Fourth Picture in Quantum Mechanics

Abstract

A quantum‐mechanical counterpart to the classical mechanical variation of constants method is derived, with initial values of coordinates and momenta as "constants." Use is made of a formal operator solution for nonautonomous or autonomous systems in classical mechanics, which we published earlier, and of the correspondence between Poisson brackets and commutators. An alternative unified Lie‐algebraic derivation is also given. It is shown that the Schrödinger, Heisenberg, and interaction pictures in quantum mechanics do not correspond directly to the method of classical mechanical variation of these "constants." A fourth picture, termed "mixed interaction," is introduced and shown to so correspond. It complements the previous three in a symmetrical manner, bearing the same relation to the Heisenberg picture that the Schrödinger picture bears to the interaction one. The group‐theoretic relationship to the interaction picture is noted, as is the relation to the usual variation‐of‐constants method in wave mechanics. For completeness, the classical counterparts of the Heisenberg and interaction pictures are also given. The present results arose from a comparison of quantum and classical treatments of collisions.

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