Born series explained
The Born series[1] is the expansion of different scattering quantities in quantum scattering theory in the powers of the interaction potential
(more precisely in powers of
where
is the free particle
Green's operator). It is closely related to
Born approximation, which is the first order term of the Born series. The series can formally be understood as
power series introducing the
coupling constant by substitution
. The speed of convergence and
radius of convergence of the Born series are related to
eigenvalues of the operator
. In general the first few terms of the Born series are good approximation to the expanded quantity for "weak" interaction
and large collision energy.
Born series for scattering states
The Born series for the scattering states reads
|\psi\rangle=|\phi\rangle+G0(E)V|\phi\rangle+[G0(E)V]2|\phi\rangle+[G0(E)V]3|\phi\rangle+...
It can be derived by iterating the
Lippmann–Schwinger equation|\psi\rangle=|\phi\rangle+G0(E)V|\psi\rangle.
Note that the
Green's operator
for a free particle can be retarded/advanced or standing wave operator for retarded
advanced
or standing wave scattering states
.The first iteration is obtained by replacing the full scattering solution
with free particle wave function
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
, giving the formal solution to Lippmann-Schwinger equation in the form
|\psi\rangle=[I-G0(E)V]-1|\phi\rangle=[V-VG0(E)V]-1V|\phi\rangle.
Born series for T-matrix
The Born series can also be written for other scattering quantities like the T-matrix which is closely related to the scattering amplitude. Iterating Lippmann-Schwinger equation for the T-matrix we get
T(E)=V+VG0(E)V+V[G0(E)V]2+V[G0(E)V]3+...
For the T-matrix
stands only for retarded
Green's operator
. The standing wave Green's operator would give the K-matrix instead.
Born series for full Green's operator
The Lippmann-Schwinger equation for Green's operator is called the resolvent identity,
Its solution by iteration leads to the Born series for the full Green's operator
G(E)=G0(E)+G0(E)VG0(E)+[G0(E)V]2G0(E)+[G0(E)V]3G0(E)+...
Bibliography
- Book: Joachain, Charles J. . Quantum collision theory . North Holland . 1983 . 978-0-7204-0294-0 . Joachain.
- Book: Taylor, John R. . Scattering Theory: The Quantum Theory on Nonrelativistic Collisions . John Wiley . 1972 . 978-0-471-84900-1.
- Book: Roger G. Newton
. Roger G. Newton . Newton, Roger G. . Scattering Theory of Waves and Particles . Dover Publications, inc. . 2002 . 978-0-486-42535-1.
Notes and References
- Born . Max . Zeitschrift für Physik. 1926. 38. 11–12 . 803–827. 10.1007/bf01397184 . Quantenmechanik der Stoßvorgänge. 1926ZPhy...38..803B . 126244962 .