We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property).We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers.Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries.For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number.A Lie-algebra approach for non-linear theories is developed.We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property).We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers.Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries.For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number.A Lie-algebra approach for non-linear theories is developed.

Journal of Nonlinear Mathematical PhysicsJan 1, 19956 citations

SEPARATION OF VARIABLES FOR THE N-ELECTRON SCHRODINGER EQUATION

Evgenii A. Smolenskii, N. S. Zefirov, Patricia A. Millard

Doklady PhysicsMar 1, 19931 citation

An application of Heun functions in the quantum mechanics of a constrained particle

Alexandre G. M. Schmidt, Matheus E. Pereira

Using the thin-layer quantization, we formulate the problem of a Schrödinger particle constrained to move along a coordinate surface of the bi-spherical coordinate system. In three-dimensional space, the free Schrödinger equation is not separable in this coordinate system. However, when we consider the equation for a particle constrained to a given surface, there are only two degrees of freedom. One has to introduce a geometrical potential to attach the particle to the surface. This well-known potential has two contributions: one from Gauss’ curvature and the other from the mean curvature. The Schrödinger equation leads to a general Heun equation. We solve it exactly and present the eigenfunctions and plots of the probability densities, and, as an application of this methodology, we study the problem of an electric charge propagating along these coordinate surfaces in the presence of a uniform magnetic field.Using the thin-layer quantization, we formulate the problem of a Schrödinger particle constrained to move along a coordinate surface of the bi-spherical coordinate system. In three-dimensional space, the free Schrödinger equation is not separable in this coordinate system. However, when we consider the equation for a particle constrained to a given surface, there are only two degrees of freedom. One has to introduce a geometrical potential to attach the particle to the surface. This well-known potential has two contributions: one from Gauss’ curvature and the other from the mean curvature. The Schrödinger equation leads to a general Heun equation. We solve it exactly and present the eigenfunctions and plots of the probability densities, and, as an application of this methodology, we study the problem of an electric charge propagating along these coordinate surfaces in the presence of a uniform magnetic field.

Journal of Mathematical PhysicsApr 1, 20235 citations

An approach to the solution of the Schroedinger equation

Boris V. Alekseev, A. I. Аbakumov

DoSSRJan 1, 19820 citations

Independent eigenstates of angular momentum in a quantumN-body system

Xiaoyan Gu, Bin Duan, Zhong-Qi Ma

The global rotational degrees of freedom in the Schr\"odinger equation for an N-body system are completely separated from the internal ones. After removing the motion of the center of mass, we find a complete set of $(2\mathcal{l}+1)$ independent base functions with angular momentum $\mathcal{l}.$ 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 $(3N\ensuremath{-}6)$ internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.The global rotational degrees of freedom in the Schr\"odinger equation for an N-body system are completely separated from the internal ones. After removing the motion of the center of mass, we find a complete set of $(2\mathcal{l}+1)$ independent base functions with angular momentum $\mathcal{l}.$ 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 $(3N\ensuremath{-}6)$ internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.

arXiv:physics/0108029Physical Review ASep 13, 200112 citations

(2\mathcal{l}+1)

independent base functions with angular momentum

\mathcal{l}.

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 $(3N\ensuremath{-}6)$ internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.