Schwinger variational principle

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

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

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.

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.

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

• Newton, Roger G. (2002). Scattering Theory of Waves and Particles. Dover Publications, inc. ISBN 978-0-486-42535-1.
• Taylor, John R. (1972). Scattering Theory: The Quantum Theory on Nonrelativistic Collisions. John Wiley. ISBN 978-0-471-84900-1.
• Schwinger, Julian (1947), Harward University lectures (unpublished)
• Schwinger, J. (1947). "Minutes of the Meeting at Stanford University, California July 11-12, 1947". Phys. Rev. 72 (8): 742. Bibcode:1947PhRv...72..738.. doi:10.1103/PhysRev.72.738.
• Lippmann, B. A.; Schwinger, J. (1950). "Variational Principles for Scattering Processes. I". Phys. Rev. 79 (3): 469–480. Bibcode:1950PhRv...79..469L. doi:10.1103/physrev.79.469.