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arXiv:2512.09109

Islands of Instability in Nonlinear Wavefunction Models in the Continuum: A Different Route to "Chaos"

W. David Wick

quant-phSubmitted Dec 9, 2025• Announced Dec 11, 2025• 0 citations
arXivHTMLPDF

Abstract

In two previous papers the author described ``Islands of Instability" that may appear in wavefunction models with nonlinear evolution (of a type proposed originally in the context of the Measurement Problem). Such ``IsoI" represent a new scenario for Hamiltonian systems implying so-called ``chaos". Criteria was derived for, and shown to be fulfilled in, some finite-dimensional (multi-qubit) models, and generalized in the second paper to continuum models. But the only example produced of the latter was a model whose linear Schrodinger equation was exactly-solvable. As exact solutions of many-body problems are rare, here I show that the instability criteria can be verified by plugging test-functions into certain computable expressions, bypassing the solvability blockade. The method can accommodate realistic inter-molecular potentials and so may be relevant to instabilities in fluids and gasses.

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Chaos in a Nonlinear Wavefunction Model: An Alternative to Born's Probability Hypothesis

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In a prior paper, the author described an instability in a nonlinear wavefunction model. Proposed in connection with the Measurement Problem, the model contained an external potential creating a ``classical'' instability. However, it is interesting to ask whether such models possess an intrinsic randomness -- even ``chaos" -- independent of external potentials. In this work, I investigate the criterion analytically and simulate from a small (``3 qubit") model, demonstrating that the Lyapunov exponent -- a standard measure of ``chaos" -- is positive. I also extend the instability criterion to models in the continuum. These results suggest that the boundary between classical and wavefunction physics may also constitute the threshold of chaos, and present an alternative to Max Born's ad hoc probability hypothesis: random outcomes in experiments result not from ``wave-particle duality" or ``the existence of the quantum," but from sensitive dependence on initial conditions, as is common in the other sciences.

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Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.

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Nonlinear Quantum Dynamics

Salman Habib, Bhattacharya, Tanmoy, Andrew C. Doherty +8 more

The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.

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Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.

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Exact solution to a kind of multi-level Rosen-Zener models

Shao-Wu Yao, Hong Cao, Li-Xiang Cen

Exact solution to the driven quantum system with an explicitly time-dependent Hamiltonian is not only an issue of fundamental importance to quantum mechanics itself, but also a ubiquitous problem in the design for quantum control. In particular, the nonadiabatic transition induced by the time-dependent external field is often involved in order to target the quantum state for the atomic and molecular systems. In this paper we investigate the exact dynamics and the associated nonadiabatic transition in a typical driven model, the Rosen-Zener model and its multi-level extension, by virtue of the algebraic dynamical method. Previously, this kind of driven models, especially of the two-level case, were solved by converting the corresponding Schrödinger equation to a hypergeometric equation. The property of the dynamical transition of the system was then achieved by the asymptotic behavior of the yielded hypergeometric function. A critical drawback related to such methods is that they are very hard to be developed so as to treat the multi-level extension of the driven model. Differing from the above mentioned method, we demonstrate that the particular kind of the Rosen-Zener model introduced here could be solved analytically via a canonical transformation or a gauge transformation approach. In comparison, we show that the present method at least has two aspects of advantages. Firstly, the method enables one to describe the evolution of the wavefunction of the system analytically over any time interval of the pulse duration. Moreover, we show that the method could be exploited to deal with the multi-level extensions of the model. The explicit expression of the dynamical basis states, including the three-level system and the four-level system, is presented and the transition probabilities induced by the nonadiabatic evolution among different levels are then characterized for the model during the time evolution. In addition, our study reveals further that the dual model of the driven system can be constructed. Since the dynamical invariant of a solvable system can always be obtained within the framework of the algebraic dynamical method, the general connection between the dual model and the original one, including the solvability and their dynamical invariants, are established and characterized distinctly.

Acta Physica Sinica•Jan 1, 2019•0 citations

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The authors prove, by considering a simple, exactly solvable model, that quantum systems under external perturbations may display a chaotic motion analogous to the corresponding classical ones.

Physical Review Letters•Feb 28, 1983•27 citations
In two previous papers the author described Islands of Instability" that may appear in wavefunction models with nonlinear evolution (of a type proposed originally in the context of the Measurement Problem). Such IsoI" represent a new scenario for Hamiltonian systems implying so-called ``chaos". Criteria was derived for, and shown to be fulfilled in, some finite-dimensional (multi-qubit) models, and generalized in the second paper to continuum models. But the only example produced of the latter was a model whose linear Schrodinger equation was exactly-solvable. As exact solutions of many-body problems are rare, here I show that the instability criteria can be verified by plugging test-functions into certain computable expressions, bypassing the solvability blockade. The method can accommodate realistic inter-molecular potentials and so may be relevant to instabilities in fluids and gasses.
In a prior paper, the author described an instability in a nonlinear wavefunction model. Proposed in connection with the Measurement Problem, the model contained an external potential creating a ``classical'' instability. However, it is interesting to ask whether such models possess an intrinsic randomness -- even ``chaos" -- independent of external potentials. In this work, I investigate the criterion analytically and simulate from a small (``3 qubit") model, demonstrating that the Lyapunov exponent -- a standard measure of ``chaos" -- is positive. I also extend the instability criterion to models in the continuum. These results suggest that the boundary between classical and wavefunction physics may also constitute the threshold of chaos, and present an alternative to Max Born's ad hoc probability hypothesis: random outcomes in experiments result not from ``wave-particle duality" or ``the existence of the quantum," but from sensitive dependence on initial conditions, as is common in the other sciences.
Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.
Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.
Exact solution to the driven quantum system with an explicitly time-dependent Hamiltonian is not only an issue of fundamental importance to quantum mechanics itself, but also a ubiquitous problem in the design for quantum control. In particular, the nonadiabatic transition induced by the time-dependent external field is often involved in order to target the quantum state for the atomic and molecular systems. In this paper we investigate the exact dynamics and the associated nonadiabatic transition in a typical driven model, the Rosen-Zener model and its multi-level extension, by virtue of the algebraic dynamical method. Previously, this kind of driven models, especially of the two-level case, were solved by converting the corresponding Schrödinger equation to a hypergeometric equation. The property of the dynamical transition of the system was then achieved by the asymptotic behavior of the yielded hypergeometric function. A critical drawback related to such methods is that they are very hard to be developed so as to treat the multi-level extension of the driven model. Differing from the above mentioned method, we demonstrate that the particular kind of the Rosen-Zener model introduced here could be solved analytically via a canonical transformation or a gauge transformation approach. In comparison, we show that the present method at least has two aspects of advantages. Firstly, the method enables one to describe the evolution of the wavefunction of the system analytically over any time interval of the pulse duration. Moreover, we show that the method could be exploited to deal with the multi-level extensions of the model. The explicit expression of the dynamical basis states, including the three-level system and the four-level system, is presented and the transition probabilities induced by the nonadiabatic evolution among different levels are then characterized for the model during the time evolution. In addition, our study reveals further that the dual model of the driven system can be constructed. Since the dynamical invariant of a solvable system can always be obtained within the framework of the algebraic dynamical method, the general connection between the dual model and the original one, including the solvability and their dynamical invariants, are established and characterized distinctly.