Born Approximation

Purpose

The first Born approximation replaces the exact scattering wave inside the interaction region by the incoming free wave. With

$$\psi_{\mathbf k_i}^{(+)}(\mathbf r) \sim e^{i\mathbf k_i\cdot\mathbf r} + f(\mathbf k_f,\mathbf k_i) \frac{e^{ik_fr}}{r},$$

define the Fourier transform without any $2\pi$ prefactor:

$$\widetilde V(\mathbf q) = \int_{\mathbb R^3} d^3r\, e^{-i\mathbf q\cdot\mathbf r} V(\mathbf r).$$

For elastic scattering by a local potential,

$$f_{\mathrm B}(\mathbf q) = - \frac{\mu}{2\pi\hbar^2} \widetilde V(\mathbf q),$$

where $\mu$ is the reduced mass and

$$\mathbf q=\mathbf k_f-\mathbf k_i, \qquad q=2k\sin\frac{\theta}{2}.$$

For scattering from a fixed infinitely heavy target, $\mu$ reduces to the projectile mass. The derivation belongs to First Born Approximation. This card emphasizes normalization, calculation shortcuts, and evidence for or against validity.

At a glance

Task	Formula
Fourier convention	$\widetilde V(\mathbf q)=\int d^3r\,e^{-i\mathbf q\cdot\mathbf r}V(\mathbf r)$
First Born amplitude	$f_{\mathrm B}(\mathbf q)=-\mu\widetilde V(\mathbf q)/(2\pi\hbar^2)$
Elastic differential cross section	$d\sigma_{\mathrm B}/d\Omega=\mu^2\lvert\widetilde V(\mathbf q)\rvert^2/(4\pi^2\hbar^4)$
Central potential	$f_{\mathrm B}(q)=-(2\mu/\hbar^2)\int_0^\infty dr\,r^2V(r)\sin(qr)/(qr)$
Forward amplitude	$f_{\mathrm B}(0)=-(2\mu/\hbar^2)\int_0^\infty dr\,r^2V(r)$
Born scattering length	$a_{\mathrm B}=(2\mu/\hbar^2)\int_0^\infty dr\,r^2V(r)$
Born phase shift	$\delta_\ell^{\mathrm B}=-(2\mu k/\hbar^2)\int_0^\infty dr\,r^2V(r)j_\ell^2(kr)$
Operator statement	$T^{(+)}=V+VG_0^{(+)}V+\cdots$, so $T_{\mathrm B}=V$

The phase-shift row is a leading weak-phase formula for a real central potential. The scattering-length row additionally assumes the low-energy short-range limit and the convention $f(k\to0)=-a$.

Derivation checkpoint

The outgoing Lippmann–Schwinger equation is

$$\lvert\psi_{\mathbf k_i}^{(+)}\rangle = \lvert\mathbf k_i\rangle + G_0^{(+)}(E) V \lvert\psi_{\mathbf k_i}^{(+)}\rangle,$$

with

$$G_0^{(+)}(E) = \frac{1}{E-H_0+i0}.$$

In coordinate space, the exact amplitude in the convention of this card is

$$\begin{aligned} f(\mathbf k_f,\mathbf k_i) = - \frac{\mu}{2\pi\hbar^2} \int d^3r\, e^{-i\mathbf k_f\cdot\mathbf r} V(\mathbf r) \psi_{\mathbf k_i}^{(+)}(\mathbf r). \end{aligned}$$

The first Born replacement is

$$\psi_{\mathbf k_i}^{(+)}(\mathbf r) \longrightarrow e^{i\mathbf k_i\cdot\mathbf r}.$$

The phase in the integral becomes

$$e^{-i\mathbf k_f\cdot\mathbf r} e^{i\mathbf k_i\cdot\mathbf r} = e^{-i\mathbf q\cdot\mathbf r},$$

which produces the Fourier transform of $V$.

Equivalently, the transition operator obeys

$$T^{(+)} = V + VG_0^{(+)}V + VG_0^{(+)}VG_0^{(+)}V + \cdots.$$

First Born means $T^{(+)}\approx V$: the particle interacts once in the amplitude. Repeated interactions are organized at Born Series.

Plane-wave and transform conventions

Take momentum eigenstates normalized by

$$\langle\mathbf r\vert\mathbf k\rangle = \frac{e^{i\mathbf k\cdot\mathbf r}}{(2\pi)^{3/2}}.$$

Then

$$\langle\mathbf k_f\vert V\vert\mathbf k_i\rangle = \frac{ \widetilde V(\mathbf q) }{(2\pi)^3},$$

and the on-shell relation to the amplitude is

$$f(\mathbf k_f,\mathbf k_i) = - \frac{4\pi^2\mu}{\hbar^2} \langle\mathbf k_f\vert T^{(+)}(E) \vert\mathbf k_i\rangle.$$

Combining these equations with $T_{\mathrm B}=V$ recovers the displayed Born prefactor.

If a source defines its Fourier transform with $(2\pi)^{-3/2}$ or $(2\pi)^{-3}$ factors, its prefactor changes. Compare the complete pair “transform definition plus amplitude formula,” not either line in isolation.

Dimensional analysis provides a quick audit:

$$[\widetilde V] = \text{energy}\cdot\text{length}^3,$$

so

$$\left[ \frac{\mu\widetilde V}{\hbar^2} \right] = \text{length}.$$

This is the required dimension of $f$ in the outgoing spherical-wave convention.

Central potentials

For $V(\mathbf r)=V(r)$, choose the polar axis along $\mathbf q$. The angular integral is

$$\int d\Omega_{\mathbf r}\, e^{-iqr\cos\vartheta} = 4\pi \frac{\sin(qr)}{qr}.$$

Therefore

$$\widetilde V(q) = 4\pi \int_0^\infty dr\, r^2V(r) \frac{\sin(qr)}{qr},$$

and

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

The equivalent form

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

is convenient for analytic transforms but hides the regular $q\to0$ limit. For numerical work, the sinc form is usually safer near forward scattering.

For elastic kinematics,

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

Small angles probe long spatial scales, while large angles require large-$q$ Fourier components and are sensitive to short-distance structure. This interpretation is clearest when the approximation is valid over the angular range being discussed.

Cross sections

For a single elastic channel,

$$\frac{d\sigma_{\mathrm B}}{d\Omega} = \lvert f_{\mathrm B}(\mathbf q)\rvert^2 = \frac{\mu^2}{4\pi^2\hbar^4} \lvert\widetilde V(\mathbf q)\rvert^2.$$

The integrated elastic result is

$$\sigma_{\mathrm B} = \int d\Omega\, \lvert f_{\mathrm B}\rvert^2.$$

For an inelastic transition between internal states, replace $V$ by the channel matrix element:

$$V_{\beta\alpha}(\mathbf r) = \langle\chi_\beta\vert V(\mathbf r,\xi) \vert\chi_\alpha\rangle.$$

With the corresponding spherical-wave normalization,

$$\frac{d\sigma_{\beta\leftarrow\alpha}^{\mathrm B}}{d\Omega} = \frac{v_\beta}{v_\alpha} \left\lvert f_{\beta\alpha}^{\mathrm B} \right\rvert^2.$$

The channel-speed factor is independent of the Born approximation; it comes from flux. See the Scattering Cross Section Formula Card for spin sums, identical-particle counting, and acceptance integrals.

Example: Gaussian potential

For

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

the transform is

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

The amplitude is

$$f_{\mathrm B}(q) = - \frac{ \mu V_0\sqrt{\pi}\,a^3 }{ 2\hbar^2 } e^{-a^2q^2/4},$$

and

$$\frac{d\sigma_{\mathrm B}}{d\Omega} = \frac{ \pi\mu^2V_0^2a^6 }{ 4\hbar^4 } e^{-a^2q^2/2}.$$

With

$$b=k^2a^2,$$

angular integration gives

$$\sigma_{\mathrm B} = \frac{ \pi^2\mu^2V_0^2a^6 }{ 2\hbar^4 } \frac{ 1-e^{-2b} }{b}.$$

As $k\to0$, $(1-e^{-2b})/b\to2$, so

$$\sigma_{\mathrm B} \longrightarrow \frac{ \pi^2\mu^2V_0^2a^6 }{ \hbar^4 }.$$

This equals $4\pi a_{\mathrm B}^2$ with the Born scattering length computed below. The algebra checks the angular measure, transform coefficient, and low-energy normalization simultaneously.

Example: Yukawa potential

For

$$V(r) = g\frac{e^{-\kappa r}}{r}, \qquad \kappa>0,$$

the transform is

$$\widetilde V(q) = \frac{4\pi g}{q^2+\kappa^2}.$$

Thus

$$f_{\mathrm B}(q) = - \frac{2\mu g}{\hbar^2} \frac{1}{q^2+\kappa^2},$$

and

$$\frac{d\sigma_{\mathrm B}}{d\Omega} = \frac{4\mu^2g^2}{\hbar^4} \frac{1}{(q^2+\kappa^2)^2}.$$

The screening scale $\kappa$ regulates the forward direction. The full observable calculation, including total and transport cross sections, is at Yukawa Potential in the Born Approximation.

Forward limit and scattering length

For an integrable central potential,

$$\lim_{q\to0} \frac{\sin(qr)}{qr} = 1,$$

so

$$f_{\mathrm B}(0) = - \frac{2\mu}{\hbar^2} \int_0^\infty dr\, r^2V(r).$$

The exact low-energy $s$-wave convention is

$$f(k\to0)=-a_s.$$

Consequently,

$$a_{\mathrm B} = \frac{2\mu}{\hbar^2} \int_0^\infty dr\, r^2V(r).$$

A weak repulsive potential has $a_{\mathrm B}>0$ in this convention. A weak attractive potential has $a_{\mathrm B}<0$. This first-order sign rule fails nonperturbatively when an attractive potential is tuned through a new near-threshold bound state: the exact scattering length then diverges and changes sign.

Partial-wave diagnostic

For a real central potential, the leading Born phase shift is

$$\delta_\ell^{\mathrm B}(k) = - \frac{2\mu k}{\hbar^2} \int_0^\infty dr\, r^2V(r) j_\ell^2(kr).$$

This formula gives several checks:

• $\delta_\ell^{\mathrm B}$ is dimensionless;
• for weak repulsion it is negative;
• for $\ell=0$ and $k\to0$, $\delta_0^{\mathrm B}\sim-ka_{\mathrm B}$;
• a necessary perturbative warning sign is $\lvert\delta_\ell^{\mathrm B}\rvert\not\ll1$ in a partial wave that contributes materially to the observable.

Small first-order phases are useful evidence, not a universal convergence proof. An accidental zero in one phase shift or angle can hide a large higher-order correction elsewhere.

Order counting and the optical theorem

Introduce a bookkeeping coupling $V\to\lambda V$. Then

$$f = \lambda f^{(1)} + \lambda^2f^{(2)} + O(\lambda^3),$$

while

$$\frac{d\sigma}{d\Omega} = \lambda^2 \lvert f^{(1)}\rvert^2 + 2\lambda^3 \operatorname{Re} \left[ f^{(1)*}f^{(2)} \right] + O(\lambda^4).$$

For a real local potential, the forward first Born amplitude is real:

$$\operatorname{Im}f^{(1)}(0)=0.$$

This does not contradict the optical theorem. Its right-hand side begins at order $\lambda^2$, and perturbative unitarity requires

$$\operatorname{Im}f^{(2)}(0) = \frac{k}{4\pi} \int d\Omega\, \lvert f^{(1)}(\Omega)\rvert^2$$

for one-channel elastic scattering. The on-shell imaginary part of $VG_0^{(+)}V$ supplies this term.

Therefore a bare first Born amplitude is not exactly unitary. Testing it by inserting $f^{(1)}$ into both sides of the exact optical theorem mixes perturbative orders. The exact relation and its channel form are collected at Optical Theorem.

Validity diagnostics

The central question is whether

$$G_0^{(+)}V \lvert\mathbf k_i\rangle$$

is small compared with the incident wave where the potential acts, and for the observable and accuracy being claimed. There is no single necessary and sufficient scalar inequality for every potential.

For a potential of characteristic magnitude $V_0$ and range $R$, define

$$\kappa_R=kR, \qquad g = \frac{2\mu\lvert V_0\rvert R^2}{\hbar^2}.$$

Useful scale screens are

$$g\ll1 \qquad \text{when }kR\lesssim1,$$

and

$$\frac{g}{kR}\ll1 \qquad \text{when }kR\gg1.$$

The first asks whether the interaction is weak on the localization energy $\hbar^2/(2\mu R^2)$. The second estimates the accumulated high-energy phase through the interaction region. Shape-dependent factors and cancellations make these diagnostics rather than sharp universal bounds.

A stronger audit uses one or more of:

• compare $f^{(2)}$ with $f^{(1)}$ over the full energy and angular range;
• solve the integral equation or radial Schrödinger equation independently;
• compare Born and exact partial-wave phase shifts;
• scale $V\to\lambda V$ and verify the expected powers of $\lambda$;
• inspect nearby bound-state, virtual-state, threshold, and resonance poles;
• vary numerical cutoffs, quadrature, and the range used to truncate $V$;
• test perturbative unitarity at consistent orders.

The canonical guide is Validity of the Born Approximation.

Failure modes

Strong distortion or repeated scattering

If $G_0^{(+)}V$ is not small in the interaction region, the incident plane wave is a poor source for the first-order amplitude and successive Born terms need not decrease.

Threshold poles and resonances

A shallow bound state, virtual state, or resonance amplifies repeated scattering. The exact amplitude can approach a partial-wave unitarity limit even when the pointwise potential does not appear large.

Low energy

High incident energy often helps for a smooth short-range potential, but the converse is important: at low energy the scattering length can be nonperturbatively large. The first Born value cannot reproduce a pole in $a_s$ at finite coupling.

Long-range interactions

For unscreened Coulomb scattering, free plane waves are not the correct asymptotic states and the short-range Born derivation does not apply. A regulated transform can reproduce the Rutherford magnitude, but it misses the full Coulomb phase and does not validate the ordinary Born series.

Singular or contact interactions

Higher Born terms can be ultraviolet divergent or cutoff dependent. A formal first transform is not enough; the interaction must be regularized and its parameters matched or renormalized.

Diffraction zeros

Near a zero of $f^{(1)}$, the ratio $f^{(2)}/f^{(1)}$ can be large even if both terms are small, and a tiny absolute correction can shift the zero substantially. Assess the observable with an absolute error scale as well as a relative one.

Better starting points

If a known part of the interaction is strong or long ranged, split

$$H=H_{\mathrm{ref}}+W$$

and expand in the residual interaction $W$ using exact or controlled scattering states of $H_{\mathrm{ref}}$. This distorted-wave Born approximation incorporates important asymptotic or background physics before the perturbative step.

Other options include solving Lippmann–Schwinger directly, using a converged partial-wave calculation, or constructing an effective low-energy interaction matched to scattering data. A resummation is not automatically reliable; its poles, regulator dependence, and power counting still require checks.

Calculation workflow

1. State $\mu$, the energy, the incident and final wave vectors, and the amplitude convention.
2. Write the Fourier-transform convention on the same page as the prefactor.
3. Determine $\mathbf q=\mathbf k_f-\mathbf k_i$ and, for elastic scattering, check $q=2k\sin(\theta/2)$.
4. Use symmetry to reduce the transform; retain the sinc form near $q=0$.
5. Check dimensions and at least one analytic limit, such as $q\to0$.
6. Convert to a cross section with channel-speed, spin, and identical-particle factors as needed.
7. State the energy, angular range, observable, and target accuracy of the Born claim.
8. Apply scale, phase-shift, next-order, or independent numerical diagnostics.
9. Exclude or treat separately poles, thresholds, long-range tails, and singular interactions.

Common mistakes

• Using a Fourier transform with an incompatible $2\pi$ prefactor.
• Replacing the reduced mass by one constituent mass without taking the fixed-target limit.
• Using physical momentum transfer in an exponent that expects wave-vector transfer, or vice versa.
• Dropping the sinc limit at $q=0$ and introducing a spurious singularity.
• Calling $V_0/E\ll1$ a universal validity proof.
• Assuming high energy repairs every long-range or resonant problem.
• Applying first Born near a large scattering length or partial-wave unitarity saturation.
• Demanding that the first-order forward amplitude alone satisfy the exact optical theorem.
• Comparing $f^{(2)}/f^{(1)}$ only at a diffraction zero.
• Treating the Born result as an upper or lower bound.
• Applying the free-wave Born series to unscreened Coulomb scattering.
• Equating Born order in a potential equation with loop order in quantum field theory.

Exercises

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

Solution

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

$$\mathbf q\cdot\mathbf r = qr\cos\vartheta$$

and

$$\int d\Omega_{\mathbf r}\, e^{-iqr\cos\vartheta} = 4\pi\frac{\sin(qr)}{qr}.$$

Therefore

$$\widetilde V(q) = 4\pi \int_0^\infty dr\, r^2V(r) \frac{\sin(qr)}{qr}.$$

Multiplication by $-\mu/(2\pi\hbar^2)$ gives

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

2. Verify the Fourier transform and forward amplitude for $V(r)=V_0e^{-r^2/a^2}$.

Solution

The three Cartesian Gaussian integrals factorize:

$$\begin{aligned} \widetilde V(\mathbf q) &= V_0 \prod_{j=x,y,z} \int_{-\infty}^{\infty} dr_j\, e^{-r_j^2/a^2-iq_jr_j} \\ &= \pi^{3/2}a^3V_0 e^{-a^2q^2/4}. \end{aligned}$$

Thus

$$f_{\mathrm B}(q) = - \frac{ \mu V_0\sqrt{\pi}\,a^3 }{ 2\hbar^2 } e^{-a^2q^2/4}.$$

At $q=0$,

$$f_{\mathrm B}(0) = - \frac{ \mu V_0\sqrt{\pi}\,a^3 }{ 2\hbar^2 }.$$

The same result follows from $\int_0^\infty r^2e^{-r^2/a^2}dr=\sqrt{\pi}a^3/4$ in the central formula.

3. Derive the Born scattering length and state its sign for weak repulsive and attractive central potentials of one sign.

Solution

At threshold,

$$f_{\mathrm B}(0) = - \frac{2\mu}{\hbar^2} \int_0^\infty dr\, r^2V(r).$$

Using $f(0)=-a_s$ gives

$$a_{\mathrm B} = \frac{2\mu}{\hbar^2} \int_0^\infty dr\, r^2V(r).$$

For $V(r)>0$, the integral is positive and $a_{\mathrm B}>0$. For $V(r)<0$, it is negative and $a_{\mathrm B}<0$. The latter statement is only perturbative; the exact attractive-potential scattering length changes nonanalytically near a threshold bound state.

4. Explain why $\operatorname{Im}f_{\mathrm B}(0)=0$ for a real central potential does not imply a zero leading cross section or a contradiction with unitarity.

Solution

With $V\to\lambda V$, the first Born amplitude is order $\lambda$, while

$$\sigma^{(2)} = \lambda^2 \int d\Omega\, \lvert f^{(1)}\rvert^2$$

begins at order $\lambda^2$. The optical theorem at that order is

$$\operatorname{Im}f^{(2)}(0) = \frac{k}{4\pi} \int d\Omega\, \lvert f^{(1)}\rvert^2.$$

The required imaginary part comes from the on-shell component of the second Born term $VG_0^{(+)}V$. Setting $\operatorname{Im}f^{(1)}(0)$ equal to the order-$\lambda^2$ cross section would compare different perturbative orders.

Canonical links

• Lippmann–Schwinger Equation
• Born Series
• First Born Approximation
• Validity of the Born Approximation
• Scattering Cross Section Formula Card
• Yukawa Potential in the Born Approximation

References

• M. Born, “Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik 38, 803–827 (1926), doi:10.1007/BF01397184.
• B. A. Lippmann and J. Schwinger, “Variational Principles for Scattering Processes. I,” Physical Review 79, 469–480 (1950), doi:10.1103/PhysRev.79.469.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006, Chs. 4 and 9.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982, Chs. 8–9.
• C. J. Joachain, Quantum Collision Theory, 3rd ed., North-Holland, 1983, Ch. 6.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, Ch. 19.