Born Approximation

The Born approximation is the leading-order treatment of quantum scattering for a weak potential: it gives the scattering amplitude as the Fourier transform of the potential with respect to the momentum transferred in the collision.

Definition

The Born approximation is the first term of the perturbative Born series for the scattering amplitude, equal to the Fourier transform of the scattering potential evaluated at the momentum transfer, valid when the potential perturbs the incident wave only weakly.

Scope

The topic covers the integral form of the Schrodinger equation using the free Green's function, the Born series generated by iterating it, the first Born approximation in which the amplitude is the Fourier transform of the potential over the momentum transfer, the conditions of weak potential or high energy under which it is valid, and standard examples such as Yukawa and Coulomb scattering.

Core questions

• How does the integral form of the Schrodinger equation lead to the Born series?
• Why is the first Born amplitude the Fourier transform of the potential?
• Under what conditions is the Born approximation accurate?
• How does it reproduce known results such as Rutherford scattering?

Key concepts

• integral Schrodinger equation
• free Green's function
• Born series
• momentum transfer
• Fourier transform of the potential
• high-energy validity

Key theories

Born series

Writing the Schrodinger equation in integral form with the free-particle Green's function and iterating generates a series in powers of the potential, each term adding one more scattering event; truncating at first order gives the Born approximation.

First Born amplitude

To leading order the scattering amplitude is proportional to the Fourier transform of the potential evaluated at the momentum transferred in the collision, so the angular distribution directly reflects the spatial structure of the potential, the principle behind diffraction-based structure determination.

Clinical relevance

The Born approximation underlies the interpretation of diffraction and scattering experiments: it relates measured angular distributions in electron, neutron, and X-ray scattering to the Fourier transform of the scattering density, providing the basis for determining the structure of atoms, molecules, and condensed matter.

History

Born introduced the approximation in his 1926 paper founding the probabilistic interpretation of quantum mechanics; it was quickly applied to atomic collisions by Bethe and others and became the standard first estimate for scattering cross sections.

Key figures

• Max Born
• Hans Bethe
• Ernest Rutherford

Seminal works

• taylor2006
• sakurai2017

Frequently asked questions

When does the Born approximation break down?

It fails when the potential is strong enough to substantially distort the incident wave, such as at low energies, near resonances, or for long-range potentials; higher Born terms or non-perturbative methods like partial-wave analysis are then required.

Why is the Born amplitude a Fourier transform of the potential?

To leading order the incident plane wave scatters once off the potential and emerges as an outgoing wave, and summing the phase contributions from all points of the potential is mathematically a Fourier transform over the momentum transferred in the collision.