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## The Born approximation

Equation (923) is not particularly useful, as it stands, because the quantity depends on the unknown ket . Recall that is the solution of the integral equation
 (924)

where is the wave-function of the incident state. According to the above equation the total wave-function is a superposition of the incident wave-function and lots of spherical-waves emitted from the scattering region. The strength of the spherical-wave emitted at a given point is proportional to the local value of the scattering potential, , as well as the local value of the wave-function, .

Suppose that the scattering is not particularly strong. In this case, it is reasonable to suppose that the total wave-function, , does not differ substantially from the incident wave-function, . Thus, we can obtain an expression for by making the substitution

 (925)

This is called the Born approximation.

The Born approximation yields

 (926)

Thus, is proportional to the Fourier transform of the scattering potential with respect to the wave-vector .

For a spherically symmetric potential,

 (927)

giving
 (928)

Note that is just a function of for a spherically symmetric potential. It is easily demonstrated that
 (929)

where is the angle subtended between the vectors and . In other words, is the angle of scattering. Recall that the vectors and have the same length by energy conservation.

Consider scattering by a Yukawa potential

 (930)

where is a constant and measures the range'' of the potential. It follows from Eq. (928) that
 (931)

since
 (932)

Thus, in the Born approximation, the differential cross-section for scattering by a Yukawa potential is
 (933)

given that
 (934)

The Yukawa potential reduces to the familiar Coulomb potential as , provided that . In this limit the Born differential cross-section becomes

 (935)

Recall that is equivalent to , so the above equation can be rewritten
 (936)

where is the kinetic energy of the incident particles. Equation (936) is the classical Rutherford scattering cross-section formula.

The Born approximation is valid provided that is not too different from in the scattering region. It follows, from Eq. (907), that the condition for in the vicinity of is

 (937)

Consider the special case of the Yukawa potential. At low energies, (i.e., ) we can replace by unity, giving
 (938)

as the condition for the validity of the Born approximation. The condition for the Yukawa potential to develop a bound state is
 (939)

where is negative. Thus, if the potential is strong enough to form a bound state then the Born approximation is likely to break down. In the high- limit, Eq. (937) yields
 (940)

This inequality becomes progressively easier to satisfy as increases, implying that the Born approximation is more accurate at high incident particle energies.

Next: Partial waves Up: Scattering theory Previous: The Lipmann-Schwinger equation
Richard Fitzpatrick 2006-02-16