Quantum Mechanics Course

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← Part VIII/Born Approximation

2. Born Approximation

Reading time: ~32 minutes | Pages: 8

Perturbative approach to scattering: weak potentials treated as perturbations on plane waves.

The Basic Idea

For weak potential $V(\vec{r})$:

  • Treat $V$ as perturbation
  • Incident wave barely distorted
  • Calculate scattering amplitude to first order
  • Analogous to first-order time-independent perturbation theory

First Born Approximation

Scattering amplitude:

$$f(\theta, \phi) = -\frac{m}{2\pi\hbar^2}\int V(\vec{r})e^{i\vec{q}\cdot\vec{r}}d^3r$$

where $\vec{q} = \vec{k}_i - \vec{k}_f$ is momentum transfer

Magnitude of momentum transfer:

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

For elastic scattering: $|\vec{k}_i| = |\vec{k}_f| = k$

Fourier Transform Interpretation

Scattering amplitude = Fourier transform of potential:

$$f(\vec{q}) \propto \tilde{V}(\vec{q})$$

Physical insight:

  • Large $q$ (large angle) → probes short-range structure
  • Small $q$ (small angle) → probes long-range structure
  • Scattering "measures" Fourier components of potential

Spherically Symmetric Potential

For $V = V(r)$, angular integration gives:

$$f(q) = -\frac{2m}{\hbar^2 q}\int_0^\infty V(r)r\sin(qr)dr$$

Depends only on $q = 2k\sin(\theta/2)$, not on $\phi$

Differential cross section:

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

Example: Yukawa Potential

Screened Coulomb potential:

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

First Born approximation:

$$f(\theta) = -\frac{2mV_0}{\hbar^2(q^2 + \mu^2)}$$

Cross section:

$$\frac{d\sigma}{d\Omega} = \frac{4m^2V_0^2}{\hbar^4(4k^2\sin^2(\theta/2) + \mu^2)^2}$$

Limit $\mu \to 0$: Rutherford formula

Example: Hard Sphere

Square well potential $V(r) = -V_0$ for $r < a$, zero otherwise:

$$f(\theta) = \frac{2mV_0}{\hbar^2 q^3}[qa\cos(qa) - \sin(qa)]$$

Features:

  • Oscillations in $\theta$ (diffraction pattern)
  • First minimum at $qa \approx 4.49$
  • Angular width $\Delta\theta \sim \lambda/a$

Validity Conditions

Born approximation valid when:

  1. Weak potential: $|V| \ll E$
  2. High energy: $k \gg \sqrt{2m|V|}/\hbar$
  3. Quantitative criterion:
    $$\left|\frac{2m}{\hbar^2}\int V(\vec{r})e^{i\vec{q}\cdot\vec{r}}d^3r\right| \ll 1$$

Fails for:

  • Strong potentials (nuclear forces)
  • Low energies
  • Long-range Coulomb (logarithmic divergence)

Higher Born Approximations

Second Born approximation:

$$f^{(2)} = f^{(1)} + \left(\frac{-m}{2\pi\hbar^2}\right)^2\int\int V(\vec{r}_1)V(\vec{r}_2)\frac{e^{ik|\vec{r}_1-\vec{r}_2|}}{|\vec{r}_1-\vec{r}_2|}e^{i\vec{q}\cdot\vec{r}_1}d^3r_1 d^3r_2$$

Accounts for double scattering

Born series: Systematic expansion in powers of $V$

Coulomb Scattering

Pure Coulomb $V = Ze^2/(4\pi\epsilon_0 r)$:

Problem: Long range causes divergence

Solution: Screen at large distance or use exact result

Remarkably, first Born gives exact Rutherford formula:

$$\frac{d\sigma}{d\Omega} = \left(\frac{Ze^2}{16\pi\epsilon_0 E}\right)^2\frac{1}{\sin^4(\theta/2)}$$

Accident of Coulomb 1/r form

Form Factors

For extended target with charge distribution $\rho(\vec{r})$:

$$F(q) = \int \rho(\vec{r})e^{i\vec{q}\cdot\vec{r}}d^3r$$

Modified cross section:

$$\frac{d\sigma}{d\Omega} = \left(\frac{d\sigma}{d\Omega}\right)_{\text{point}} |F(q)|^2$$

Applications:

  • Nuclear charge distribution from electron scattering
  • Proton structure functions
  • Crystal structure from X-ray/neutron diffraction

Inelastic Scattering

Born approximation extends to inelastic processes:

$$f_{i\to f} = -\frac{m}{2\pi\hbar^2}\langle f|V|i\rangle$$

Matrix element between initial and final states

Examples:

  • Atomic excitation by electron impact
  • Nuclear inelastic scattering
  • Compton scattering (requires relativistic treatment)

Advantages and Limitations

Advantages:

  • Simple analytical formula (often tractable integrals)
  • Physical intuition (Fourier transform)
  • Systematic improvement (Born series)
  • Works for complex potentials

Limitations:

  • Only valid for weak potentials or high energies
  • Misses resonances entirely
  • Poor for low-energy or strong coupling
  • Coulomb requires special treatment

For strong potentials or low energies, use partial wave analysis instead

← Cross SectionsPartial Wave Analysis →