8 research outputs found
Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds
There are two classes of quantum integrable systems on a manifold with
quadratic integrals, the Liouville and the Lie integrable systems as it happens
in the classical case. The quantum Liouville quadratic integrable systems are
defined on a Liouville manifold and the Schr\"odinger equation can be solved by
separation of variables in one coordinate system. The Lie integrable systems
are defined on a Lie manifold and are not generally separable ones but the can
be solved. Therefore there are superintegrable systems with two quadratic
integrals of motion not necessarily separable in two coordinate systems. The
quantum analogues of the two dimensional superintegrable systems with quadratic
integrals of motion on a manifold are classified by using the quadratic
associative algebra of the integrals of motion. There are six general
fundamental classes of quantum superintegrable systems corresponding to the
classical ones. Analytic formulas for the involved integrals are calculated in
all the cases. All the known quantum superintegrable systems are classified as
special cases of these six general classes. The coefficients of the associative
algebra of the general cases are calculated. These coefficients are the same as
the coefficients of the classical case multiplied by plus quantum
corrections of order and .Comment: LaTeX file, 25 page
Robotank biathlon
In this article the competition of “biathlon of “robotanks” considers as formation of engineering-technical staff for high-tech economy of Russia. What is more, it develops children's engineering and algorithmic intellectionВ статье рассматривается соревнование «Роботанковый биатлон» как формирование инженерно-технических кадров для высокотехнологичной экономики России, а так же развитие у детей инженерного и алгоритмического мышлени
Fine-dispersed and nano mineral particles in mining and metallurgical wastes after commercial and laboratory grinding
The paper presents the results of defining the content of fine-grained fractions and nanofractions in mining wastes and the correlation equations for express estimate of powder dispensability. The authors discuss the non-uniqueness of the “fine-grained mineral particles and nanoparticles” notions in scientific literatureyesBelgorod State Universit