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Applicability of Hamilton’s equations in the quantum soliton problem

David W. Brown, Katja Lindenberg, Bruce J. West

We test the validity of Hamilton-equation methods for determining the time evolution of trial state vectors in quantum mechanics. Given a trial state vector, we are able to construct a differential operator under which a scalar Hamilton function must be invariant. State vectors composed of single-particle states, coherent-state products, and mixed single-particle states and coherent-state products are considered explicitly. In the latter category, we consider state vectors of the form proposed by Davydov in his treatment of the quantum soliton problem. We find that Davydov's wave vector, as determined by the Hamilton-equation method, is not a solution of the Schr\"odinger equation for the Fr\"ohlich Hamiltonian except under very restrictive circumstances. The theoretical justification for a number of conclusions about soliton transport in Fr\"ohlich-type systems is thus called into question.We test the validity of Hamilton-equation methods for determining the time evolution of trial state vectors in quantum mechanics. Given a trial state vector, we are able to construct a differential operator under which a scalar Hamilton function must be invariant. State vectors composed of single-particle states, coherent-state products, and mixed single-particle states and coherent-state products are considered explicitly. In the latter category, we consider state vectors of the form proposed by Davydov in his treatment of the quantum soliton problem. We find that Davydov's wave vector, as determined by the Hamilton-equation method, is not a solution of the Schr\"odinger equation for the Fr\"ohlich Hamiltonian except under very restrictive circumstances. The theoretical justification for a number of conclusions about soliton transport in Fr\"ohlich-type systems is thus called into question.

Physical Review AJun 1, 1986121 citations

On Hamiltonian Formulations of the Schrödinger System

László Árpád Gergely

arXiv:hep-th/0301052Annals of PhysicsJun 1, 200214 citations

Comment on ‘‘Applicability of Hamilton’s equations in the quantum soliton problem’’

M. J. Škrinjaŕ, D. Kapor, S. D. Stojanović

We have shown by consistent application of the Schr\"odinger and Heisenberg equations that the differential invariance condition imposed by Brown, Lindenberg, and West [Phys. Rev. A 33, 4104 (1986)] on Hamilton's function, which was obtained by averaging Fr\"ohlich's Hamiltonian over Davydov soliton states, is not a correct one. The conclusion that Davydov made an erroneous application of Hamilton's equations is still valid, based on completely different arguments.

Physical Review AJan 1, 19888 citations

Hamiltonian Dynamics and the Schrodinger Equation

Leon Bess

Progress of Theoretical PhysicsJul 1, 19745 citations

Variational study of polaron dynamics with the Davydov Ansätze

Bin Luo, Jun Ye, Yang Zhao

Abstract We derive explicit expressions for the amplitudes of the deviation vectors for the time‐dependent Davydov D1, D2 and $ {\tilde {\rm D}} $ Ansätze from the exact solution of the Schrödinger equation for the Holstein polaron. By comparing the deviation amplitudes with the system energies, a systematic study is carried out on the validity of the Davydov Ansätze. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Physica status solidi. C, Conferences and critical reviews/Physica status solidi. C, Current topics in solid state physicsDec 2, 201022 citations

Mathematical Justification of the Aharonov-Bohm Hamiltonian

César R. de Oliveira, Marciano Pereira

arXiv:0810.0684Journal of Statistical PhysicsNov 10, 200831 citations

Davydov Ansatz and Proper Solutions of Schrödinger Equation for Fröhlich Hamiltonian

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