The schwinger variational principles for one-dimensional quantum scattering

Alexander L. Zubarev

Radosław Szmytkowski

M. I. Lomonosov, A. E. Oganyan, K.P. Cherkasova

Der-Ruenn Su

Variational principles for quantum theory of scattering are given. Four variational principles give the matrix elements of transition operator and two variational principles give the matrix elements of reaction operator. All the forms of the variational principles are similiar to the Schwinger variational principle for the scattering amplitude in the stationary collision theory.

G. H. Goedecke

Toshiharu Kawai

Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian Lc(\dotx, x) = (M/2) f-1(x)\dotx2 - v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lagrangian operator is essential in our formulation.