Perturbation foundation of q-deformed dynamics

Abstract

In the q-deformed theory the perturbation approach can be expressed in terms of two pairs of undeformed position and momentum operators. There are two configuration spaces. Correspondingly there are two q-perturbation Hamiltonians, one originates from the perturbation expansion of the potential in one configuration space, the other one originates from the perturbation expansion of the kinetic energy in another configuration space. In order to establish a general foundation of the q-perturbation theory, two perturbation equivalence theorems are proved: (I) Equivalence theorem I: Perturbation expressions of the q-deformed uncertainty relations calculated by two pairs of undeformed operators are the same, and the two q-deformed uncertainty relations undercut Heisenberg's minimal one in the same style. (II) The general equivalence theorem II: for any potential (regular or singular) the expectation values of two q-perturbation Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent to all orders of the perturbation expansion. As an example of singular potentials the perturbation energy spectra of the q-deformed Coulomb potential are studied. §

Green synthesis of 3,4‐disubstituted isoxazol‐5(4H)‐ones using ZnO@Fe3O4 core–shell nanocatalyst in water

“…A number of catalysts have been used in one‐pot multicomponent reactions for the synthesis of isoxazol‐5(4 H )‐one scaffolds including boric acid, citric acid, molecular iodine, KI, Ag/SiO 2 , 4‐( N , N‐ dimethylamino) pyridinium acetate, tetrabutylammonium perchlorate/glycine/sodium oxalate, amine‐modified montmorillonite nanoclay, antimony trichloride, mesolite, DABCO functionalized dicationic ionic liquid, nano‐MgO, sodium silicate pentahydrate, citrus fruit juice, pyridinium p‐ toluenesulfonate, cerium chloride heptahydrate, pyridine, nickel (II) acetate and many more . Also, advanced techniques such as microwave heating, grinding, ultrasonic irradiation and visible light in the presence of sodium acetate in aqueous EtOH have also been reported for the preparation of isoxazol‐5(4 H )‐ones, Although many of these protocols suffer from drawbacks and limitations such as low yields/long reaction times, strongly acidic or basic conditions and the use of toxic reagents, stringent reaction conditions, the use of expensive catalyst, multistep synthetic sequence and difficult work up procedures that restrict their scope in practical applications. Considering the above points, we report an efficient and green method for the synthesis of a wide variety of isoxazol‐5(4 H )‐one derivatives via the one‐pot, three‐component process catalyzed by ZnO@Fe 3 O 4 core–shell nanoparticles under aqueous conditions (Scheme ).…”

Dissolution and utilization of chitosan in a 1‐carboxymethyl‐3‐methylimidazolium hydrochloride ionic salt aqueous solution

“…The absorption band at 1156 cm 21 was characteristic of the asymmetric stretching of the CAOAC bridge. 23 The band at 1079 cm 21 was attributed to the skeletal vibration involving CAO stretching. 23 The band at 897 cm 21 was assigned to the absorption peaks of b-(1-4)glycosidic bonds in chitosan.…”

“…23 The band at 1079 cm 21 was attributed to the skeletal vibration involving CAO stretching. 23 The band at 897 cm 21 was assigned to the absorption peaks of b-(1-4)glycosidic bonds in chitosan. 24 The FTIR spectra suggested that the important functional groups were still present, and the main polysaccharide chain structure remained after regeneration.…”

Radioactive Bimetallic Gold–Silver Nanoparticles Production in a Research Nuclear Reactor

“…[1,2,[5][6][7] Many different methods of synthesis are available to produce bimetallic nanoparticles including chemical processes (such as sol-gel, impregnation method, liquid reduction method and others), biological process, physical process, and radiochemical reduction processes. [2,5,[8][9][10][11][12][13] This work demonstrates the production of radioactive BMNPs by water radiolysis using a research nuclear reactor. The reaction of water radiolysis can be defined as shown in Equation 1:…”