The Heisenberg and Interaction Pictures

Krishna Thyagarajan

Brian P. Anderson

Alireza R. Baghai-Wadji

This chapter outlines the governing equations in non-relativistic quantum physics. The Schrödinger equation (the Schrödinger picture) looms large. Its intuitive simplicity and elegance render it particularly appealing to aspiring engineers and materials scientists. The other two equivalent pictures presented are the Heisenberg picture and the interaction picture. The Heisenberg picture emphasizes time-dependent operators, and the interaction picture splits the so-called interaction Hamiltonian from the total Hamiltonian of the system considered. Much attention has been paid in this chapter to establish relationships between the pictures, as meaningful as possible at this level of treatment.

R. A. Marcus

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.

Giancarlo Calvanese Strinati

In the Schrödinger and Heisenberg representations, the time-evolution operator depends on the full time-dependent Hamiltonian, which includes an external time-dependent potential. Out of this full-time dependence, it is useful to isolate the time dependences due to either the full system Hamiltonian or its noninteracting part. These two cases, referred to as the Heisenberg and interaction pictures, respectively, are considered separately.