scholarly journals Star-quantization of an infinite wall

2006 ◽  
Vol 84 (6-7) ◽  
pp. 557-563 ◽  
Author(s):  
S Kryukov ◽  
M A Walton
Keyword(s):  
Quantum Mechanics ◽  
Deformation Quantization ◽  
Quantum Particle ◽  
Equation Of Motion ◽  
One Dimension ◽  
Mass Treatment ◽  
Single Quantum ◽  
Weyl Transform ◽  
Infinite Wall ◽  
03.65 Ge

In deformation quantization (a.k.a. the Wigner–Weyl–Moyal formulation of quantum mechanics), we consider a single quantum particle moving freely in one dimension, except for the presence of one infinite potential wall. Dias and Prata pointed out that, surprisingly, its stationary-state Wigner function does not obey the naive equation of motion, i.e., the naive stargenvalue (*-genvalue) equation. We review our recent work on this problem that treats the infinite wall as the limit of a Liouville potential. Also included are some new results: (i) we show explicitly that the Wigner-Weyl transform of the usual density matrix is the physical solution, (ii) we prove that an effective-mass treatment of the problem is equivalent to the Liouville one, and (iii) we point out that self-adjointness of the operator Hamiltonian requires a boundary potential, but one apparently different from that proposed by Dias and Prata. PACS Nos.: 03.65.–w, 03.65.Ca, 03.65.Ge

Angular Momentum

2019 ◽  
pp. 45-62
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Quantum Mechanics ◽  
Angular Momentum ◽  
General Theory ◽  
Spin Chain ◽  
Quantum Particle ◽  
Quantum Spin ◽  
Single Quantum ◽  
Quantum Spin Chain ◽  
Single Spin ◽  
Quantum Spins

In order to prepare for the discussion of quantum many-particle Hamiltonians, for example the Heisenberg quantum spin chain Hamiltonian, this chapter examines the concept of angular momentum in quantum mechanics, especially the coupling of spin-2 operators for several quantum spins. It begins with the general theory of angular momentum for a single quantum particle, especially for a single spin-1, described by Pauli spin matrices, and then extends to the theory of angular momentum for several particles, again especially for several spins1.

The f-Deformation II: f-Deformed Quantum Mechanics in One Dimension

2020 ◽  
Vol 85 (3) ◽  
pp. 305-329
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
One Dimension

Diffusion and dissipation by linear momentum in spherical environment

2014 ◽  
Vol 28 (11) ◽  
pp. 1450077
Author(s):  
Werner Scheid ◽  
Aurelian Isar ◽  
Aurel Sandulescu
Keyword(s):  
Potential Energy ◽  
Density Matrix ◽  
Finite Number ◽  
Bessel Functions ◽  
Open Quantum System ◽  
Equation Of Motion ◽  
Linear Momentum ◽  
Spherical Space ◽  
Infinite Energy ◽  
Infinite Wall

An open quantum system is studied consisting of a particle moving in a spherical space with an infinite wall. With the theory of Lindblad the system is described by a density matrix which gets affected by operators with diffusive and dissipative properties depending on the linear momentum and density matrix only. It is shown that an infinite number of basis states leads to an infinite energy because of the infinite unsteadiness of the potential energy at the infinite wall. Therefore only a solution with a finite number of basis states can be performed. A slight approximation is introduced into the equation of motion in order that the trace of the density matrix remains constant in time. The equation of motion is solved by the method of searching eigenvalues. As a side-product two sums over the zeros of spherical Bessel functions are found.

scholarly journals PHYSICS OF SINGULAR POINTS IN QUANTUM MECHANICS

2003 ◽  
Vol 01 (04) ◽  
pp. 543-560 ◽  
Author(s):  
IZUMI TSUTSUI ◽  
TAMÁS FÜLÖP
Keyword(s):  
Quantum Mechanics ◽  
Berry Phase ◽  
Singular Points ◽  
One Dimension ◽  
Partition Wall ◽  
Quantum Singularities ◽  
Quantum Force ◽  
Physical Phenomena ◽  
Particle Statistics ◽  
Behavior Characteristic

Defects or junctions in materials serve as a source of interactions for particles, and in idealized limits they may be treated as singular points yielding contact interactions. In quantum mechanics, these singularities accommodate an unexpectedly rich structure and thereby provide a variety of physical phenomena, especially if their properties are controlled properly. Based on our recent studies, we present a brief review on the physical aspects of such quantum singularities in one dimension. Among the intriguing phenomena that the singularities admit, we mention strong versus weak duality, supersymmetry, quantum anholonomy (Berry phase), and a copying process by anomalous caustics. We also show that a partition wall as a singularity in a potential well can give rise to a quantum force which exhibits an interesting temperature behavior characteristic to the particle statistics.

q-Equivalent Hamiltonian Operators

1972 ◽  
Vol 50 (17) ◽  
pp. 2037-2047 ◽  
Author(s):  
M. Razavy
Keyword(s):  
Quantum Mechanics ◽  
Classical Limit ◽  
First Integral ◽  
Canonical Commutation Relation ◽  
Equation Of Motion ◽  
First Integrals ◽  
Position Operator ◽  
Integrals Of Motion ◽  
Adjoint Operators ◽  
Hamiltonian Operators

From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

scholarly journals Two-Way Communication with a Single Quantum Particle

2018 ◽  
Vol 120 (6) ◽  
Author(s):  
Flavio Del Santo ◽  
Borivoje Dakić
Keyword(s):  
Quantum Particle ◽  
Single Quantum

scholarly journals Carrying an arbitrarily large amount of information using a single quantum particle

2020 ◽  
Vol 102 (2) ◽  
Author(s):  
Li-Yi Hsu ◽  
Ching-Yi Lai ◽  
You-Chia Chang ◽  
Chien-Ming Wu ◽  
Ray-Kuang Lee
Keyword(s):  
Quantum Particle ◽  
Single Quantum ◽  
Amount Of Information

Three-dimensional quantum mechanics in a curved space based on the q-addition

2019 ◽  
Vol 34 (29) ◽  
pp. 1950177
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Coordinate System ◽  
Three Dimensional ◽  
One Dimension ◽  
Cartesian Coordinate System ◽  
Spherical Coordinate ◽  
Curved Space ◽  
Dimension Formula ◽  
Cartesian Coordinate

In this paper, we extend the theory of the [Formula: see text]-deformed quantum mechanics in one dimension[Formula: see text] into three-dimensional case. We relate the [Formula: see text]-deformed quantum theory to the quantum theory in a curved space. We discuss the diagonal metric based on [Formula: see text]-addition in the Cartesian coordinate system and core radius of neutron star. We also discuss the diagonal metric based on [Formula: see text]-addition in the spherical coordinate system and [Formula: see text]-deformed Heisenberg atom model.

scholarly journals Deformation quantization: Quantum mechanics lives and works in phase space

2014 ◽  
Vol 78 ◽  
pp. 02004
Author(s):  
Cosmas K. Zachos
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Deformation Quantization
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