scholarly journals The Modified Fundamental Equations of Quantum Mechanics in Symmetric Forms

2021 ◽  
Author(s):  
Huai-Yu Wang
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Kinetic Energy ◽  
Time Derivative ◽  
Dinger Equation ◽  
Calculated Density ◽  
One Dimensional ◽  
Klein Gordon ◽  
Fundamental Equations ◽  
First Time

Up to now, Schrödinger equation, Klein-Gordon equation (KGE) and Dirac equation are believed the fundamental equations of quantum mechanics. Schrödinger equation has a defect that there is no NKE solutions. Dirac equation has positive kinetic energy (PKE) and negative kinetic energy (NKE) branches. Both branches should have low momentum, or nonrelativistic, approximations: one is Schrödinger equation and the other is NKE Schrödinger equation. KGE has two problems: it is an equation of second time derivative, and calculated density is not definitely positive. To overcome the problems, it should be revised as PKE and NKE decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are of unitary in that everyone contains the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as matter and dark matter, of our universe. The problems of one-dimensional step potentials are resolved by means of the modified fundamental equations for a nonrelativistic particle.

scholarly journals Dirac Equation in Cosmological Inertial Frame

2021 ◽  
Author(s):  
Sangwha Yi
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Dirac Equation ◽  
Inertial Frame ◽  
Gordon Equation ◽  
Probability Amplitude ◽  
Function Type ◽  
Klein Gordon ◽  
Klein Gordon Equation

Dirac equation is a one order-wave equation. Wave function uses as a probability amplitude in quantum mechanics. We make Dirac Equation from wave function, Type A in cosmological inertial frame.The Dirac equation satisfy Klein-Gordon equation in cosmological inertial frame.

ALGEBRAIC APPROACH TO THE KLEIN–GORDON EQUATION WITH HYPERBOLIC SCARF POTENTIAL

2011 ◽  
Vol 20 (01) ◽  
pp. 55-61 ◽  
Author(s):  
SHISHAN DONG ◽  
SHI-HAI DONG ◽  
H. BAHLOULI ◽  
V. B. BEZERRA
Keyword(s):  
Quantum Mechanics ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Algebraic Approach ◽  
Supersymmetric Quantum Mechanics ◽  
Shape Invariance ◽  
One Dimensional ◽  
Scarf Potential ◽  
Klein Gordon ◽  
Klein Gordon Equation

Using the shape invariance approach we obtain exact solutions of one-dimensional Klein–Gordon equation with equal types of scalar and vector hyperbolic Scarf potentials. This is considered in the framework of supersymmetric quantum mechanics method.

Dirac and Klein–Gordon–Fock equations in Grumiller’s spacetime

2018 ◽  
Vol 15 (04) ◽  
pp. 1850051
Author(s):  
A. Al-Badawi ◽  
I. sakalli
Keyword(s):  
Black Hole ◽  
Dirac Equation ◽  
Wave Equations ◽  
Massive Scalar ◽  
One Dimensional ◽  
Effective Potentials ◽  
Central Object ◽  
Klein Gordon ◽  
Rindler Acceleration ◽  
The Waves

We study the Dirac and the chargeless Klein–Gordon–Fock equations in the geometry of Grumiller’s spacetime that describes a model for gravity of a central object at large distances. The Dirac equation is separated into radial and angular equations by adopting the Newman–Penrose formalism. The angular part of the both wave equations are analytically solved. For the radial equations, we managed to reduce them to one dimensional Schrödinger-type wave equations with their corresponding effective potentials. Fermions’s potentials are numerically analyzed by serving their some characteristic plots. We also compute the quasinormal frequencies of the chargeless and massive scalar waves. With the aid of those quasinormal frequencies, Bekenstein’s area conjecture is tested for the Grumiller black hole. Thus, the effects of the Rindler acceleration on the waves of fermions and scalars are thoroughly analyzed.

Towards a Relativistic Quantum Mechanics

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Degrees Of Freedom ◽  
Relativistic Quantum ◽  
Negative Energy ◽  
Relativistic Quantum Mechanics ◽  
Relativistic Limit ◽  
Probability Current ◽  
Klein Gordon ◽  
Relativistic Equations

Towards a relativistic quantum mechanics. Klein–Gordon and the problems of the probability current and the negative energy solutions. The Dirac equation and negative energies. P, C, and T symmetries. Positrons. The Schrödinger equation as the non-relativistic limit of relativistic equations. Majorana and Weyl equations. Relativistic corrections in hydrogen-like atoms. The Dirac equation as a quantum system with an infinite number of degrees of freedom.

scholarly journals Shannon entropy and Fisher information of the one-dimensional Klein–Gordon oscillator with energy-dependent potential

2018 ◽  
Vol 33 (06) ◽  
pp. 1850033 ◽  
Author(s):  
Abdelmalek Boumali ◽  
Malika Labidi
Keyword(s):  
Quantum Mechanics ◽  
Fisher Information ◽  
Shannon Entropy ◽  
One Dimensional ◽  
Probability Densities ◽  
Klein Gordon ◽  
Dependent Potential ◽  
The One ◽  
Energy Dependent ◽  
Ordinary Quantum

In this paper, we studied, at first, the influence of the energy-dependent potentials on the one-dimensionless Klein–Gordon oscillator. Then, the Shannon entropy and Fisher information of this system are investigated. The position and momentum information entropies for the low-lying states n = 0, 1, 2 are calculated. Some interesting features of both Fisher and Shannon densities, as well as the probability densities, are demonstrated. Finally, the Stam, Cramer–Rao and Bialynicki–Birula–Mycielski (BBM) inequalities have been checked, and their comparison with the regarding results have been reported. We showed that the BBM inequality is still valid in the form [Formula: see text], as well as in ordinary quantum mechanics.

Fermat’s principle and the effect of jerk on light motion near the Sun

2022 ◽  
Author(s):  
Rami Ahmad El-Nabulsi ◽  
Waranont Anukool
Keyword(s):  
Kinetic Energy ◽  
Time Derivative ◽  
Classical Mechanics ◽  
Time Parameter ◽  
The Sun ◽  
Deflection Of Light ◽  
Long Baseline ◽  
Electromagnetic Interactions ◽  
Vlbi Observations ◽  
First Time

In classical mechanics, in the case of gravitational and electromagnetic interactions, the force on a particle is usually proportional to its acceleration: The force acts locally on the particle. However, there are situations possible-if the particle moves through a suitable medium, for example, in which the force depends also on the first-time derivative of its acceleration, the jerk, and on its second-time derivative, the snap, and possibly also on higher-time derivatives. Such forces are called nonlocal, and this work investigates such nonlocal forces, mainly those depending on the jerk. In particular, we implement jerk and acceleration in geodesics by means of the nonlocal-in-time kinetic energy approach to spacetime physics. We describe a framework that can be used to estimate the quantum nonlocal time parameter by studying the deflection of light around the Sun. Comparing our results with long baseline interferometry (VLBI) observations, we concluded that the nonlocal time parameter [Formula: see text] s.

scholarly journals Dirac’s reduced radial equations and the problem of additional solutions

2017 ◽  
Vol 26 (07) ◽  
pp. 1750043 ◽  
Author(s):  
Anzor Khelashvili ◽  
Teimuraz Nadareishvili
Keyword(s):  
Boundary Condition ◽  
Quantum Mechanics ◽  
Dirac Equation ◽  
Radial Function ◽  
Delta Function ◽  
Two Dimensional ◽  
Radial Equation ◽  
Reduced Equations ◽  
Klein Gordon ◽  
History Of

We show that additional solutions must be ignored (in differences of the Schrödinger and Klein–Gordon equations) in the Dirac equation, where usually the second-order radial equation is passed, called the reduced equation, instead of a system. Analogously to the Schrödinger equation, in this process, the Dirac’s delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end, we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survive as a result of usual physical requirements.

scholarly journals AdS2 × S2 × CY2 solutions in Type IIB with 8 supersymmetries

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Yolanda Lozano ◽  
Carlos Nunez ◽  
Anayeli Ramirez
Keyword(s):  
Quantum Mechanics ◽  
Riemann Surface ◽  
Central Charge ◽  
Global Solutions ◽  
Infinite Family ◽  
Dimensional System ◽  
One Dimensional

Abstract We present a new infinite family of Type IIB supergravity solutions preserving eight supercharges. The structure of the space is AdS2 × S2 × CY2 × S1 fibered over an interval. These solutions can be related through double analytical continuations with those recently constructed in [1]. Both types of solutions are however dual to very different superconformal quantum mechanics. We show that our solutions fit locally in the class of AdS2 × S2 × CY2 solutions fibered over a 2d Riemann surface Σ constructed by Chiodaroli, Gutperle and Krym, in the absence of D3 and D7 brane sources. We compare our solutions to the global solutions constructed by Chiodaroli, D’Hoker and Gutperle for Σ an annulus. We also construct a cohomogeneity-two family of solutions using non-Abelian T-duality. Finally, we relate the holographic central charge of our one dimensional system to a combination of electric and magnetic fluxes. We propose an extremisation principle for the central charge from a functional constructed out of the RR fluxes.

scholarly journals Topological correlators and surface defects from equivariant cohomology

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Rodolfo Panerai ◽  
Antonio Pittelli ◽  
Konstantina Polydorou
Keyword(s):  
Quantum Mechanics ◽  
Equivariant Cohomology ◽  
Surface Defects ◽  
Star Product ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional System ◽  
The Novel ◽  
One Dimensional ◽  
Dual Quantum

Abstract We find a one-dimensional protected subsector of $$ \mathcal{N} $$ N = 4 matter theories on a general class of three-dimensional manifolds. By means of equivariant localization we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on S3. Then, we apply it to the novel case of S2× S1 and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models are naturally associated with a noncom- mutative star product, while their correlation functions are essentially topological. Finally, we couple the three-dimensional theory to general $$ \mathcal{N} $$ N = (2, 2) surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.

scholarly journals The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
P. Kim ◽  
R. Jorge ◽  
W. Dorland
Keyword(s):  
Magnetic Field ◽  
Original Work ◽  
Three Dimensional ◽  
Radial Distance ◽  
Analytical Form ◽  
One Dimensional ◽  
Modern Notation ◽  
Magnetic Well ◽  
First Time ◽  
Dimensional Integral

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

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