Lippmann-Schwinger equation

The Lippmann-Schwinger equation (after Bernard Lippmann and Julian Schwinger ) is used in quantum mechanical perturbation theory and especially in scattering theory . It has the form of an integral equation for the wave function sought and is an alternative to the direct solution of the Schrödinger equation , whereby the boundary conditions are in the definition of the Green functions used . ${\displaystyle \psi }$

In quantum mechanical perturbation theory

In general, in perturbation theory, the Hamilton operator is broken down into the "free Hamilton operator" , for which a solution is known, and a part treated as a small perturbation ( potential ) : ${\displaystyle H}$ ${\displaystyle H_{0}}$ ${\displaystyle V}$

${\displaystyle H=H_{0}+V}$

Eigenfunctions of the free Hamilton operator satisfy the equation ${\displaystyle |\phi _{0}\rangle }$

${\displaystyle \left(E-H_{0}\right)|\phi _{0}\rangle =0}$

where is the associated eigenvalue . ${\displaystyle E}$

An operator is called a "free Green function " for which the following applies: ${\displaystyle G_{0}}$

${\displaystyle G_{0}\left(E-H_{0}\right)|\phi _{0}\rangle =|\phi _{0}\rangle }$

This operator is, so to speak, an inverse function of the free Hamilton operator . A mathematically correct representation requires the consideration of as distribution . ${\displaystyle G_{0}}$

The unknown eigenfunctions of the complete Hamilton operator and its Green function are now defined in an analogous manner . ${\displaystyle |\psi \rangle }$ ${\displaystyle G}$

The Lippmann-Schwinger equation thus applies:

${\displaystyle |\psi \rangle =|\phi _{0}\rangle +G_{0}V|\psi \rangle }$

This equation is usually solved iteratively , with the restriction to the first non- trivial order being called Born's approximation .

In scattering theory

Accordingly, the Lippmann-Schwinger equation is mainly used in scattering theory. Here it is calculated how the wave function of a particle changes when scattering at a potential V, using the kinetic component for a free particle as the free Hamilton operator :

${\displaystyle {\hat {H}}_{0}={\frac {{\hat {\mathbf {p} }}^{2}}{2m}}}$

with the momentum operator . ${\displaystyle {\hat {\mathbf {p} }}}$

To derive the Lippmann-Schwinger equation for a stationary scattering problem, one starts from the Schrödinger equation:

${\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\Delta +V({\vec {r}})\right]\psi _{k}=E_{k}\psi _{k}}$

With

• the kinetic energy of a free particle ${\displaystyle E_{k}={\frac {\hbar ^{2}{\vec {k}}^{2}}{2m}}}$
• its direction of entry ${\displaystyle {\frac {\vec {k}}{k}}}$
• its direction of spread ${\displaystyle {\frac {\vec {k^{\prime }}}{k}}={\frac {\vec {r}}{r}}={\vec {n}}}$

it should be noted that it is an elastic scattering, i. H. the magnitude of the momentum vector is not changed: and for all vectors . ${\displaystyle k^{\prime }=k}$ ${\displaystyle |{\vec {v}}|=v}$

Converted and with the requirement results: ${\displaystyle E\geq 0}$

${\displaystyle \left[\Delta +k^{2}\right]\psi _{k}={\frac {2m}{\hbar ^{2}}}V({\vec {r}})\psi _{k}=:U({\vec {r}})\psi _{k}}$

This can be solved with the method of Green's functions:

${\displaystyle {\begin{aligned}\left[\Delta +k^{2}\right]\phi _{0}=0\quad &\Rightarrow \quad \phi _{0}={\frac {1}{(2\pi )^{3/2}}}e^{i{\vec {k}}{\vec {r}}}\\\left[\Delta +k^{2}\right]G({\vec {r}})=\delta {({\vec {r}})}\quad &\Rightarrow \quad G({\vec {r}})=-{\frac {1}{4\pi }}{\frac {e^{ikr}}{r}}\\\psi _{k}({\vec {r}})=\phi _{0}&+\int d^{3}{\vec {r}}^{\prime }G({\vec {r}}-{\vec {r}}^{\prime })\cdot U({\vec {r}}^{\prime })\psi _{k}({\vec {r}}^{\prime })\\\end{aligned}}}$

This results in the Lippmann-Schwinger equation of the scattering theory:

${\displaystyle \psi _{k}({\vec {r}})={\frac {1}{(2\pi )^{3/2}}}e^{i{\vec {k}}\cdot {\vec {r}}}-{\frac {2m}{4\pi \hbar ^{2}}}\int d^{3}{\vec {r}}^{\prime }\cdot {\frac {e^{ik|{\vec {r}}-{\vec {r}}^{\prime }|}}{|{\vec {r}}-{\vec {r}}^{\prime }|}}V({\vec {r}}^{\prime })\psi _{k}({\vec {r}}^{\prime })}$

The location representation was explicitly chosen here.

This equation can be solved iteratively by replacing on the right-hand side with the solution obtained up to that point and choosing as the starting value of the iteration: ${\displaystyle \psi _{k}({\vec {r}})}$

${\displaystyle \psi _{k}^{(0)}({\vec {r}})=\phi _{0}({\vec {r}})}$

The first iteration

${\displaystyle \psi _{k}^{(1)}({\vec {r}})={\frac {1}{(2\pi )^{3/2}}}e^{i{\vec {k}}{\vec {r}}}-{\frac {2m}{4\pi \hbar ^{2}}}\int d^{3}{\vec {r}}^{\prime }\cdot {\frac {e^{ik|{\vec {r}}-{\vec {r}}^{\prime }|}}{|{\vec {r}}-{\vec {r}}^{\prime }|}}V({\vec {r}}^{\prime }){\frac {1}{(2\pi )^{3/2}}}e^{i{\vec {k}}{\vec {r}}^{\prime }}}$

is then the already mentioned Born approximation in position representation.

Individual evidence

1. ^ Bernard Lippmann and Julian Schwinger : Variational principles for scattering processes. I . In: Physical Review . tape 79 , no. 3 , 1950, p. 469-480 , doi : 10.1103 / PhysRev.79.469 . Equation 1.84 on p. 475.