Schwinger variational 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.
Linear form of the functional
The T-matrix expressed in the form of stationary value of the functional reads
where and are the initial and the final states respectively, is the interaction potential and is the retarded Green's operator for collision energy . The condition for the stationary value of the functional is that the functions and satisfy the Lippmann-Schwinger equation
Fractional form of the functional
Different form of the stationary principle for T-matrix reads
The wave functions and 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 is guessed, with some free parameters, that are determined from the condition of stationarity of the functional.
- R.G. Newton, Scattering Theory of Waves and Particles
- 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- Schwinger, Julian (1947), Harward University lectures (unpublished)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.