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Mass-Imbalanced Atoms in a Hard-Wall Trap: An Exactly Solvable Model Associated with D₆ Symmetry.

ABSTRACT: We show that a system consisting of two interacting particles with mass ratio 3 or 1/3 in a hard-wall box can be exactly solved by using Bethe-type ansatz. The ansatz is based on a finite superposition of plane waves associated with a dihedral group D₆, which enforces the momentums after a series of scattering and reflection processes to fulfill the D₆ symmetry. Starting from a two-body elastic collision model in a hard-wall box, we demonstrate how a finite momentum distribution is related to the D_2n symmetry for permitted mass ratios. For a quantum system with mass ratio 3, we obtain exact eigenenergies and eigenstates by solving Bethe-type-ansatz equations for arbitrary interaction strength. A many-body excited state of the system is found to be independent of the interaction strength, i.e., the wave function looks exactly the same for non-interacting two particles or in the hard-core limit.

SUBMITTER: Liu Y

PROVIDER: S-EPMC6911986 | biostudies-literature |

REPOSITORIES: biostudies-literature

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Project description:Background Facial symmetry severely affects appearance and function. Large numbers of patients seek orthodontic treatment to improve facial symmetry. However, the correlation between hard- and soft-tissue symmetry is still unclear. Our aim was to investigate the hard- and soft-tissue symmetry in subjects with different levels of menton deviation and sagittal skeletal classes with 3D digital analysis and to investigate the relationship between the entire and individual hard- and soft-tissues. Methods A total of 270 adults (135 males and 135 females) consisting of 45 subjects of each sex in each sagittal skeletal classification group. All subjects were further classified into relative symmetry (RS), moderate asymmetry (MA) and severe asymmetry (SA) groups based on the degree of menton deviation from the mid-sagittal plane (MSP). The 3D images were segmented into anatomical structures and mirrored across the MSP after establishing a coordinate system. Original and mirrored images were registered by a best-fit algorithm, and the corresponding root mean square (RMS) values and colormap were obtained. The Mann‒Whitney U test and Spearman correlation were conducted for statistical analysis. Results The RMS increased with greater deviations with regard to the deviation of the menton in most of anatomical structures. Asymmetry was represented in the same way regardless of sagittal skeletal pattern. The soft-tissue asymmetry had a significant correlation with dentition in the RS group (0.409), while in the SA group, it was related to the ramus (0.526) and corpus (0.417) in males and was related to the ramus in the MA (0.332) and SA (0.359) groups in females. Conclusions The mirroring method combining CBCT and 3dMD provides a new approach for symmetry analysis. Asymmetry might not be influenced by sagittal skeletal patterns. Soft-tissue asymmetry might be reduced by improving the dentition in individuals with RS group, while among those with MA or SA, whose menton deviation was larger than 2 mm, orthognathic treatment should be considered. Supplementary Information The online version contains supplementary material available at 10.1186/s12903-023-03163-z.

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Mass-Imbalanced Atoms in a Hard-Wall Trap: An Exactly Solvable Model Associated with D₆ Symmetry.

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Mass-Imbalanced Atoms in a Hard-Wall Trap: An Exactly Solvable Model Associated with D6 Symmetry.

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Mass-Imbalanced Atoms in a Hard-Wall Trap: An Exactly Solvable Model Associated with D₆ Symmetry.