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Journal of Mathematical Physics

Exact propagators for quadratic Hamiltonians

Yoel Tikochinsky

Published Apr 1, 1978• 34 citations
Publisher

Abstract

The problem of finding the exact propagator for a given quadratic Hamiltonian is presented as an initial-value problem — a form suitable for both analytic and numerical computation. We illustrate the method by finding the propagator for (a) the Hamiltonian with constant coefficients and (b) and (c) Hamiltonians associated with damped harmonic motion.

DOI
10.1063/1.523718

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Physical Review A•Dec 15, 2006•4 citations
The problem of finding the exact propagator for a given quadratic Hamiltonian is presented as an initial-value problem — a form suitable for both analytic and numerical computation. We illustrate the method by finding the propagator for (a) the Hamiltonian with constant coefficients and (b) and (c) Hamiltonians associated with damped harmonic motion.
Abstract In this paper, we find the quantum propagator for a general time-dependent quadratic Hamiltonian. The method is based on the properties of the propagator and the fact that the quantum propagator fulfills two independent partial differential equations originating from Heisenberg equations for positions and momenta. As an application of the method, we find the quantum propagator for a linear chain of interacting oscillators for both periodic and Dirichlet boundary conditions. The state and excitation propagation along the harmonic chain in the absence and presence of an external classical source is studied and discussed. The location of the first maxima of the probability amplitude P(n, τ) is a straight line in the (n, τ)-plane, indicating a constant speed of excitation propagation along the chain.