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Physical Review A

Applicability of Hamilton’s equations in the quantum soliton problem

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

Published Jun 1, 1986• 121 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.33.4104

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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.

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To Applicability of Hamilton's Equations in Quantum Many‐Body Problems

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Abstract The connection between Schrödinger equation and Hamilton equations for interacting many‐particle systems is given and it is shown in which way the Hamilton function has to be constructed. Starting from an identity for the Hamilton function a criterion is derived which permits to check approximations. Applying this criterion to former calculations to the soliton problem in a modified Hubbard chain the consistency of the used approximation is shown.

physica status solidi (b)•May 1, 1987•1 citation

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

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

A response is made to the three assertions of Škrinjar, Kapov,and Stojanović [Phys. Rev. A 37, 639 (1988)].Received 24 February 1987DOI:https://doi.org/10.1103/PhysRevA.37.642©1988 American Physical Society

Physical Review A•Jan 1, 1988•9 citations

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Journal of the Optical Society of America B•Jun 1, 1989•22 citations
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.
Various equivalent representations of dynamical invariants (or constants of motion) are derived for N-level systems in mixed states with particular emphasis on so-called coherence vector invariants. They appear as certain homogeneous forms in the real solutions of the von Neumann equation with coefficients given by multilinear forms in the completely symmetric structure constants of the Lie-algebra of SU(N). The treatment is motivated by the close analogy between Lax pairs For classical dynamical systems and pairs given by density and Hamilton operators. In both cases the underlying mathematical structure is essentially determined by the properties of symmetric functions. However, in the quantum case more theorems of general validity can be derived due to the peculiar properties of density operators. In particular, the maximum number of functionally independent invariants in relation to the spectral properties is obtained, as well as bounds on their order of magnitude, the latter from an extremal property analysis. From group-theoretical methods a complete classification of all possibilities of completely incoherent state dynamics is deduced. As a by-product a simple algorithm for the explicit determination of all (N-1)-dimensional irreducible representation matrices of the symmetric group S N and an associated construction of hyperpolyhedra is worked out. Finally, the importance of invariants is stressed for the control of numerical accuracy in large-scale computations in very high dimensions which, after taking partial traces, can be used For the description of irreversible processes. In summary, the results will be of practical relevance for applications to problems of short-time dynamics in molecular laser spectroscopy, quantum optics and magnetic resonance.
The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.
The quantum nonlinear Schrödinger equation, which models the propagation of light through dispersive nonlinear waveguides, provides an integrable quantum field theory that has been solved by a number of methods. Most recently, Gutkin [ J. Func. Anal.77, 327 ( 1988)] has developed an interwining operator technique for obtaining the time evolution of the field operators. Using Gutkin’s formalism, we show how to obtain the exact time evolution of an initial Schrödinger state vector. As an example, the two-particle case is carried out explicitly. Numerical results for the time evolution of the position probability distribution are presented for the case when the two particles initially occupy the same Gaussian wave packet.