In the problem of inverse-square force, there exists a hidden SO(4) symmetry, which is manifested as a conserved quantity the Laplace-Runge-Lenz vector in the classical Kepler problem, and as the degeneracy of hydrogen atom energy levels beyond the three-dimensional rotational symmetry in quantum mechanics. This paper presents a more intuitive representation of this hidden symmetry through geometric imagery. Additionally, the paper derives in detail the relevant formulas for the raising and lowering operators corresponding to this symmetry in quantum mechanics.

This study investigates the polaron states formed by an impurity immersed in a spin-1 spinor Bose-Einstein condensate. Employing the Fröhlich Hamiltonian framework, we systematically calculated the ground-state energy and effective mass of polarons in both weak-coupling and strong-coupling regimes. For the weakly interacting regime, the analysis was conducted using second-order perturbation theory. In the strong-coupling regime, we constructed an analytical approximation for the polaron states and performed variational calculations to determine their energy and effective mass. Furthermore, we discussed the transition from the mobile state to the self-trapped state of the polaron in the strong-coupling region. We also estimate the signatures of polaron effects in spinor BEC for the future experiments.

Transport measurement is an important tool for studying the fundamental properties of matter. Among them, transport through quantum point contacts is a very interesting phenomenon in mesoscopic physics. In non-interacting or weakly interacting Fermi systems, the conductivity exhibits quantized behavior as a function of gate voltage, revealing the quantum nature of matter. In recent years, transport in cold atomic systems has become one of the frontiers in this field. Cold atomic systems have highly are controllable and exhibit many novel phases, leading to new experimental phenomena that cannot be observed in traditional material physics. This article will review some of the latest experimental progress in transport through quantum point contacts in cold atomic Fermi gases, and will also introduce the non-equilibrium field theoretical methods commonly used to study this system.