Scattering Theory

Scattering theory is a fundamental concept in quantum mechanics that describes how particles interact and deviate from their original paths. It's crucial for understanding phenomena like the interaction of electrons with atoms, the scattering of light by molecules, and the collision of subatomic particles in high-energy physics. In essence, scattering theory provides a framework to calculate the probability of a particle being deflected in a particular direction after interacting with a potential or another particle.

Differential and Total Scattering Cross Sections

When we talk about scattering, we are interested in how likely it is for a particle to be scattered into a specific direction or into any direction at all. This is quantified by cross sections.

Differential Scattering Cross Section ($\frac{d\sigma}{d\Omega}$)

The differential scattering cross section, denoted as $\frac{d\sigma}{d\Omega}$, represents the probability that an incident particle will be scattered into a specific solid angle $d\Omega$. Imagine a beam of particles hitting a target. The differential cross section tells us, for a unit incident flux, how many particles per unit time are scattered into a small cone defined by the solid angle $d\Omega$. The units of $\frac{d\sigma}{d\Omega}$ are typically area per steradian (e.g., m²/sr).

Mathematically, if $N_{in}$ is the number of incident particles per unit area per unit time, and $dN_{out}$ is the number of particles scattered into the solid angle $d\Omega$ per unit time, then:

$dN_{out} = N_{in} \frac{d\sigma}{d\Omega} d\Omega$

The differential scattering cross section is a function of the scattering angle ($\theta$) and the azimuthal angle ($\phi$), and often depends on the energy of the incident particle and the nature of the scattering potential.

Total Scattering Cross Section ($\sigma$)

The total scattering cross section, $\sigma$, is the integral of the differential scattering cross section over all possible solid angles. It represents the total probability that an incident particle will be scattered in *any* direction. If we think of the target as having an effective "size" or "area" that can cause scattering, the total cross section is a measure of this effective area. Its units are typically area (e.g., m² or barns, where 1 barn = 10⁻²⁸ m²).

It is calculated by integrating the differential cross section over the entire solid angle ($4\pi$ steradians):

$\sigma = \int \frac{d\sigma}{d\Omega} d\Omega = \int_{0}^{2\pi} \int_{0}^{\pi} \frac{d\sigma}{d\theta d\phi} \sin\theta \, d\theta \, d\phi$

A larger total cross section means a higher probability of scattering occurring.

Born Approximation

The Born approximation is a method used to approximate the scattering amplitude and cross sections, particularly useful for weak potentials or high-energy scattering where the incident wave is not significantly perturbed by the scattering potential. It assumes that the scattered wave is small compared to the incident wave.

The key idea is to treat the interaction potential $V(\mathbf{r})$ as a perturbation. The scattering amplitude $f(\theta, \phi)$ can be approximated by the first Born approximation as:

$f^{(1)}(\mathbf{q}) = -\frac{m}{2\pi\hbar^2} \int e^{-i\mathbf{q}\cdot\mathbf{r}} V(\mathbf{r}) e^{i\mathbf{k}\cdot\mathbf{r}} d^3\mathbf{r}$

Here, $m$ is the mass of the scattered particle, $\hbar$ is the reduced Planck constant, $\mathbf{k}$ is the incident wave vector, and $\mathbf{q} = \mathbf{k}' - \mathbf{k}$ is the momentum transfer vector, where $\mathbf{k}'$ is the wave vector of the scattered particle. For elastic scattering, $|\mathbf{k}| = |\mathbf{k}'|$.

The integral can be simplified by using the Fourier transform of the potential, $\tilde{V}(\mathbf{q})$:

$f^{(1)}(\mathbf{q}) = -\frac{m}{2\pi\hbar^2} \tilde{V}(\mathbf{q})$

The differential cross section is then given by $|f^{(1)}(\mathbf{q})|^2$.

When is the Born approximation valid?

The Born approximation is generally valid when the potential is weak, or when the energy of the incident particle is high. A common criterion for the validity of the first Born approximation is:

Example: Rutherford Scattering

The classical Rutherford scattering formula for alpha particles scattering off a nucleus can be derived using the Born approximation with the Coulomb potential $V(r) = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 r}$. The calculation yields the correct angular dependence, although the quantum mechanical derivation is more rigorous.

Partial Wave Analysis and Phase-Shift Analysis

For spherically symmetric potentials ($V(\mathbf{r}) = V(r)$), it is convenient to decompose the scattering problem into spherical harmonics. This leads to the partial wave analysis. The incident plane wave is expanded into a sum of spherical waves. Each spherical wave is characterized by its angular momentum quantum number $l$.

The Schrödinger equation in spherical coordinates for a potential $V(r)$ separates into a radial equation and an angular equation. The angular part yields the spherical harmonics $Y_{lm}(\theta, \phi)$, associated with the orbital angular momentum $L^2 = \hbar^2 l(l+1)$.

The radial wave function $R_{nl}(r)$ satisfies the radial equation:

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

where $u_l(r) = r R_l(r)$ and $E$ is the energy of the incident particle. The term $\frac{\hbar^2 l(l+1)}{2mr^2}$ is the effective potential due to angular momentum, often called the "centrifugal barrier".

Phase Shift ($\delta_l$)

For large $r$ (far from the scattering center, where $V(r) \to 0$), the radial wave function $u_l(r)$ behaves like:

$u_l(r) \xrightarrow{r \to \infty} A_l \sin(kr - \frac{l\pi}{2} + \delta_l)$

where $k = \sqrt{2mE}/\hbar$ is the wave number, and $\delta_l$ is the phase shift. The term $-\frac{l\pi}{2}$ comes from the asymptotic form of the spherical Bessel function.

In the absence of a potential ($V(r)=0$), the solution is $u_l^{(0)}(r) = B_l \sin(kr - \frac{l\pi}{2})$. The phase shift $\delta_l$ quantifies the change in the phase of the radial wave function due to the scattering potential. If $\delta_l = 0$, the potential has no effect on the $l$-th partial wave.

The phase shift $\delta_l$ depends on the energy $E$ and the angular momentum $l$.

Scattering Amplitude in terms of Phase Shifts

The scattering amplitude $f(\theta)$ for a spherically symmetric potential can be expressed as a sum over partial waves:

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

where $P_l(\cos\theta)$ are the Legendre polynomials.

The differential cross section is then $|f(\theta)|^2$.

Total Cross Section in terms of Phase Shifts

The total cross section can also be expressed using phase shifts:

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

This formula is very important. It shows that the total cross section is a sum of contributions from each partial wave.

Resonance Scattering: When a particular phase shift $\delta_l$ approaches $\pi/2$, $\sin^2(\delta_l)$ reaches its maximum value of 1. This leads to a peak in the total cross section, indicating a resonant scattering phenomenon. This often occurs when the energy of the incident particle matches an energy level of the compound system formed during the collision.

Low Energy Scattering ($kr \ll 1$)

At very low energies, only the $l=0$ partial wave (s-wave scattering) contributes significantly because the centrifugal barrier $\frac{\hbar^2 l(l+1)}{2mr^2}$ suppresses higher angular momentum waves. In this limit, $k \to 0$, and the scattering is isotropic.

The phase shift $\delta_0$ is related to the scattering length $a$ by $\delta_0 \approx -ka$ for $ka \ll 1$. The total cross section then becomes $\sigma \approx 4\pi a^2$.

Relativistic Wave Equations

The Schrödinger equation is non-relativistic. For particles moving at speeds comparable to the speed of light, or when dealing with antiparticles, relativistic wave equations are necessary. These equations incorporate special relativity into quantum mechanics.

Klein–Gordon Equation

The Klein–Gordon equation is a relativistic wave equation that describes spin-0 particles (bosons). It is derived by combining the relativistic energy-momentum relation $E^2 = (pc)^2 + (m_0c^2)^2$ with the quantum mechanical operators for energy ($E \to i\hbar\frac{\partial}{\partial t}$) and momentum ($p \to -i\hbar\nabla$).

Substituting these operators into the energy-momentum relation:

$(i\hbar\frac{\partial}{\partial t})^2 \psi = (-i\hbar c\nabla)^2 \psi + (m_0c^2)^2 \psi$

This simplifies to the Klein–Gordon equation:

$\frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi + \frac{m_0^2 c^2}{\hbar^2} \psi = 0$

Or, using natural units where $\hbar=c=1$:

$\partial_\mu \partial^\mu \psi + m_0^2 \psi = 0$

where $\partial_\mu = (\frac{1}{c}\frac{\partial}{\partial t}, \nabla)$ and $\partial^\mu = ( \frac{1}{c}\frac{\partial}{\partial t}, -\nabla)$. The operator $\partial_\mu \partial^\mu$ is the d'Alembertian operator.

Challenges with Klein–Gordon Equation:

Initially, the Klein–Gordon equation was disfavored because it led to probability densities that could be negative, which is unphysical. This issue arises because it is a second-order equation in time, similar to classical wave equations, and does not naturally yield a conserved positive-definite probability current. However, it is now understood as the correct equation for spin-0 particles.

Free-Particle Solutions of the Klein–Gordon Equation

Consider a free particle, meaning there is no potential, $V=0$. The equation becomes:

$\frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi + \frac{m_0^2 c^2}{\hbar^2} \psi = 0$

We look for plane wave solutions of the form $\psi(\mathbf{r}, t) = A e^{i(\mathbf{p}\cdot\mathbf{r} - E t)/\hbar}$. Substituting this into the equation:

$\frac{1}{c^2} (-i\hbar E/\hbar)^2 A e^{i(\mathbf{p}\cdot\mathbf{r} - E t)/\hbar} - (i\mathbf{p}/\hbar \cdot i\mathbf{p}/\hbar) A e^{i(\mathbf{p}\cdot\mathbf{r} - E t)/\hbar} + \frac{m_0^2 c^2}{\hbar^2} A e^{i(\mathbf{p}\cdot\mathbf{r} - E t)/\hbar} = 0$

$-\frac{E^2}{c^2} - \frac{p^2}{\hbar^2} + \frac{m_0^2 c^2}{\hbar^2} = 0$

Multiplying by $-\hbar^2$:

$\frac{E^2}{c^2} + p^2 - m_0^2 c^2 = 0$

$E^2 = p^2c^2 + m_0^2c^4$

This is precisely the relativistic energy-momentum relation. However, the equation $E^2 = p^2c^2 + m_0^2c^4$ has two possible solutions for $E$:

$E = \pm \sqrt{p^2c^2 + m_0^2c^4}$

This implies that the Klein–Gordon equation describes particles with both positive and negative energies. The positive energy solutions correspond to particles, while the negative energy solutions were initially problematic but are now interpreted as describing antiparticles.

Dirac Equation

The Dirac equation was developed by Paul Dirac to describe relativistic spin-1/2 particles (fermions), such as electrons. Dirac sought a first-order relativistic wave equation in both space and time that would reduce to the Schrödinger equation in the non-relativistic limit and naturally incorporate spin.

Dirac's insight was to linearize the relativistic energy-momentum relation $E^2 = p^2c^2 + m_0^2c^4$. He proposed an equation of the form:

$i\hbar\frac{\partial \psi}{\partial t} = (\boldsymbol{\alpha} \cdot \mathbf{p}c + \boldsymbol{\beta}m_0c^2) \psi$

Here, $\psi$ is not a simple scalar function but a four-component column vector (a spinor), and $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ are matrices. For the equation to be consistent with the relativistic energy-momentum relation, these matrices must satisfy certain commutation and anti-commutation relations.

The Dirac matrices are:

• $\boldsymbol{\alpha} = (\alpha_1, \alpha_2, \alpha_3)$
• $\boldsymbol{\beta}$

These are $4 \times 4$ matrices. The standard representation is:

$\boldsymbol{\alpha}_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix}$, $\boldsymbol{\beta} = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}$

where $\sigma_i$ are the $2 \times 2$ Pauli matrices and $I$ and $0$ are the $2 \times 2$ identity and zero matrices, respectively.

The Dirac equation in covariant form is:

$(i\hbar\gamma^\mu \partial_\mu - m_0c)\psi = 0$

where $\gamma^\mu = (\gamma^0, \gamma^1, \gamma^2, \gamma^3)$ are the Dirac gamma matrices, related to $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ by $\gamma^0 = \boldsymbol{\beta}$ and $\gamma^i = \boldsymbol{\beta}\boldsymbol{\alpha}_i$ for $i=1,2,3$. The $\gamma^\mu$ matrices also satisfy anti-commutation relations: $\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}I$, where $\eta^{\mu\nu}$ is the Minkowski metric tensor.

Spin and Antiparticles: The Dirac equation naturally predicts the existence of spin-1/2 and also leads to the existence of antiparticles. For every particle described by the Dirac equation, there is a corresponding antiparticle with the same mass but opposite charge. For example, the Dirac equation predicts the positron (anti-electron).

Free-Particle Solutions of the Dirac Equation

For a free particle (no potential, so $\mathbf{p}$ is a constant momentum), the Dirac equation is:

$(i\hbar\frac{\partial}{\partial t} - m_0c^2)\psi = \boldsymbol{\alpha} \cdot \mathbf{p}c \psi$

We look for solutions of the form $\psi(\mathbf{r}, t) = u(\mathbf{p}) e^{i(\mathbf{p}\cdot\mathbf{r} - Et)/\hbar}$, where $u(\mathbf{p})$ is a constant four-component spinor. Substituting this yields:

$(E - \boldsymbol{\alpha} \cdot \mathbf{p}c)\psi = m_0c^2\psi$

This means $(E - \boldsymbol{\alpha} \cdot \mathbf{p}c)u(\mathbf{p}) = m_0c^2 u(\mathbf{p})$.

The eigenvalues of $\boldsymbol{\alpha} \cdot \mathbf{p}c$ are $\pm pc$. So, the equation becomes:

$(E - (\pm pc)) u(\mathbf{p}) = m_0c^2 u(\mathbf{p})$

This leads to two sets of solutions for the energy $E$:

1. $E_+ = pc + m_0c^2$
2. $E_- = -pc - m_0c^2$

These solutions correspond to the positive and negative energy states.

The solutions are four-component spinors. For each momentum $\mathbf{p}$, there are two linearly independent solutions corresponding to positive energy ($E_+$) and two corresponding to negative energy ($E_-$). These correspond to the particle's spin. For example, for an electron, these represent spin-up and spin-down states for both positive and negative energy solutions.

The negative energy solutions, with energies $E_- = -pc - m_0c^2$, are interpreted as antiparticles. For instance, the negative energy solutions for the electron describe the positron. The Dirac sea concept was an early attempt to explain the stability of matter in the presence of these negative energy states.