Differential representation of the delta function in two-dimensional quantum mechanics

2018 ◽  
Vol 56 (5) ◽  
pp. 2547-2552
Author(s):  
Kok An Wong ◽  
Su-Long Nyeo
Keyword(s):  
Quantum Mechanics ◽  
Delta Function ◽  
Two Dimensional

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 Schwarzian quantum mechanics as a Drinfeld-Sokolov reduction of BF theory

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Fridrich Valach ◽  
Donald R. Youmans
Keyword(s):  
Quantum Mechanics ◽  
Partition Function ◽  
Gauge Group ◽  
Path Integral ◽  
Coadjoint Orbits ◽  
Two Dimensional ◽  
Edge States ◽  
Class Constraint ◽  
Action Functional ◽  
Bf Theory

Abstract We give an interpretation of the holographic correspondence between two-dimensional BF theory on the punctured disk with gauge group PSL(2, ℝ) and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence of certain edge states imposing a first class constraint on the model. The constrained path integral localizes over exceptional Virasoro coadjoint orbits. The reduced theory is governed by the Schwarzian action functional generating a Hamiltonian S1-action on the orbits. The partition function is given by a sum over topological sectors (corresponding to the exceptional orbits), each of which is computed by a formal Duistermaat-Heckman integral.

FOURIER ANALYSIS OVER HYPERBOLIC ALGEBRA, PSEUDO-DIFFERENTIAL OPERATORS, AND HYPERBOLIC DEFORMATION OF CLASSICAL MECHANICS

2007 ◽  
Vol 10 (03) ◽  
pp. 421-438 ◽  
Author(s):  
ANDREI KHRENNIKOV
Keyword(s):  
Quantum Mechanics ◽  
Fourier Analysis ◽  
Poisson Bracket ◽  
Commutative Algebra ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Quantum Deformation ◽  
Pseudo Differential Operators

We develop Fourier analysis over hyperbolic algebra (the two-dimensional commutative algebra with the basis e1 = 1, e2 = j, where j2 = 1). We demonstrated that classical mechanics has, besides the well-known quantum deformation over complex numbers, another deformation — so-called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h → 0 not only of the ordinary Moyal bracket, but also a hyperbolic analogue of the Moyal bracket.

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.

scholarly journals Integrable and Superintegrable Systems with Higher Order Integrals of Motion: Master Function Formalism

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Z. Alizadeh ◽  
H. Panahi
Keyword(s):  
Quantum Mechanics ◽  
Higher Order ◽  
Supersymmetric Quantum Mechanics ◽  
Two Dimensional ◽  
Integrals Of Motion ◽  
Function Formalism ◽  
Superintegrable Systems ◽  
Master Function

We construct two-dimensional integrable and superintegrable systems in terms of the master function formalism and relate them to Mielnik’s and Marquette’s construction in supersymmetric quantum mechanics. For two different cases of the master functions, we obtain two different two-dimensional superintegrable systems with higher order integrals of motion.

ON p-ADIC FUNCTIONAL INTEGRALS

1988 ◽  
Vol 03 (06) ◽  
pp. 639-643 ◽  
Author(s):  
GIORGIO PARISI
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Free Field ◽  
Two Dimensional ◽  
Functional Integrals

In this letter we present a possible form of quantum mechanics in the case where the dynamical variables (or the time) are p-adic numbers. We also present a possible formulation of two-dimensional free field theory on a two-dimensional p-adic space.

scholarly journals CONSISTENT DEFORMED BOSONIC ALGEBRA IN NONCOMMUTATIVE QUANTUM MECHANICS

2008 ◽  
Vol 23 (09) ◽  
pp. 1393-1403 ◽  
Author(s):  
JIAN-ZU ZHANG
Keyword(s):  
Quantum Mechanics ◽  
Weyl Algebra ◽  
Fock Space ◽  
General Relation ◽  
Noncommutative Space ◽  
Two Dimensional ◽  
Noncommutative Quantum Mechanics ◽  
Commutative Space ◽  
Creation Operators ◽  
Noncommutative Quantum

In two-dimensional noncommutative space for the case of both position–position and momentum–momentum noncommuting, the consistent deformed bosonic algebra at the nonperturbation level described by the deformed annihilation and creation operators is investigated. A general relation between noncommutative parameters is fixed from the consistency of the deformed Heisenberg–Weyl algebra with the deformed bosonic algebra. A Fock space is found, in which all calculations can be similarly developed as if in commutative space and all effects of spatial noncommutativity are simply represented by parameters.

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