The model of two particles on a sphere is extended to two particles on concentric spheres (POCS). The quantum states are found for two electrons, one on a sphere of radius 10 a.u. (simulating the shell $n=3$ in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities $\ensuremath{\rho}({\ensuremath{\theta}}_{12})$ exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.The model of two particles on a sphere is extended to two particles on concentric spheres (POCS). The quantum states are found for two electrons, one on a sphere of radius 10 a.u. (simulating the shell $n=3$ in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities $\ensuremath{\rho}({\ensuremath{\theta}}_{12})$ exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.

Physical Review AOct 1, 198357 citations

Quantum-elastic bump on a surface

Victor Atanasov, Rossen Dandoloff

Abstract We use an exact solution of the elastic membrane shape equation, representing the curvature, which will serve as a quantum potential in the quantum mechanical two dimensional Schrödinger equation for a (quasi-) particle on the surface of the membrane. Surface curvature in the quasi one-dimensional case is related to an unexpected static formation: on one hand the elastic energy has a maximum where surface curvature has a maximum and on the other hand the concentration of the expectation value to find the (quasi-) particle is again where the elastic energy is concentrated, namely where surface curvature has a maximum. This represents a particular form of a conformon.Abstract We use an exact solution of the elastic membrane shape equation, representing the curvature, which will serve as a quantum potential in the quantum mechanical two dimensional Schrödinger equation for a (quasi-) particle on the surface of the membrane. Surface curvature in the quasi one-dimensional case is related to an unexpected static formation: on one hand the elastic energy has a maximum where surface curvature has a maximum and on the other hand the concentration of the expectation value to find the (quasi-) particle is again where the elastic energy is concentrated, namely where surface curvature has a maximum. This represents a particular form of a conformon.

arXiv:1402.2078European Journal of PhysicsNov 21, 20167 citations

Quantum mechanical treatment of a constrained particle on two dimensional sphere

L. Jahangiri, H. Panahi

Annals of PhysicsOct 25, 20164 citations

Interacting quantum gases in confined space: Two- and three-dimensional equations of state

Wu-Sheng Dai, Mi Xie

In this paper, we calculate the equations of state and the thermodynamic quantities for two- and three-dimensional hard-sphere Bose and Fermi gases in finite-size containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hard-sphere interaction to a kind of boundary effect, and then the problem of a confined hard-sphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on d-dimensional ideal quantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low temperatures and high densities, the ratios of the effect of the boundary to the effect of the interparticle interaction in two dimensions are essentially different to those in three dimensions: in two dimensions, the ratios for Bose systems and for Fermi systems are the same and are independent of temperatures, while in three dimensions, the ratio for Bose systems depends on temperatures, but the ratio for Fermi systems is independent of temperatures. Moreover, for three-dimensional Fermi cases, compared with the contributions from the boundary, the contributions from the interparticle interaction to entropies and specific heats are negligible.In this paper, we calculate the equations of state and the thermodynamic quantities for two- and three-dimensional hard-sphere Bose and Fermi gases in finite-size containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hard-sphere interaction to a kind of boundary effect, and then the problem of a confined hard-sphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on d-dimensional ideal quantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low temperatures and high densities, the ratios of the effect of the boundary to the effect of the interparticle interaction in two dimensions are essentially different to those in three dimensions: in two dimensions, the ratios for Bose systems and for Fermi systems are the same and are independent of temperatures, while in three dimensions, the ratio for Bose systems depends on temperatures, but the ratio for Fermi systems is independent of temperatures. Moreover, for three-dimensional Fermi cases, compared with the contributions from the boundary, the contributions from the interparticle interaction to entropies and specific heats are negligible.

Journal of Mathematical PhysicsDec 1, 200739 citations

Constrained particles in quantum mechanics

R. C. T. da Costa

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoMar 1, 19837 citations

n=3

in He) and the other, on spheres of 10, 15, 25, 50, and 100 a.u. The eigenvalues and densities $\ensuremath{\rho}({\ensuremath{\theta}}_{12})$ exhibit a transition from collective, moleculelike behavior to independent-particle-like behavior with Russell-Saunders coupling. The parallel problem of two particles with electron masses interacting via a repulsive Gaussian potential is also treated and a similar transition from collective to independent-particle behavior found. The principal difference between the two cases is only the region of radius of the larger sphere where the transition occurs.