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Physical Review A

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

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

Published Jan 1, 1988• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.37.639

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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.

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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

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.

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Comment on ' Computational method for the quantum Hamilton-Jacobi equation: Bound states in one dimension' [J. Chem. Phys. 125, 174103 (2006)]

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Some difficulties, both numerical and conceptual, of the method to compute one dimensional wave functions by numerically integrating the quantum Hamilton-Jacobi equation, presented in the paper mentioned in the title, are analyzed. The origin of these difficulties is discussed, and it is shown how they can be avoided by means of another approach, based on different solutions of the same equation. Results for the same potentials, obtained by this latter method are presented and a comparison is made.

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Davydov Ansatz and Proper Solutions of Schrödinger Equation for Fröhlich Hamiltonian

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

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Schrodinger Equation and Hamilton's Equations

Ruggeri, Francesco R.

In (1), it is suggested that one may obtain the Schrodinger equation by using Hamilton’s equations as a starting point, i.e dp/dt = - dH/dq and dq/dt = dH/dp. (1) defines new variables phi(x,t) = ap+ibq and phi*(x,t) and writes: iC dphi/dt partial = {phi, H} and iC dphi*/dt partial ={ phi*, H}, where {} are the Poisson brackets. (1) then changes phi(x,t) into a stochastic variable which still follows these Hamilton-like relations iC dphi/dt = Integral dx H1(x1,x) phi(x) + b(x,t) where H= Integral dx1dx H1 phi*(x1)phi(x) and b(x,t is a random function. In other words, Hamilton’s theory with an added stochastic b(x,t) is applied to a stochastic variable in order to obtain the Schrodinger equation. The catch, however, is that (1) introduces a number of conditions and assumptions to simplify the problem and actually obtains iC dphi/dt = Integral dx H1(x1,x)phi(x) with b(x,t) gone. We suggest that this implies that one may apply the Hamilton-like form directly to a stochastic variable without b(x,t) present. To strengthen this argument, one may consider a bound state in which b(x,t) may be said to average to 0. Basically, the idea of (1) seems to be that quantum mechanics follows from introducing stochastic physics linked to usual Newtonian trajectory physics, in this case Hamilton’s theory. We argue that this is very similar to the classical action approach we used in (2) to obtain exp(-iEt+ipx) from the classical free particle action applied to stochastic motion and so we compare the two approaches.

•Dec 12, 2025•0 citations
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.
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.
Some difficulties, both numerical and conceptual, of the method to compute one dimensional wave functions by numerically integrating the quantum Hamilton-Jacobi equation, presented in the paper mentioned in the title, are analyzed. The origin of these difficulties is discussed, and it is shown how they can be avoided by means of another approach, based on different solutions of the same equation. Results for the same potentials, obtained by this latter method are presented and a comparison is made.
In (1), it is suggested that one may obtain the Schrodinger equation by using Hamilton’s equations as a starting point, i.e dp/dt = - dH/dq and dq/dt = dH/dp. (1) defines new variables phi(x,t) = ap+ibq and phi*(x,t) and writes: iC dphi/dt partial = {phi, H} and iC dphi*/dt partial ={ phi*, H}, where {} are the Poisson brackets. (1) then changes phi(x,t) into a stochastic variable which still follows these Hamilton-like relations iC dphi/dt = Integral dx H1(x1,x) phi(x) + b(x,t) where H= Integral dx1dx H1 phi*(x1)phi(x) and b(x,t is a random function. In other words, Hamilton’s theory with an added stochastic b(x,t) is applied to a stochastic variable in order to obtain the Schrodinger equation. The catch, however, is that (1) introduces a number of conditions and assumptions to simplify the problem and actually obtains iC dphi/dt = Integral dx H1(x1,x)phi(x) with b(x,t) gone. We suggest that this implies that one may apply the Hamilton-like form directly to a stochastic variable without b(x,t) present. To strengthen this argument, one may consider a bound state in which b(x,t) may be said to average to 0. Basically, the idea of (1) seems to be that quantum mechanics follows from introducing stochastic physics linked to usual Newtonian trajectory physics, in this case Hamilton’s theory. We argue that this is very similar to the classical action approach we used in (2) to obtain exp(-iEt+ipx) from the classical free particle action applied to stochastic motion and so we compare the two approaches.