arXiv subjects
Publications and source records attributed to Zhong-Qi Ma.
We summarize some predictions from the model of supermassive object with magnetic monopoles which match up with recent astronomical observations quantitatively. They may be the signals for existence of magnetic monopoles in the supermassive objects, such as one at the Galactic Center.
The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors, where the coefficients are calculated explicitly in this paper.
We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional $κ$-component fermions with repulsive and with attractive delta-function interaction in strong and weak coupling regimes. We thus obtain a highly accurate result for the ground state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair correlation functions shed light on the quantum statistic effects of the $κ$-component interacting fermions. For the balanced spin case with repulsive interaction, the ground state energy obtained confirms Yang and You's result [Chin. Phys. Lett. {\bf 28}, 020503 (2011)] that the energy per particle as $κ\to \infty$ is the same as for spinless Bosons.
The Fredholm equations for one-dimensional two-component Fermions with repulsive and with attractive delta-function interactions are solved by an asymptotic expansion for A) strong repulsion, B) weak repulsion, C) weak attraction and D) strong attraction. Consequently, we obtain the first few terms of the expansion of ground state energy for the Fermi gas with polarization for these regimes. We also prove that the two sets of the Fredhom equations for weakly repulsive and attractive interactions are identical as long as the integration boundaries match each other between the two sides. Thus the asymptotic expansions of the energies of the repulsive and attractive Fermions are identical to all orders in this region. The identity of the asymptotic expansions may not mean that the energy analytically connects.
In these two papers, we solve the N body 1D harmonically trapped spinless Boson or spin 1/2 Fermions with repulsive delta function interaction in the limit $N\to \infty$.
Conjectures are made for the ground state energy of a large spin 1/2 Fermion system trapped in a 1D harmonic trap with delta function interaction. States with different spin J are separately studied. The Thomas-Fermi method is used as an effective test for the conjecture.
We present an exact analytical solution of the fundamental systems of quasi-one-dimensional spin-1/2 fermions with infinite repulsion for arbitrary confining potential. The eigenfunctions are constructed by the combination of Gireardeau's hard-core contacting boundary condition and group theoretical method which guarantees the obtained states to be simultaneously the eigenstates of $S$ and $S_z$ and fulfill the antisymmetry under odd permutation. We show that the total ground-state density profile behaves like the polarized noninteracting fermions, whereas the spin-dependent densities display different properties for different spin configurations. We also discuss the splitting of the ground states for large but finite repulsion.
An exact quantization rule for the Schrödinger equation is presented. In the exact quantization rule, in addition to $Nπ$, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Pöschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom.
An exact quantization rule for the bound states of the one-dimensional Schrödinger equation is presented and is generalized to the three-dimensional Schrödinger equation with a spherically symmetric potential.
Complete spectrum of exact interdimensional degeneracies for a quantum $N$-body system in $D$-dimensions is presented by the method of generalized spherical harmonic polynomials. In an $N$-body system all the states with angular momentum $[μ+n]$ in $(D-2n)$ dimensions are degenerate where $[μ]$ and $D$ are given and $n$ is an arbitrary integer if the representation $[μ+n]$ exists for the SO($D-2n$) group and $D-2n\geq N$. There is an exceptional interdimensional degeneracy for an $N$-body system between the state with zero angular momentum in $D=N-1$ dimensions and the state with zero angular momentum in $D=N+1$ dimensions.
By the method of generalized spherical harmonic polynomials, the Schrödinger equation for a four-body system in $D$-dimensional space is reduced to the generalized radial equations where only six internal variables are involved. The problem on separating the rotational degrees of freedom from the internal ones for a quantum $N$-body system in $D$ dimensions is generally discussed.
A new approach is developed to derive the complete spectrum of exact interdimensional degeneracies for a quantum three-body system in D-dimensions. The new method gives a generalization of previous methods.
The Dirac equation is generalized to $D+1$ space-time.The conserved angular momentum operators and their quantum numbers are discussed. The eigenfunctions of the total angular momenta are calculated for both odd $D$ and even $D$ cases. The radial equations for a spherically symmetric system are derived. The exact solutions for the system with a Coulomb potential are obtained analytically. The energy levels and the corresponding fine structure are also presented.
The independent eigenstates of the total orbital angular momentum operators for a three-body system in an arbitrary D-dimensional space are presented by the method of group theory. The Schrödinger equation is reduced to the generalized radial equations satisfied by the generalized radial functions with a given total orbital angular momentum denoted by a Young diagram $[μ,ν,0,...,0]$ for the SO(D) group. Only three internal variables are involved in the functions and equations. The number of both the functions and the equations for the given angular momentum is finite and equal to $(μ-ν+1)$.
In this Letter the bound states of (2+1) Dirac equation with the cylindrically symmetric $δ(r-r_{0})$-potential are discussed. It is surprisingly found that the relation between the radial functions at two sides of $r_{0}$ can be established by an SO(2) transformation. We obtain a transcendental equation for calculating the energy of the bound state from the matching condition in the configuration space. The condition for existence of bound states is determined by the Sturm-Liouville theorem.
The global rotational degrees of freedom in the Schrödinger equation for an $N$-body system are completely separated from the internal ones. After removing the motion of center of mass, we find a complete set of $(2\ell+1)$ independent base functions with the angular momentum $\ell$. These are homogeneous polynomials in the components of the coordinate vectors and the solutions of the Laplace equation, where the Euler angles do not appear explicitly. Any function with given angular momentum and given parity in the system can be expanded with respect to the base functions, where the coefficients are the functions of the internal variables. With the right choice of the base functions and the internal variables, we explicitly establish the equations for those functions. Only (3N-6) internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.
We demonstrate how to separate the rotational degrees of freedom in a quantum N-body problem completely from the internal ones. It is shown that any common eigenfunction of the total orbital angular momentum ($\ell$) and the parity in the system can be expanded with respect to $(2\ell+1)$ base-functions, where the coefficients are the functions of the internal variables. We establish explicitly the equations for those functions, called the generalized radial equations, which are $(2\ell+1)$ coupled partial differential equations containing only $(3N-6)$ internal variables.
The exact solution to the Schrödinger equation for the rigid body with the given angular momentum and parity is obtained. Since the quantum rigid body can be thought of as the simplest quantum three-body problem where the internal motion is frozen, this calculation method is a good starting point for solving the quantum three-body problems.