First Born Approximation

The first Born approximation is the leading weak-potential approximation to the scattering amplitude. In transition-operator language it sets $T^{(+)}(E)\approx V$; equivalently, it replaces the exact scattering wave inside the potential region by the incoming free plane wave. T-Matrix owns the exact operator, Born Series owns the repeated-scattering expansion, and Validity of the Born Approximation owns quantitative control tests.

With the amplitude convention

$$\psi_{\mathbf k}^{(+)}(\mathbf r) \sim e^{i\mathbf k\cdot\mathbf r} + f(\theta,\phi) \frac{e^{ikr}}{r},$$

the exact Lippmann-Schwinger amplitude is

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

The first Born approximation sets

$$\psi_{\mathbf k}^{(+)}(\mathbf r) \approx e^{i\mathbf k\cdot\mathbf r},$$

giving

$$f_{\mathrm B}(\mathbf q) = - \frac{m}{2\pi\hbar^2} \int d^3r\, e^{-i\mathbf q\cdot\mathbf r} V(\mathbf r),$$

where

$$\mathbf q = \mathbf k'-\mathbf k$$

is the momentum transfer.

Assumptions

The approximation is controlled when the scattered wave generated by the potential is small compared with the incident wave in the region where $V$ is important. This often means:

• the potential is weak,
• the incident energy is high enough,
• no bound state or resonance lies close to the scattering energy,
• the interaction is sufficiently short-ranged,
• repeated scattering from the potential is negligible.

These are diagnostics, not a single universal inequality. The Born approximation is a perturbative expansion in the effect of $V$ on the scattering state. The range-and-strength parameters, high-energy phase criterion, and pole warnings are developed at Validity of the Born Approximation.

Derivation from Lippmann-Schwinger

The Lippmann-Schwinger equation is

$$\lvert\psi^{(+)}\rangle = \lvert\phi\rangle + G_0^{(+)}V\lvert\psi^{(+)}\rangle.$$

Iterating once gives

$$\lvert\psi^{(+)}\rangle = \lvert\phi\rangle + G_0^{(+)}V\lvert\phi\rangle + \cdots.$$

The first Born approximation keeps only the incoming free state inside the amplitude matrix element. In coordinate language, this is exactly the replacement $\psi^{(+)}\to e^{i\mathbf k\cdot\mathbf r}$ in the integral.

Momentum Transfer

For elastic scattering,

$$|\mathbf k'|=|\mathbf k|=k.$$

The magnitude of the momentum transfer is

$$q = 2k\sin\frac{\theta}{2}.$$

Thus a central potential produces a Born amplitude that depends only on $q$, or equivalently only on the scattering angle $\theta$.

Central Potentials

If $V(\mathbf r)=V(r)$, then the three-dimensional Fourier transform reduces to

$$\int d^3r\, e^{-i\mathbf q\cdot\mathbf r} V(r) = 4\pi \int_0^\infty r^2dr\, V(r) \frac{\sin(qr)}{qr}.$$

Therefore

$$f_{\mathrm B}(q) = - \frac{2m}{\hbar^2} \int_0^\infty r^2dr\, V(r) \frac{\sin(qr)}{qr}.$$

This is why scattering at different angles probes different spatial Fourier components of the potential.

Gaussian Potential

For

$$V(r)=V_0e^{-r^2/a^2},$$

the Fourier transform is

$$\int d^3r\, e^{-i\mathbf q\cdot\mathbf r} V_0e^{-r^2/a^2} = V_0\pi^{3/2}a^3 e^{-q^2a^2/4}.$$

Thus

$$f_{\mathrm B}(q) = - \frac{mV_0\sqrt{\pi}a^3}{2\hbar^2} e^{-q^2a^2/4}.$$

A broad smooth potential suppresses large momentum transfer, so large-angle scattering is small.

Yukawa Potential

For a Yukawa potential

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

the Fourier transform is

$$\int d^3r\, e^{-i\mathbf q\cdot\mathbf r} g\frac{e^{-\mu r}}{r} = \frac{4\pi g}{q^2+\mu^2}.$$

The Born amplitude is

$$f_{\mathrm B}(q) = - \frac{2mg}{\hbar^2} \frac{1}{q^2+\mu^2}.$$

This example foreshadows how exchange of a massive mediator produces a momentum-space denominator.

Yukawa Potential in the Born Approximation carries this model through its differential, total, transport, threshold, and partial-wave observables, and states the limits of the tree-exchange analogy.

Validity Checks

The first Born approximation should be checked rather than trusted automatically. At minimum, state the energy and angular range, inspect nearby thresholds or poles, and compare one piece of omitted physics with the target tolerance.

Validity of the Born Approximation is the canonical guide to interaction-region bounds, weak and high-energy scales, partial-wave tests, low-energy $s$-wave failure, long-range interactions, and numerical residuals.

Born Approximation Numerical Test turns those diagnostics into a converged coupling-and-momentum sweep for a repulsive Gaussian potential.

Common Mistakes

• Treating the Born approximation as a general scattering formula rather than a weak-potential approximation.
• Forgetting that the result depends on Fourier-transform convention.
• Ignoring long-range Coulomb subtleties.
• Using Born scattering near a resonance or shallow bound state.
• Comparing $f_{\mathrm B}$ directly to exact amplitudes without matching normalization conventions.

Exercises

1. Derive the central-potential Born formula from the three-dimensional Fourier transform.

Solution

Choose the polar axis along $\mathbf q$. Then

$$\mathbf q\cdot\mathbf r=qr\cos\theta.$$

The angular integral is

$$\int d\Omega\,e^{-iqr\cos\theta} = 4\pi\frac{\sin(qr)}{qr}.$$

Substitution gives

$$f_{\mathrm B}(q) = - \frac{m}{2\pi\hbar^2} 4\pi \int_0^\infty r^2dr\, V(r) \frac{\sin(qr)}{qr},$$

or

$$f_{\mathrm B}(q) = - \frac{2m}{\hbar^2} \int_0^\infty r^2dr\, V(r) \frac{\sin(qr)}{qr}.$$

2. For a repulsive Gaussian potential with $V_0>0$, what is the sign of the Born amplitude at $q=0$?

Solution

At $q=0$,

$$f_{\mathrm B}(0) = - \frac{mV_0\sqrt{\pi}a^3}{2\hbar^2}.$$

For $V_0>0$, this is negative.

3. Why is the Born approximation expected to improve at high incident energy for many short-range potentials?

Solution

At higher energy the incident wavelength is shorter and the particle spends less time being distorted by a localized weak potential. Repeated scattering from the same potential region is often reduced. This is a heuristic, not a theorem; resonances and long-range interactions can still invalidate a naive Born estimate.

References

• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982.