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McGuire, James; Dirk, Charlotte

doi:10.1023/a:1004815406443

“…[21,41] for a more detailed discussion. Note that certain extensions to the Bethe ansatz suggest that additional solvable cases exist [42,43]. However, our numerical analysis does not find any traces of solvability beyond the two limiting cases, and supports the widely accepted idea that almost any one-dimensional problem with mass imbalance is non-integrable (notable exceptions include Refs.…”

Section: Formulationsupporting

confidence: 82%

Morphology of three-body quantum states from machine learning

Huber,

Marchukov,

Hammer

et al. 2021

Preprint

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The relative motion of three impenetrable particles on a ring, in our case two identical fermions and one impurity, is isomorphic to a triangular quantum billiard. Depending on the ratio κ of the impurity and fermion masses, the billiards can be integrable or non-integrable (also referred to in the main text as chaotic). To set the stage, we first investigate the energy level distributions of the billiards as a function of 1/κ ∈ [0, 1] and find no evidence of integrable cases beyond the limiting values 1/κ = 1 and 1/κ = 0. Then, we use machine learning tools to analyze properties of probability distributions of individual quantum states. We find that convolutional neural networks can correctly classify integrable and non-integrable states. The decisive features of the wave functions are the normalization and a large number of zero elements, corresponding to the existence of a nodal line. The network achieves typical accuracies of 97%, suggesting that machine learning tools can be used to analyze and classify the morphology of probability densities obtained in theory and experiment.

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“…Solving the (hyper)spherical Helmholtz equation on an angular sector Ω p with Dirichlet boundary conditions is an example of quantum billiards. The problem of quantum and classical billiards in planar triangles is wellstudied [53][54][55][56][57][58][59][60][61], and the integrability and solvability of the dynamics depends critically on the domain shape of the billiards. For example, the only three triangular billiards in a plane that have classically-integrable dynamics are the three triangles with distinguishable sides that tile the plane under reflections, without gaps or overlaps (see footnote 3 of [33]).…”

Section: Model and Symmetrymentioning

confidence: 99%

Integrable Families of Hard-Core Particles with Unequal Masses in a One-Dimensional Harmonic Trap

Harshman

¹

,

Olshanii

²

,

Dehkharghani

³

et al. 2017

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We show that the dynamics of particles in a one-dimensional harmonic trap with hard-core interactions can be solvable for certain arrangements of unequal masses. For any number of particles, there exist two families of unequal mass particles that have integrable dynamics, and there are additional exceptional cases for three, four and five particles. The integrable mass families are classified by Coxeter reflection groups and the corresponding solutions are Bethe ansatz-like superpositions of hyperspherical harmonics in the relative hyperangular coordinates that are then restricted to sectors of fixed particle order. We also provide evidence for superintegrability of these Coxeter mass families and conjecture maximal superintegrability.

show abstract

Morphology of three-body quantum states from machine learning

Huber

¹

,

Marchukov

²

,

Hammer

³

et al. 2021

View full text Add to dashboard Cite

The relative motion of three impenetrable particles on a ring, in our case two identical fermions and one impurity, is isomorphic to a triangular quantum billiard. Depending on the ratio κ of the impurity and fermion masses, the billiards can be integrable or non-integrable (also referred to in the main text as chaotic). To set the stage, we first investigate the energy level distributions of the billiards as a function of 1/κ ∈ [0, 1] and find no evidence of integrable cases beyond the limiting values 1/κ = 1 and 1/κ = 0. Then, we use machine learning tools to analyze properties of probability distributions of individual quantum states. We find that convolutional neural networks can correctly classify integrable and non-integrable states. The decisive features of the wave functions are the normalization and a large number of zero elements, corresponding to the existence of a nodal line. The network achieves typical accuracies of 97%, suggesting that machine learning tools can be used to analyze and classify the morphology of probability densities obtained in theory or experiment.

show abstract

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Cited by 4 publications

References 2 publications

Morphology of three-body quantum states from machine learning

Morphology of three-body quantum states from machine learning

Integrable Families of Hard-Core Particles with Unequal Masses in a One-Dimensional Harmonic Trap

Morphology of three-body quantum states from machine learning

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