Equation (938) is not particularly useful, as it stands, because the quantity depends on the unknown ket . Recall that is the solution of the integral equation
where is the wavefunction of the incident state. According to the above equation, the total wavefunction is a superposition of the incident wavefunction 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 wavefunction, .
Suppose that the scattering is not particularly strong. In this case, it is reasonable to suppose that the total wavefunction, , does not differ substantially from the incident wavefunction, . Thus, we can obtain an expression for by making the substitution
This is called the Born approximation.
The Born approximation yields
Thus, is proportional to the Fourier transform of the scattering potential with respect to the wavevector .
For a spherically symmetric potential,
Note that is just a function of for a spherically symmetric potential. It is easily demonstrated that
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, as a consequence of energy conservation.
Consider scattering by a Yukawa potential
where is a constant, and measures the ``range'' of the potential. It follows from Equation (943) that
Thus, in the Born approximation, the differential cross-section for scattering by a Yukawa potential is
The Yukawa potential reduces to the familiar Coulomb potential as , provided that . In this limit, the Born differential cross-section becomes
Recall that is equivalent to , so the above equation can be rewritten
where is the kinetic energy of the incident particles. Equation (950) is identical to 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 Equation (922), that the condition for in the vicinity of is
Consider the special case of the Yukawa potential. At low energies, (i.e., ) we can replace by unity, giving
as the condition for the validity of the Born approximation. The condition for the Yukawa potential to develop a bound state is
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, Equation (951) yields
This inequality becomes progressively easier to satisfy as \(k\) increases, implying that the Born approximation is more accurate at high incident particle energies.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)