arXiv subjects
Publications and source records attributed to Xiao-Yan Gu.
In the recent experiment, the phenomena of superluminal and slow-light propagation in dispersive medium were found, and there are various explanation in theory. We find the phenomenon can be explained by multiple-photon interaction. Otherwise, the multiple-photon interaction can also appear other optical phenomenon: doubling frequency, sum frequency, difference frequency which are found in nonlinear optics.
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 $(μ-ν+1)$.
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.