Nested interaction representations in time dependent quantum mechanics

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Abstract
Two significant developments in the formulation of the equations of motion in the interaction representation (IR) are introduced in this paper. These developments have direct bearing on the efficient propagation in the interaction representation. The first of these developments involves the concept of sequential, or ‘‘nested’’ interaction transformations on the Hamiltonian and the wave function. Two such nested IRs are described. The first is a momentum- or ‘‘P-adapted’’ IR which, in addition to removing wave packet spreading, systematically maintains the average momentum 〈P〉 at zero. This facilitates a grid which not only has a minimal range, but whose points are maximally sparse. Evaluating the Hamiltonian operator in the P-adapted interaction representation involves only one simple algorithmic modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈P〉t/m) eiH0t/ℏ, i.e., using the potential energy function which is downstream from the wave function by the classical propagation distance. A PR-adapted IR is also defined, which systematically maintains both the average position 〈R〉 and momentum 〈P〉 at zero. The PR-adapted IR also involves only one modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈R〉 +〈P〉t/m)eiH0t/ℏ. The second formal development consists of the application of the operator equation eiH0t/ℏV(R)e−iH0t /ℏ=V(R+Pt/m) in conjunction with the nested IRs. This formal identity, which is essentially the Heisenberg representation for V(R), provides a convenient route to propagation in the interaction representation without reconstructing the Schrödinger wave function as an intermediate. The time propagation in all representations is performed using an iterative Lanczos reduction scheme combined with a second- order Magnus expansion. Numerical results for the exponential and the 1/R potential are presented, and illustrate that the new representations can easily lead to a savings of an order of magnitude in the size of the grid required for the propagation.