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Physica Scripta

The role of avoided crossings in Bohmian dynamics

Athanasios C. Tzemos, G. Contopoulos

Published May 21, 2026• 0 citations
Publisher

Abstract

Abstract We study the Bohmian trajectories of a perturbed two-dimensional quantum harmonic oscillator and examine how avoided crossings in the quantum energy spectrum influence their dynamical behaviour both in non-resonant and resonant cases. Using superpositions of a finite number of eigenstates of the perturbed Hamiltonian, we show that avoided crossings strongly enhance the prevalence of chaotic trajectories within the Born-rule distribution. The use of eigenstates of the full Hamiltonian allows ordered Bohmian trajectories to persist, in contrast to constructions based on unperturbed states. We further compare the Bohmian behaviour with the corresponding classical dynamics at comparable energies, identifying parameter regimes in which the two descriptions exhibit markedly different sensitivity to perturbations.

DOI
10.1088/1402-4896/ae6a82

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Bohmian mechanics is a causal interpretation of quantum mechanics in which there are well-defined particle trajectories guided by the wave function. Within this framework, it is shown that certain classical integrable systems can exhibit both periodic and chaotic behaviours. In this paper, we extend the work of Borondo et al. by investigating the behaviours of commensurate two-dimensional harmonic oscillator systems with different ratios of frequencies. We show that the system may produce chaotic behavior of Bohmian trajectories to be dependenton frequencies ratio. This result is illustrated numerically in computer experiments displaying the Bohmian trajectories.
We study in detail the trajectories, ordered and chaotic, of two entangled Bohmian qubits when their initial preparation satisfies (or not) Born's rule for various amounts of quantum entanglement. For any non zero value of entanglement ordered and chaotic trajectories coexist and the proportion of ordered trajectories increases with the decrease of the entanglement. In the extreme cases of zero and maximum entanglement we have only ordered and chaotic trajectories correspondingly. The chaotic trajectories of this model are ergodic, for any given value of entanglement, namely the limiting distribution of their points does not depend on their initial conditions. Consequently it is the ratio between ordered and chaotic trajectories which is responsible for the dynamical establishment (or not) of Born's rule.