Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2

1999 ◽  
Vol 40 (10) ◽  
pp. 5026-5057 ◽  
Author(s):  
Manuel F. Rañada ◽  
Mariano Santander
Keyword(s):  
Hyperbolic Plane ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Superintegrable Systems

scholarly journals A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

2014 ◽  
Vol 47 (34) ◽  
pp. 345204 ◽  
Author(s):  
Ángel Ballesteros ◽  
Alfonso Blasco ◽  
Francisco J Herranz ◽  
Fabio Musso
Keyword(s):  
Hyperbolic Plane ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Anisotropic Oscillator

scholarly journals GEOMETRIC MOMENTUM IN THE MONGE PARAMETRIZATION OF TWO-DIMENSIONAL SPHERE

2013 ◽  
Vol 10 (03) ◽  
pp. 1220031 ◽  
Author(s):  
D. M. XUN ◽  
Q. H. LIU
Keyword(s):  
Quantum Mechanics ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Constrained Motion ◽  
Geometric Invariant ◽  
Gradient Operator ◽  
The Laplace Operator

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

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