scholarly journals On the singular solution of Schrödinger equation for the hydrogen atom

2019 ◽  
Vol 45 (0) ◽  
pp. 85-91
Author(s):  
A. I. Haysak ◽  
I. I. Haysak
Keyword(s):  
Hydrogen Atom ◽  
Singular Solution ◽  
Dinger Equation

scholarly journals The Quantum Black Hole as a Gravitational Hydrogen Atom

2019 ◽  
Author(s):  
Christian Corda ◽  
Fabiano Feleppa
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Energy Spectrum ◽  
Hydrogen Atom ◽  
Gravitational Potential ◽  
Particle System ◽  
Energy Levels ◽  
Planck Scale ◽  
Dinger Equation ◽  
The One

In this paper only one basic assumption has been made: if we try to describe black holes, their behavior should be understood in the same language as the one we use for particles; black holes should be treated just like atoms. They must be quantum forms of matter, moving in accordance with Schrödinger equations just like everything else. In particular, Rosen’s quantization approach to the gravitational collapse is applied in the simple case of a pressureless “star of dust” by finding the gravitational potential, the Schrödinger equation and the solution for the collapse’s energy levels. By applying the constraints for a Schwarzschild black hole (BH) and by using the concept of BH effective state, previously introduced by one of the authors (CC), the BH quantum gravitational potential, Schrödinger equation and the BH energy spectrum are found. Remarkably, such an energy spectrum is in agreement (in its absolute value) with the one which was conjectured by Bekenstein in 1974 and consistent with other ones in the literature. This approach also permits to find an interesting quantum representation of the Schwarzschild BH ground state at the Planck scale. Moreover, two fundamental issues about black hole quantum physics are addressed by this model: the area quantization and the singularity resolution. As regards the former, a result similar to the one obtained by Bekenstein, but with a different coefficient, has been found. About the latter, it is shown that the traditional classical singularity in the core of the Schwarzschild BH is replaced, in a full quantum treatment, by a two-particle system where the two components strongly interact with each other via a quantum gravitational potential. The two-particle system seems to be non-singular from the quantum point of view and is analogous to the hydrogen atom because it consists of a “nucleus” and an “electron”.

Measurement of the Lamb shift in the hydrogen atom

1982 ◽  
Vol 138 (10) ◽  
pp. 347 ◽  
Author(s):  
Yurii L. Sokolov ◽  
V.P. Yakovlev
Keyword(s):  
Hydrogen Atom ◽  
Lamb Shift

EXCITON INSULATOR STATES IN MATERIALS WITH ‘MEXICAN HAT’ BAND STRUCTURE DISPERSION AND IN GRAPHENE

2015 ◽  
Vol 11 (1) ◽  
pp. 2927-2949
Author(s):  
Lyubov E. Lokot
Keyword(s):  
Band Structure ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Electron Hole ◽  
Dinger Equation ◽  
Fermi Sphere ◽  
Zinc Blende ◽  
Bulk Gan ◽  
Structure Dispersion ◽  
Excitonic States

In the paper a theoretical study the both the quantized energies of excitonic states and their wave functions in grapheneand in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN is presented. An integral twodimensionalSchrödinger equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection isexactly solved. The solutions of Schrödinger equation in momentum space in studied materials by projection the twodimensionalspace of momentum on the three-dimensional sphere are found exactly. We analytically solve an integral twodimensionalSchrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. Instudied materials the electron-hole pairing leads to the exciton insulator states. Quantized spectral series and lightabsorption rates of the excitonic states which distribute in valence cone are found exactly. If the electron and hole areseparated, their energy is higher than if they are paired. The particle-hole symmetry of Dirac equation of layered materialsallows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone andhence driving the Cooper instability. The solutions of Coulomb problem of electron-hole pair does not depend from a widthof band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of theequation of motion for a microscopic dipole of graphene where >1 s r . The absorption spectrums for the zinc-blendeGaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are presented. Comparison with availableexperimental data shows good agreement.

Direct Cyclization of Alkenyl Thioester via Transition Metal‐hydrogen Atom Transfer/radical‐polar Crossover Mechanism

2019 ◽  
Author(s):  
Shiori Date ◽  
Kensei Hamasaki ◽  
Karen Sunagawa ◽  
Hiroki Koyama ◽  
Chikayoshi Sebe ◽  
...  
Keyword(s):  
Natural Products ◽  
Hydrogen Atom ◽  
Transition Metal ◽  
Hydrogen Atom Transfer ◽  
Synthetic Method ◽  
Atom Transfer ◽  
Reaction Conditions ◽  
Hydride Hydrogen ◽  
Sulfur Heterocycles ◽  
One Step

<div>We report here a catalytic, Markovnikov selective, and scalable synthetic method for the synthesis of saturated sulfur heterocycles, which are found in the structures of pharmaceuticals and natural products, in one step from an alkenyl thioester. Unlike a potentially labile alkenyl thiol, an alkenyl thioester is stable and easy to prepare. The powerful Co catalysis via a cobalt hydride hydrogen atom transfer and radical-polar crossover mechanism enabled simultaneous cyclization and deprotection. The substrate scope was expanded by the extensive optimization of the reaction conditions and tuning of the thioester unit.</div>

Synthesis of Spongidine A, D and Petrosaspongiolide L Methyl Ester Using Pyridine C-H Functionalization

2019 ◽  
Author(s):  
Florian Bartels ◽  
Manuela Weber ◽  
Mathias Christmann
Keyword(s):  
Natural Products ◽  
Methyl Ester ◽  
Hydrogen Atom ◽  
Phospholipase A2 ◽  
Late Stage ◽  
Hydrogen Atom Transfer ◽  
Ring System ◽  
Tetracyclic Core ◽  
Intramolecular Hydrogen ◽  
Phospholipase A2 Inhibitors

<div>An efficient strategy for the synthesis of the potent phospholipase A2 inhibitors spongidine A and D is presented. The tetracyclic core of the natural products was assembled via an intramolecular hydrogen atom transfer‐initiated Minisci reaction. A divergent late‐stage functionalization of the tetracyclic ring system was also used to achieve a concise synthesis of petrosaspongiolide L methyl ester.</div>

Catalytic Synthesis of Cyclic Guanidines via Hydrogen Atom Transfer and Radical-Polar Crossover

2020 ◽  
Author(s):  
Shunya Ohuchi ◽  
Hiroki Koyama ◽  
Hiroki Shigehisa
Keyword(s):  
Hydrogen Atom ◽  
Functional Group ◽  
Hydrogen Atom Transfer ◽  
Catalytic Synthesis ◽  
Membered Ring ◽  
Biologically Active ◽  
Atom Transfer ◽  
Powerful Method ◽  
Protective Groups ◽  
The Common

A catalytic synthesis of cyclic guanidines, which are found in many biologically active compounds and natu-ral products, was developed, wherein transition-metal hydrogen atom transfer and radical-polar crossover were employed. This mild and functional-group tolerant process enabled the cyclization of alkenyl guanidines bearing common protective groups, such as Cbz and Boc. This powerful method not only provided the common 5- and 6-membered rings but also an unusual 7-membered ring. The derivatization of the products afforded various heterocycles. We also investigated the se-lective cyclization of mono-protected or hetero-protected (TFA and Boc) alkenyl guanidines and their further derivatiza-tions.

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