Schwinger variational principle

Not to be confused with Schwinger's quantum action principle, Schwinger's quantum action principle.

Schwinger variational principle is a variational principle which expresses the scattering T-matrix as a functional depending on two unknown wave functions. The functional attains stationary value equal to actual scattering T-matrix. The functional is stationary if and only if the two functions satisfy the Lippmann-Schwinger equation. The development of the variational formulation of the scattering theory can be traced to works of L. Hultén and J. Schwinger in 1940s.^[1]

Linear form of the functional

The T-matrix expressed in the form of stationary value of the functional reads

${\displaystyle \langle \phi '|T(E)|\phi \rangle =T[\psi ',\psi ]\equiv \langle \psi '|V|\phi \rangle +\langle \phi '|V|\psi \rangle -\langle \psi '|V-VG_{0}^{(+)}(E)V|\psi \rangle ,}$

where ${\displaystyle \phi }$ and ${\displaystyle \phi '}$ are the initial and the final states respectively, ${\displaystyle V}$ is the interaction potential and ${\displaystyle G_{0}^{(+)}(E)}$ is the retarded Green's operator for collision energy ${\displaystyle E}$. The condition for the stationary value of the functional is that the functions ${\displaystyle \psi }$ and ${\displaystyle \psi '}$ satisfy the Lippmann-Schwinger equation

${\displaystyle |\psi \rangle =|\phi \rangle +G_{0}^{(+)}(E)V|\psi \rangle }$

and

${\displaystyle |\psi '\rangle =|\phi '\rangle +G_{0}^{(-)}(E)V|\psi '\rangle .}$

Fractional form of the functional

Different form of the stationary principle for T-matrix reads

${\displaystyle \langle \phi '|T(E)|\phi \rangle =T[\psi ',\psi ]\equiv {\frac {\langle \psi '|V|\phi \rangle \langle \phi '|V|\psi \rangle }{\langle \psi '|V-VG_{0}^{(+)}(E)V|\psi \rangle }}.}$

The wave functions ${\displaystyle \psi }$ and ${\displaystyle \psi '}$ must satisfy the same Lippmann-Schwinger equations to get the stationary value.

Application of the principle

The principle may be used for the calculation of the scattering amplitude in the similar way like the variational principle for bound states, i.e. the form of the wave functions ${\displaystyle \psi ,\psi '}$ is guessed, with some free parameters, that are determined from the condition of stationarity of the functional.

See also

• Lippmann-Schwinger equation
• Quantum scattering theory
• T-matrix method
• Green's operator