \frac{\hbar^2 l(l+1)}{2mr^2}$$

The centrifugal barrier $\hbar^2 l(l+1)/(2mr^2)$ provides the barrier for $l \geq 1$. For $l = 0$, a resonance requires the potential itself to have a barrier (e.g., a potential with both attractive and repulsive regions).

⚠️ Common Misconception: Not every potential supports resonances. A purely attractive potential with no barrier (like the finite square well with $l = 0$) does not produce resonances, though it can produce bound states. Resonances require a mechanism for temporary trapping, which means a barrier.

Worked Example: $p$-Wave Resonance in a Square Well

Consider a square well $V(r) = -V_0$ for $r < a$, $V = 0$ for $r > a$, with parameters chosen so that a $p$-wave ($l = 1$) resonance occurs. The effective potential for $l = 1$ is:

$$V_{\text{eff}}(r) = \begin{cases} -V_0 + \hbar^2 \cdot 2/(2mr^2) & r < a \\ \hbar^2 \cdot 2/(2mr^2) & r > a \end{cases}$$

The centrifugal barrier peaks at $r = a$ with height $V_{\text{barrier}} = \hbar^2/(ma^2)$ (in natural units, this is $1/a^2$). For a resonance to exist, we need:

1. The well must be deep enough to support a quasi-bound state: $V_0 > V_{\text{barrier}}$
2. The incident energy must match the quasi-bound state energy: $E \approx E_r$

At the resonance, the $l = 1$ phase shift passes through $\pi/2$, and the $p$-wave cross section reaches its maximum:

$$\sigma_1^{\max} = \frac{12\pi}{k_r^2}$$

The factor of 12 comes from $(2l+1) = 3$ and the factor of $4\pi$. If $k_r a \sim 1$, this gives $\sigma \sim 12\pi a^2$ --- twelve times the geometric cross section. The resonance width $\Gamma$ depends on the tunneling probability through the centrifugal barrier, which decreases exponentially with the barrier width and height.

For concrete numbers: with $V_0 = 50\;\text{(natural units)}$, $a = 2$, the first $p$-wave resonance occurs near $E_r \approx 2.1$ with width $\Gamma \approx 0.3$, corresponding to a lifetime of $\tau = \hbar/\Gamma \approx 3.3$ (natural units). The phase shift $\delta_1(E)$ rises from near 0 to near $\pi$ over an energy range of order $\Gamma$ centered on $E_r$.

Connection to Unstable Particles

The Breit-Wigner formula is not merely a formula for nuclear cross sections --- it is the universal description of any unstable quantum state. In particle physics, every resonance is listed in the Particle Data Group's Review of Particle Physics with its mass (= $E_r$) and width (= $\Gamma$). The $\Delta(1232)$ baryon, the first hadronic resonance discovered, appears as a peak in the $\pi N$ cross section with $M = 1232\;\text{MeV}$ and $\Gamma = 117\;\text{MeV}$. The $Z$ boson appears as a Breit-Wigner peak in the $e^+e^- \to f\bar{f}$ cross section with $M_Z = 91.188\;\text{GeV}$ and $\Gamma_Z = 2.495\;\text{GeV}$.

The profound insight is that there is no sharp boundary between a "particle" and a "resonance." A stable particle is a resonance with $\Gamma = 0$ (infinite lifetime); a short-lived "particle" like the $\Delta$ is a broad resonance. The S-matrix treats them all on the same footing: stable particles are poles on the real axis in the complex energy plane, while resonances are poles with nonzero imaginary part.

22.8 The Optical Theorem

Statement

The optical theorem is one of the most elegant results in scattering theory. It relates the total cross section to the forward ($\theta = 0$) scattering amplitude:

$$\boxed{\sigma_{\text{tot}} = \frac{4\pi}{k}\operatorname{Im}[f(0)]}$$

Proof from Partial Waves

The proof is straightforward using the partial wave expansion. At $\theta = 0$, $P_l(1) = 1$ for all $l$, so:

$$f(0) = \frac{1}{k}\sum_{l=0}^{\infty}(2l+1)e^{i\delta_l}\sin\delta_l$$

The imaginary part is:

$$\operatorname{Im}[f(0)] = \frac{1}{k}\sum_{l=0}^{\infty}(2l+1)\sin^2\delta_l$$

Comparing with $\sigma_{\text{tot}} = (4\pi/k^2)\sum_l(2l+1)\sin^2\delta_l$:

$$\sigma_{\text{tot}} = \frac{4\pi}{k}\operatorname{Im}[f(0)] \qquad \checkmark$$

Physical Interpretation

The optical theorem expresses conservation of probability (unitarity). The total cross section measures the total number of particles removed from the forward beam. These particles must go somewhere --- they are scattered into other angles. The forward scattering amplitude encodes this removal through its imaginary part, which represents the interference between the incident wave and the scattered wave in the forward direction.

💡 Key Insight: The optical theorem is the quantum analogue of the extinction theorem in classical optics, which states that a beam passing through a medium is attenuated by destructive interference between the forward-propagating incident wave and the forward-scattered wave from the medium. The name "optical theorem" reflects this lineage.

The real part of $f(0)$, by contrast, describes a phase shift of the forward wave without removing particles from the beam --- a refractive index effect. This is related to the scattering length and low-energy effective range theory.

The Optical Theorem as a Consistency Check

In practical calculations, the optical theorem serves as a powerful consistency check. If you compute $f(\theta)$ by some method (Born approximation, partial waves, numerical solution), you can verify self-consistency by checking whether:

$$\frac{4\pi}{k}\operatorname{Im}[f(0)] \stackrel{?}{=} \int |f(\theta)|^2\,d\Omega$$

If these are not equal, something has gone wrong. In the first Born approximation, the left side is zero (because $f_{\text{Born}}$ is real for a real potential), while the right side is nonzero. This tells us that the first Born approximation violates unitarity --- it is only approximately correct and must be supplemented by higher-order terms to restore the optical theorem.

🔴 Warning: The failure of the first Born approximation to satisfy the optical theorem is not a bug --- it is an inherent limitation of any finite truncation of the Born series. Only the full series respects unitarity exactly.

The Optical Theorem in Practice

The optical theorem is used in several important ways:

1. Measuring total cross sections. Rather than placing detectors at every angle (which is experimentally challenging), one can measure the forward scattering amplitude using interference methods and extract $\sigma_{\text{tot}}$ from the optical theorem. This technique, called the Coulomb-nuclear interference method, is standard in high-energy hadron physics.

2. Constraining theoretical calculations. Any approximate calculation of $f(\theta)$ must satisfy the optical theorem to be physically consistent. If it does not, the approximation is violating unitarity, which tells you how trustworthy the calculation is and where it might go wrong.

3. Dispersion relations. The optical theorem, combined with analyticity of the scattering amplitude, leads to dispersion relations --- integral relations connecting the real and imaginary parts of $f(\theta)$. These relations, which are the scattering theory analogues of the Kramers-Kronig relations in optics, provide powerful constraints on the scattering amplitude even when the potential is unknown.

🧪 Experiment: At the Large Hadron Collider, the TOTEM experiment measures the total proton-proton cross section using the optical theorem. By measuring the rate of proton-proton scattering at very small angles (the forward amplitude) and using $\sigma_{\text{tot}} = (4\pi/k)\operatorname{Im}[f(0)]$, TOTEM has determined $\sigma_{\text{tot}}(pp) = 110.6 \pm 3.4\;\text{mb}$ at $\sqrt{s} = 13\;\text{TeV}$. This is one of the few truly model-independent measurements possible at the LHC.

22.9 The S-Matrix: An Introduction

Definition and Properties

The S-matrix (scattering matrix) is the central object in quantum scattering theory. For a central potential, it is diagonal in the angular momentum basis:

$$\langle l'm'|S|lm\rangle = S_l\delta_{ll'}\delta_{mm'}$$

where $S_l = e^{2i\delta_l}$ as we introduced in Section 22.5. The key properties are:

1. Unitarity: $S^\dagger S = I$, which for partial waves means $|S_l| = 1$. This is probability conservation.

2. Analyticity: When continued to complex energy (or momentum), $S_l(k)$ is an analytic function whose singularities encode the physics: - Bound states correspond to poles on the positive imaginary $k$-axis (i.e., $k = i\kappa$ with $\kappa > 0$, giving $E = -\hbar^2\kappa^2/2m < 0$). - Resonances correspond to poles in the lower half of the complex $k$-plane (or equivalently, complex energy poles with $\operatorname{Im}(E) < 0$), located at $E = E_r - i\Gamma/2$.

3. Crossing symmetry and reciprocity: $S_l(k) = S_l(-k)^*$ for real potentials, which ensures time-reversal invariance.

The T-Matrix

The T-matrix (transition matrix) is defined by $S = I + 2iT$, so:

$$S_l = 1 + 2iT_l, \qquad T_l = \frac{S_l - 1}{2i} = \frac{e^{2i\delta_l} - 1}{2i} = e^{i\delta_l}\sin\delta_l$$

The scattering amplitude is:

$$f(\theta) = \frac{1}{k}\sum_l(2l+1)T_l P_l(\cos\theta)$$

and the optical theorem in terms of $T_l$ reads:

$$\operatorname{Im}(T_l) = |T_l|^2 \quad \Leftrightarrow \quad \sin^2\delta_l = \operatorname{Im}(e^{i\delta_l}\sin\delta_l)$$

This is the unitarity condition for each partial wave.

🔗 Connection: The S-matrix formulation becomes the primary framework in relativistic quantum field theory and particle physics, where one often has no direct access to the potential (or there is no potential in the usual sense). The program of computing S-matrix elements directly from symmetry principles, analyticity, and unitarity --- without ever writing down a Hamiltonian --- was championed by Heisenberg in the 1940s and later developed into the S-matrix bootstrap program of the 1960s. We will encounter the S-matrix again in Part VI (Quantum Field Theory) and Part VII (Advanced Topics).

Inelastic Scattering

For inelastic scattering, particles can be absorbed (converted to other channels). The S-matrix element then satisfies $|S_l| \leq 1$. Writing $S_l = \eta_l e^{2i\delta_l}$ with $0 \leq \eta_l \leq 1$ (the inelasticity parameter), the elastic, inelastic (reaction), and total cross sections are:

$$\sigma_{\text{el}} = \frac{\pi}{k^2}\sum_l(2l+1)|1 - S_l|^2$$

$$\sigma_{\text{reac}} = \frac{\pi}{k^2}\sum_l(2l+1)(1 - |S_l|^2)$$

$$\sigma_{\text{tot}} = \frac{2\pi}{k^2}\sum_l(2l+1)(1 - \operatorname{Re}[S_l])$$

The optical theorem continues to hold: $\sigma_{\text{tot}} = (4\pi/k)\operatorname{Im}[f(0)]$.

Bound States as S-Matrix Poles

One of the most elegant results of S-matrix theory is the connection between bound states and poles. Consider the $l = 0$ S-matrix element continued to imaginary momentum $k = i\kappa$ (with $\kappa > 0$, corresponding to negative energy $E = -\hbar^2\kappa^2/(2m)$). The outgoing wave $e^{ikr} = e^{-\kappa r}$ is now exponentially decaying, which is exactly the asymptotic behavior of a bound state wavefunction. The S-matrix has a pole whenever this purely decaying solution satisfies the boundary conditions at the origin --- i.e., whenever a bound state exists.

For a square well potential, the $s$-wave S-matrix is:

$$S_0(k) = e^{-2ika}\frac{K\cot(Ka) + ik}{K\cot(Ka) - ik}$$

where $K = \sqrt{k^2 + 2mV_0/\hbar^2}$. Setting $k = i\kappa$ and requiring the denominator to vanish: $K'\cot(K'a) = -\kappa$ where $K' = \sqrt{2m(V_0 - |E|)/\hbar^2}$. This is exactly the transcendental equation for the bound state energies that one obtains by matching wavefunctions at $r = a$ --- confirming that bound states are indeed poles of the S-matrix.

This result generalizes beautifully: for any potential, every bound state corresponds to a pole of $S_l(k)$ on the positive imaginary axis of the complex $k$-plane. Resonances, by contrast, correspond to poles in the lower half of the complex $k$-plane (or equivalently, at complex energies $E = E_r - i\Gamma/2$ with $\Gamma > 0$). The closer a resonance pole is to the real axis, the narrower the resonance and the longer the quasi-bound state lives.

22.10 Rutherford Scattering from Quantum Mechanics

The Coulomb Problem

We now apply our machinery to the crown jewel of scattering: the Coulomb potential:

$$V(r) = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 r} = \frac{\alpha}{r}$$

where $\alpha = Z_1 Z_2 e^2/(4\pi\epsilon_0)$ for two charged particles with charges $Z_1 e$ and $Z_2 e$.

The Coulomb potential is special (and problematic) because it falls off as $1/r$ --- too slowly for the standard scattering formalism to apply directly. The plane wave plus outgoing spherical wave decomposition breaks down because the long-range Coulomb tail distorts the wavefunction at all distances.

The Born Approximation for Coulomb

Despite the formal difficulties, the first Born approximation for the Coulomb potential can be carried through. We regularize by taking the Yukawa result and letting $\mu \to 0$:

$$f_{\text{Coulomb}}^{\text{Born}}(\theta) = -\frac{2m\alpha}{\hbar^2}\frac{1}{4k^2\sin^2(\theta/2)} = -\frac{\eta}{2k\sin^2(\theta/2)}$$

where $\eta = m\alpha/(\hbar^2 k)$ is the Sommerfeld parameter (also called the Coulomb parameter).

The differential cross section is:

$$\boxed{\frac{d\sigma}{d\Omega}\bigg|_{\text{Rutherford}} = \left(\frac{\alpha}{4E}\right)^2 \frac{1}{\sin^4(\theta/2)} = \left(\frac{Z_1 Z_2 e^2}{16\pi\epsilon_0 E}\right)^2\frac{1}{\sin^4(\theta/2)}}$$

This is the Rutherford scattering formula.

🔵 Historical Note: Ernest Rutherford derived this formula classically in 1911 to explain the large-angle scattering of alpha particles from gold foil observed by Geiger and Marsden. It is one of the remarkable coincidences of physics that the classical and quantum Coulomb cross sections are identical for point charges. This coincidence does not hold for other potentials --- it is a special property of the $1/r$ potential, related to the hidden $SO(4)$ symmetry of the Kepler/Coulomb problem.

Exact Quantum Treatment

The exact quantum solution for Coulomb scattering can be obtained using parabolic coordinates. The scattering amplitude is:

$$f_{\text{Coulomb}}(\theta) = -\frac{\eta}{2k\sin^2(\theta/2)}\exp\left[-i\eta\ln\sin^2(\theta/2) + 2i\arg\Gamma(1 + i\eta)\right]$$

The modulus squared gives exactly the Rutherford formula --- no corrections. The extra phase factor (involving $\ln\sin^2(\theta/2)$ and the gamma function) modifies the phase of $f$ but not $|f|^2$. This exact agreement is the coincidence noted above.

Where Rutherford Breaks Down

Rutherford scattering breaks down when:

1. The projectile penetrates the nucleus: At high energies or small impact parameters, the alpha particle enters the nuclear interior where the strong force dominates. This produces deviations from the $1/\sin^4(\theta/2)$ law at large angles and was how the nuclear radius was first measured.

2. The projectile's de Broglie wavelength is comparable to the nuclear size: Diffraction effects appear, producing oscillations in the cross section as a function of angle.

3. Relativistic effects: At very high energies, the Mott cross section (which includes spin and relativistic corrections) replaces Rutherford.

4. Screening by atomic electrons: For atoms (not bare nuclei), the outer electrons screen the nuclear charge at large distances, effectively cutting off the Coulomb potential and regularizing the total cross section.

📊 By the Numbers: The total Rutherford cross section is infinite: $\sigma_{\text{tot}} = \int |f|^2 d\Omega = \infty$, because $|f|^2 \propto 1/\sin^4(\theta/2)$ diverges at $\theta = 0$. This is the famous "Coulomb divergence" --- physically, it reflects the infinite range of the $1/r$ potential. Every particle is deflected by at least a tiny angle, no matter how far away it passes. In practice, screening by atomic electrons or a finite beam size regularizes this divergence.

✅ Checkpoint: The Rutherford formula diverges as $\theta \to 0$. What is the physical reason for this divergence? How does it differ from the behavior of the Yukawa cross section at $\theta = 0$?

Numerical Example: Alpha-Gold Scattering

Let us work through Rutherford's historical experiment quantitatively. For 5.0 MeV alpha particles ($Z_1 = 2$, $m_\alpha = 3727\;\text{MeV}/c^2$) scattering off gold ($Z_2 = 79$):

$$\frac{Z_1 Z_2 e^2}{4\pi\epsilon_0} = 2 \times 79 \times 1.44\;\text{MeV}\cdot\text{fm} = 227.5\;\text{MeV}\cdot\text{fm}$$

$$\frac{d\sigma}{d\Omega}\bigg|_{\theta = 60°} = \left(\frac{227.5}{4 \times 5.0}\right)^2 \frac{1}{\sin^4(30°)} = (11.375)^2 \times \frac{1}{(0.5)^4} = 129.4 \times 16 = 2070\;\text{fm}^2/\text{sr}$$

$$\frac{d\sigma}{d\Omega}\bigg|_{\theta = 150°} = (11.375)^2 \frac{1}{\sin^4(75°)} = 129.4 \times \frac{1}{(0.9659)^4} = 129.4 \times 1.146 = 148.3\;\text{fm}^2/\text{sr}$$

The ratio of cross sections at $60°$ and $150°$ is about 14:1. The strong forward peaking reflects the dominance of large-impact-parameter (glancing) collisions, which produce small deflections. The relatively large backscattering cross section at $150°$ is what surprised Geiger and Marsden --- a uniform "plum pudding" distribution would have produced essentially zero backscattering.

The distance of closest approach for a head-on collision ($\theta = 180°$) is:

$$d_{\min} = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 \cdot 2E} = \frac{227.5}{2 \times 5.0} = 22.75\;\text{fm}$$

The gold nuclear radius is $R_{\text{Au}} \approx 1.2 \times 197^{1/3} \approx 7.0\;\text{fm}$. Since $d_{\min} \gg R_{\text{Au}}$, the alpha particle never reaches the nuclear surface, and the Rutherford formula is valid. To probe the nuclear interior, one would need alpha energies above $E \gtrsim Z_1 Z_2 e^2/(4\pi\epsilon_0 \cdot 2R_{\text{Au}}) \approx 16\;\text{MeV}$.

Connecting to the Hydrogen Atom

The Coulomb scattering calculation brings us full circle to our anchor example. The hydrogen atom's Coulomb potential $V(r) = -e^2/(4\pi\epsilon_0 r)$ determines both the bound state spectrum (Chapter 5) and the scattering behavior. Bound states exist for $E < 0$: they are the discrete hydrogen energy levels $E_n = -13.6/n^2\;\text{eV}$. For $E > 0$, the same potential produces Rutherford scattering with a continuous spectrum of scattering states.

In the S-matrix language, the bound states are poles of $S(k)$ on the positive imaginary $k$-axis, while the scattering states form the continuum along the real $k$-axis. The two are not separate phenomena but different aspects of the same analytic function. This deep connection between bound states and scattering --- encoded in the analyticity of the S-matrix --- is one of the central organizing principles of quantum mechanics.

🔗 Connection: The hydrogen Coulomb scattering cross section applies not just to electron-proton scattering but (with appropriate charge substitutions) to any two charged particles: alpha-nucleus (Rutherford's experiment), electron-electron (Moller scattering, which requires quantum field theory for the full treatment), and proton-proton (which involves Mott scattering corrections for identical particles). The Coulomb potential is truly universal.

22.11 Summary, Key Results, and Project Checkpoint

Comparing the Two Approaches: Born vs. Partial Waves

Having developed both methods, we are now in a position to compare them systematically.

In practice, the two methods are complementary. At high energies, the Born approximation gives quick results and intuitive pictures (the cross section as a "photograph" of the potential's Fourier transform). At low energies, partial wave analysis is essential --- only a few terms contribute, and resonances can be identified and characterized.

For intermediate energies and potentials of moderate strength, both methods can be applied and compared. Their agreement (or disagreement) serves as a consistency check on the calculation.

Conceptual Summary

Scattering theory provides the mathematical framework for the most fundamental type of physics experiment: throwing things at each other. We have developed two complementary approaches:

The Born approximation is perturbative: the scattering amplitude is (proportional to) the Fourier transform of the potential, evaluated at the momentum transfer. It works best for weak potentials or high energies and provides intuition about the relationship between the potential's shape and the angular distribution of scattered particles.

Partial wave analysis is exact: it decomposes the scattering amplitude into contributions from each angular momentum channel $l$. Each partial wave is characterized by a phase shift $\delta_l$ that encodes the effect of the potential on that channel. Resonances appear as rapid passage of $\delta_l$ through $\pi/2$, producing Breit-Wigner peaks in the cross section. This approach works best at low energies where only a few partial waves contribute.

The optical theorem ties everything together: $\sigma_{\text{tot}} = (4\pi/k)\operatorname{Im}[f(0)]$, expressing probability conservation as a relation between the total cross section and the imaginary part of the forward scattering amplitude.

The S-matrix provides the fundamental description, relating incoming to outgoing states. Its unitarity encodes probability conservation, its poles encode bound states and resonances, and its structure persists even when no potential exists (as in quantum field theory).

Key Equations at a Glance

New Terms Introduced

Techniques Introduced

1. Born approximation: Computing scattering amplitudes via Fourier transforms of the potential
2. Partial wave decomposition: Expanding the scattering amplitude in angular momentum eigenstates
3. Phase shift extraction: Numerically solving the radial equation and matching to free-particle asymptotics
4. Resonance identification: Recognizing Breit-Wigner peaks and extracting resonance parameters

Progressive Project Checkpoint

In this chapter, you add the scattering module to your quantum toolkit:

# New functions for quantum_toolkit/scattering.py
born_approximation(V_func, k, theta)     # First Born amplitude for given potential
partial_waves(V_func, E, l_max, r_max)   # Phase shifts for l = 0, ..., l_max
cross_section(delta_l, k, theta)          # Differential cross section from phase shifts
breit_wigner(E, E_r, Gamma, l, k)        # Breit-Wigner resonance cross section
optical_theorem_check(f_forward, sigma)   # Verify the optical theorem
rutherford(E, Z1, Z2, theta)             # Rutherford cross section

See code/project-checkpoint.py for the full implementation and code/example-01-scattering.py for worked examples with visualization.

⚖️ Interpretation: Scattering theory raises a subtle interpretive question. The scattering amplitude $f(\theta)$ is a property of the stationary state $\psi(\mathbf{r})$, which is time-independent. Yet scattering is inherently a time-dependent process: a wave packet approaches, interacts, and departs. How do we reconcile these? The resolution is that for narrow-bandwidth wave packets, the stationary-state cross section gives the correct scattering probability at each energy, and the time-dependent picture and time-independent picture agree. This is the scattering-theory version of the general principle that stationary states describe time-averaged behavior.

Looking Ahead

Scattering theory is our last topic in approximation methods. With perturbation theory (Chapters 17--18), the variational principle (Chapter 19), WKB (Chapter 20), time-dependent perturbation theory (Chapter 21), and now scattering theory under our belt, we have assembled a formidable toolkit for attacking any quantum problem --- exactly where possible, approximately where necessary.

It is worth pausing to appreciate the intellectual structure we have built. In Parts I and II, we set up the formalism --- wave mechanics, Dirac notation, operators, and symmetries. In Part III, we developed angular momentum and spin, the internal quantum numbers that classify states. In Part IV, we have learned how to solve problems that are too complicated for exact solution: perturbation theory for nearly solvable problems, the variational principle for ground states, WKB for the semiclassical regime, time-dependent perturbation theory for transitions, and now scattering theory for collisions. These tools, taken together, can address essentially any problem in non-relativistic quantum mechanics.

In Part V, we turn to the conceptual frontiers of quantum mechanics: the density matrix and mixed states (Chapter 23), entanglement and Bell's theorem (Chapter 24), and the foundations of quantum information (Chapter 25). The S-matrix formalism we introduced in Section 22.9 will reappear in Chapter 29 (Dirac equation) and Chapter 34 (second quantization), where scattering theory becomes the language of particle physics. The Born approximation, generalized to quantum field theory via Feynman diagrams, is the computational backbone of the Standard Model. In a very real sense, the ideas in this chapter --- incident states, scattered states, cross sections, unitarity --- are the foundation on which all of modern particle physics is built.