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American Journal of Physics

Heisenberg picture in quantum mechanics and linear evolutionary systems

Pasquale Nardone

Published Mar 1, 1993• 19 citations
Publisher

Abstract

A straightforward procedure is presented to derive the propagator in simple quantum systems. This approach enables us to define a state which behaves as a generalization of the well-known coherent state of the harmonic oscillator.
DOI
10.1119/1.17296

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Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
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We study how catlike superpositions of generalized Heisenberg algebras (GHA) nonlinear coherent states behave under dissipative decoherence. Two cases are presented: the infinite square well potential and systems whose spectra, given by infinite strictly increasing sequences of nonnegative real numbers, can be considered perturbations of the harmonic oscillator. The decoherence effect caused by the interaction of these systems with a thermal bath is analyzed: from their fidelity behavior we see that a region always exists in the parameter space where the quantum coherence is better preserved as compared to the harmonic oscillator. Moreover, we show that the qualitative behavior of GHA systems under the studied mechanism of decoherence can be inferred from the algebraic structure via the analysis of their GHA characteristic functions.