Elastic scattering

): $$-\frac{\hbar^2}{2m} \nabla^2 \psi(\vec{r}) + V(\vec{r}) \psi(\vec{r}) = E \psi(\vec{r}),$$−2mℏ2​∇2ψ(r

)+V(r

)ψ(r

)=Eψ(r

), where $E = \frac{\hbar^2 k^2}{2m}$E=2mℏ2k2​ is the energy of the incident particle with wave number $k$k. The unperturbed solution, representing an incident plane wave propagating along the $z$z-direction, is $\psi_i(\vec{r}) = e^{i k z}$ψi​(r

)=eikz. The total wave function is then expressed as $\psi(\vec{r}) = \psi_i(\vec{r}) + \psi_s(\vec{r})$ψ(r

)=ψi​(r

)+ψs​(r

), where $\psi_s(\vec{r})$ψs​(r

) is the scattered wave. Far from the scattering region (as $r \to \infty$r→∞), the scattered wave takes the asymptotic form $$\psi_s(\vec{r}) \sim f(\theta) \frac{e^{i k r}}{r},$$ψs​(r

)∼f(θ)reikr​, with $f(\theta)$f(θ) denoting the scattering amplitude, which depends on the scattering angle $\theta$θ between the incident and outgoing directions. This form captures the outgoing spherical wave nature of the scattered particle, modulated by the amplitude $f(\theta)$f(θ). The scattering amplitude $f(\theta)$f(θ) encapsulates the quantum mechanical probability for scattering into direction $\hat{r}$r^. The differential cross-section, which gives the probability per unit solid angle for scattering into $\theta$θ, is directly related to the modulus squared of the amplitude: $$\frac{d\sigma}{d\Omega} = |f(\theta)|^2.$$dΩdσ​=∣f(θ)∣2. This relation generalizes the classical differential cross-section to the quantum regime, where interference effects are inherently included through the wave function. Integrating over all angles yields the total cross-section $\sigma = \int |f(\theta)|^2 d\Omega$σ=∫∣f(θ)∣2dΩ. The asymptotic behavior ensures that the flux of scattered particles matches experimental observables in far-field detectors. To solve for $\psi(\vec{r})$ψ(r

) and thus $f(\theta)$f(θ), the Lippmann-Schwinger equation provides an integral formulation equivalent to the Schrödinger equation. It expresses the total wave function in terms of the incident wave and the potential: $$\psi(\vec{r}) = \phi(\vec{r}) + \frac{2m}{\hbar^2} \frac{1}{4\pi} \int G(\vec{r}, \vec{r}') V(\vec{r}') \psi(\vec{r}') d^3 r',$$ψ(r

)=ϕ(r

)+ℏ22m​4π1​∫G(r

,r

′)V(r

′)ψ(r

′)d3r′, where $\phi(\vec{r}) = e^{i k z}$ϕ(r

)=eikz is the incident plane wave, and $G(\vec{r}, \vec{r}') = -\frac{e^{i k |\vec{r} - \vec{r}'|}}{|\vec{r} - \vec{r}'|}$G(r

,r

′)=−∣r

−r

′∣eik∣r

−r

′∣​ is the outgoing Green's function for the Helmholtz equation $(\nabla^2 + k^2) G = \delta(\vec{r} - \vec{r}')$(∇2+k2)G=δ(r

−r

′). This equation is particularly useful for perturbative solutions, as it resums multiple scattering events through the integral term. The scattering amplitude can be extracted from the asymptotic limit of this equation. The formalism was developed by Lippmann and Schwinger in 1950 as a variational approach to dynamic problems in quantum mechanics. A key perturbative method to approximate $f(\theta)$f(θ) is the first-order Born approximation, obtained by replacing $\psi(\vec{r}')$ψ(r

′) in the Lippmann-Schwinger equation with the incident wave $\phi(\vec{r}')$ϕ(r

′). This yields $$f(\theta) \approx -\frac{m}{2\pi \hbar^2} \int e^{-i \vec{q} \cdot \vec{r}} V(\vec{r}) \, d^3 r,$$f(θ)≈−2πℏ2m​∫e−iq

​⋅r

V(r

)d3r, where $\vec{q} = \vec{k} - \vec{k}'$q

​=k

−k

′ is the momentum transfer vector with $|\vec{k}| = |\vec{k}'| = k$∣k

∣=∣k

′∣=k and $\vec{k}' = k \hat{r}$k

′=kr^, so $q = 2 k \sin(\theta/2)$q=2ksin(θ/2). The integral represents the Fourier transform of the potential, linking the scattering amplitude directly to its spatial structure. This approximation was introduced by Max Born in 1926 as part of early quantum collision theory. The Born approximation holds for weak potentials where $|V(\vec{r})| \ll E$∣V(r

)∣≪E, ensuring higher-order terms in the Born series are negligible, and for slowly varying potentials where the potential changes little over the de Broglie wavelength $\lambda = 2\pi / k$λ=2π/k, corresponding to $q$q times the potential range being small. These conditions are typically met in high-energy scattering or dilute systems, but fail for strong or rapidly oscillating potentials, such as in low-energy atomic collisions.^[20]

Partial Wave Analysis