Morphology of three-body quantum states from machine learning

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 $\kappa$ 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 $1/\kappa\in [0,1]$ and find no evidence of integrable cases beyond the limiting values $1/\kappa=1$ and $1/\kappa=0$. 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.