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Molecular Physics

Adiabatic and diabatic representations, coordinate and unitary transformations: coupled oscillators

Jeremy Graham Frey

Published Oct 20, 1990• 0 citations
Publisher

Abstract

The relationships between the adiabatic approximation and a transformation of the coordinates or the Hamiltonian are investigated for a two-dimensional coupled oscillator. The Smith transformation is used systematically to obtain a transformation returning the adiabatic form of the Hamiltonian to its original ‘diabatic’ form and thus provides a method of obtaining the unitary transformation that is equivalent to the coordinate transformation.

DOI
10.1080/00268979000102041

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The relationships between the adiabatic approximation and a transformation of the coordinates or the Hamiltonian are investigated for a two-dimensional coupled oscillator. The Smith transformation is used systematically to obtain a transformation returning the adiabatic form of the Hamiltonian to its original ‘diabatic’ form and thus provides a method of obtaining the unitary transformation that is equivalent to the coordinate transformation.
In view of a recent paper [F. Michelot, Phys. Rev. A 45, 4271 (1992)] tackling the Hamiltonian of an arbitrary number of harmonically coupled oscillators, we present a coordinate representation of the unitary operator that can diagonalize the Hamiltonian. The normally ordered form of the unitary operator, which manifestly connects two Fock spaces associated with the uncoupled and coupled oscillators, is also derived by virtue of the technique of integration within an ordered product of operators.
An algebraic (matrix) method is presented for canonically transforming the Hamiltonians quadratic in coordinate and momentum operators into the Hamiltonian of non-interacting harmonic oscillators. The method is illustrated by transforming the Hamiltonian of a harmonic oscillator in a constant magnetic field into the Hamiltonian of the two-dimensional, in general anisotropic, harmonic oscillator. The results are found to be in agreement with those obtained previously for the same Hamiltonian using a different technique.