Born Series
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The Born series is the expansion of different scattering quantities in quantum scattering theory in the powers of the interaction potential V (more precisely in powers of G_0 V, where G_0 is the free particle Green's operator). It is closely related to
Born approximation Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named a ...
, which is the first order term of the Born series. The series can formally be understood as
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
introducing the
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
by substitution V \to \lambda V . The speed of convergence and
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
of the Born series are related to
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of the operator G_0 V . In general the first few terms of the Born series are good approximation to the expanded quantity for "weak" interaction V and large collision energy.


Born series for scattering states

The Born series for the scattering states reads : , \psi\rangle = , \phi \rangle + G_0(E) V , \phi\rangle + _0(E) V2 , \phi\rangle + _0(E) V3 , \phi\rangle + \dots It can be derived by iterating the
Lippmann–Schwinger equation The Lippmann–Schwinger equation (named after Bernard Lippmann and Julian Schwinger) is one of the most used equations to describe particle collisions – or, more precisely, scattering – in quantum mechanics. It may be used in scatt ...
: , \psi\rangle = , \phi \rangle + G_0(E) V , \psi\rangle. Note that the Green's operator G_0 for a free particle can be retarded/advanced or standing wave operator for retarded , \psi^\rangle advanced , \psi^\rangle or standing wave scattering states , \psi^\rangle . The first iteration is obtained by replacing the full scattering solution , \psi\rangle with free particle wave function , \phi\rangle on the right hand side of the Lippmann-Schwinger equation and it gives the first
Born approximation Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named a ...
. 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 In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each succ ...
with the common ratio equal to the operator G_0 V , giving the formal solution to Lippmann-Schwinger equation in the form : , \psi\rangle = - G_0(E) V , \phi \rangle = - VG_0(E) V V , \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 In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.Lippmann-Schwinger equation for the T-matrix we get : T(E) = V + V G_0(E) V + V _0(E) V2 + V _0(E) V3 + \dots For the T-matrix G_0 stands only for retarded Green's operator G_0^(E) . 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, : G(E) = G_0(E) + G_0(E) V G(E). Its solution by iteration leads to the Born series for the full Green's operator G(E)=(E-H+i\epsilon)^ : G(E) = G_0(E) + G_0(E) V G_0(E) + _0(E) V2 G_0(E) + _0(E) V3 G_0(E) + \dots


Bibliography

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References

{{Reflist Scattering Max Born