Morphological image analysis is applied to the time evolution of the probability distribution of a quantum particle moving in two- and three-dimensional billiards. It is shown that the time-averaged Euler characteristic of the probability distribution provides a well defined quantity to distinguish between classically integrable and nonintegrable billiards. In three dimensions the time-averaged mean breadth of the probability distribution may also be used for this purpose.
Morphological image analysis is applied to the time evolution of the probability distribution of a quantum particle moving in two- and three-dimensional billiards. It is shown that the time-averaged Euler characteristic of the probability distribution provides a well defined quantity to distinguish between classically integrable and nonintegrable billiards. In three dimensions the time-averaged mean breadth of the probability distribution may also be used for this purpose.
Abstract 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.
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 thewave 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.
Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.
The Schrödinger equation governs the probabilistic behavior of quantum particles through the wave function. Microlocal singularities denote regions with significantly high probability density or abrupt changes therein. By visualizing the probability distribution in time and space, we discern regions with higher probability density, indicative of potential microlocal singularities. These regions probably correspond to areas with a greater probability of particle presence. Such analysis aligns with Theorem 1, predicting characteristics of microlocal singularities of wave functions. Furthermore, Theorem 2 postulates that semiclassical path integrals along these singularities contribute significantly to solving the Schrödinger equation. Interpreting the temporal evolution of the probability density in the probability distribution visualization reveals the propagation of the particle over time. Regions of high density mean likely presence of particles at specific times, aligning with the predictions of Theorem 2. Consequently, the analysis of the contribution of high-density regions to the temporal evolution of the wave function resembles semi-classical path integral calculations. Thus, our findings demonstrate that visualization of probability distributions obtained from the numerical resolution of the Schrödinger equation allows a comprehensive interpretation of the behavior of quantum particles, consistent with the theorems.
Quantum billiards has attracted much interest recent years.There are many works on the two-dimensional billiards(2D).Compare with the 2D billiards,the three-dimensional(3D) billiards approaches with the actual system better.Taking the cubic billiards as an example,using the closed orbit theory(COT),we find the classical open orbits and study the correspondence between the quantum spectra and the length of classical open orbits in the 3D billiard system.The peaks' locations of the quantum spectra match with the length of the classical open orbits very well.This will help us to analyze the transport property through an open billiards.By this method,the physical image is clear,the computation is easier and it can help us to understand the character of some chaotic systems better.This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.
The motion of a quantum particle in a time-dependent circular billiard is studied on the basis of the Schrödinger equation with time-dependent boundary conditions.The cases of monotonically expanding (contracting), non-harmonically, harmonically breathing circles the case when billiard wall suddenly disappears are explored in detail.The exact analytical solutions for monotonically expanding and contracting circles are obtained.For all cases, the time-dependence of the quantum average energy is calculated.It is found that for an harmonically breathing circle, the average energy is time-periodic in the adiabatic regime with the same period as that of the oscillation.For intermediate frequencies which are comparable with the initial frequency of the particle in unperturbed billiard, such periodicity is broken.However, for very high frequencies, the average energy once again becomes periodic.A qualitative analysis of the border between adiabatic and non-adiabatic regimes is provided.