This paper introduces a unified fractal wavefunction model by incorporatinga dynamic fractal dimension Df (t), providing novel explanations for wavefunc?tion geometrization, decoherence, and time irreversibility in quantum mechanics.By reinterpreting the physical essence of the wavefunction from the perspectiveof fractal geometry, we propose a fractal Schr¨odinger equation and demonstratethe extensive applicability of fractal wavefunctions in quantum systems throughexperimental validation and numerical simulations. The primary innovations in?clude the introduction of dynamic fractal dimensions, mathematical modeling offractal wavefunctions, and a geometric interpretation of decoherence and time irre?versibility. With supplementary theoretical foundations, experimental designs, andnumerical analyses, this work enhances the academic rigor and credibility of theproposed model.
Number of zeros seen by a particle around small clusters of other particles is encoded in the root partition, and partly characterizes the correlations in fractional quantum Hall trial wavefunctions. We explore a generalization wherein we consider the counting of zeros seen by a cluster of particles on another cluster. Numbers of such zeros between clusters in the Laughlin wavefunctions are fully determined by the root partition. However, such a counting is unclear for general Jain states where a polynomial expansion is difficult. Here we consider the simplest state beyond the Laughlin wavefunction, namely a state containing a single quasiparticle of the Laughlin state. We show numerically and analytically that in the trial wavefunction for the quasiparticle of the Laughlin state, counting of zeros seen by a cluster on another cluster depends on the relative dimensions of the two clusters. We further ask if the patterns in the counting of zeros extend, in at least an approximate sense, to wavefunctions beyond the trial states. Using numerical computations in systems up to
N=9, we present results for the statistical distribution of zeros around particle clusters at the center of an FQH droplet in the ground state of a Hamiltonian that is perturbed away from the
V1 interaction (short-range repulsion). Evolution of this distribution with the strength of the perturbation shows that the counting of zeros is altered by even a weak perturbation away from the parent Hamiltonian, though the perturbations do not change the phase of the system.