Born series

Born series for scattering states

The Born series for the scattering states reads

${\displaystyle |\psi ⟩=|\varphi ⟩+G_0(E)V|\varphi ⟩+[G_0(E)V{]}^2|\varphi ⟩+[G_0(E)V{]}^3|\varphi ⟩+\dots }$

It can be derived by iterating the Lippmann–Schwinger equation

${\displaystyle |\psi ⟩=|\varphi ⟩+G_0(E)V|\psi ⟩.}$

Note that the Green's operator ${\displaystyle G_0}$ for a free particle can be retarded/advanced or standing wave operator for retarded ${\displaystyle |{\psi }^{(+)}⟩}$ advanced ${\displaystyle |{\psi }^{(-)}⟩}$ or standing wave scattering states ${\displaystyle |{\psi }^{(P)}⟩}$. The first iteration is obtained by replacing the full scattering solution ${\displaystyle |\psi ⟩}$ with free particle wave function ${\displaystyle |\varphi ⟩}$ on the right hand side of the Lippmann-Schwinger equation and it gives the first Born approximation. The second iteration substitutes the first Born approximation in the right hand side and the result is called the second Born approximation. In general the n-th Born approximation takes n-terms of the series into account. The second Born approximation is sometimes used, when the first Born approximation vanishes, but the higher terms are rarely used. The Born series can formally be summed as geometric series with the common ratio equal to the operator ${\displaystyle G_{0}V}$, giving the formal solution to Lippmann-Schwinger equation in the form

${\displaystyle |\psi ⟩=[I-G_0(E)V{]}^{-1}|\varphi ⟩=[V-V{G}_0(E)V{]}^{-1}V|\varphi ⟩.}$