CONFORMAL PARASUPERSYMMETRY IN QUANTUM MECHANICS

1989 ◽  
Vol 04 (26) ◽  
pp. 2519-2529 ◽  
Author(s):  
STEPHANE DURAND ◽  
LUC VINET
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Symmetry Algebra ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
One Dimensional ◽  
Complete Determination ◽  
Symmetry Generators

Conformal parasupersymmetry of order 2 is exemplified using a one-dimensional quantum mechanical system. Symmetry generators are seen to realize trilinear structure relations. The relevant representations of this novel symmetry algebra are constructed and shown to allow for a complete determination of the energy spectrum and wave functions of the system.

scholarly journals Schrodinger Wave Equation

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Aaron C.H. Davey
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Quantum Theory ◽  
System Change ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
One Dimensional ◽  
Erwin Schrödinger ◽  
The One ◽  
Over Time

The father of quantum mechanics, Erwin Schrodinger, was one of the most important figures in the development of quantum theory. He is perhaps best known for his contribution of the wave equation, which would later result in his winning of the Nobel Prize for Physics in 1933. The Schrodinger wave equation describes the quantum mechanical behaviour of particles and explores how the Schrodinger wave functions of a system change over time. This project is concerned about exploring the one-dimensional case of the Schrodinger wave equation in a harmonic oscillator system. We will give the solutions, called eigenfunctions, of the equation that satisfy certain conditions. Furthermore, we will show that this happens only for particular values called eigenvalues.

EXACT WAVE FUNCTIONS OF MULTI-SPECIES ANYON QUANTUM MECHANICS

1993 ◽  
Vol 08 (29) ◽  
pp. 5101-5113 ◽  
Author(s):  
K.H. CHO ◽  
CHAIHO RIM ◽  
D.S. SOH
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Angular Momentum ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Potential Well ◽  
Quantum Mechanical System ◽  
Ladder Operators ◽  
Free System ◽  
The Many

We investigate the many-anyon quantum-mechanical system of multi-species and obtain algebraically the energy and the angular momentum eigenstates in a harmonic potential well, in a constant magnetic field and of free system using ladder operators. The eigenvalues and their multiplicity of the eigenstates are given.

scholarly journals Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

2018 ◽  
Vol 4 (1) ◽  
pp. 47-55
Author(s):  
Timothy Brian Huber
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Mechanical System ◽  
Schrodinger Equation ◽  
Supersymmetric Quantum Mechanics ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Intertwining Operator

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

WKB method for potentials unbounded from below

2016 ◽  
Vol 30 (03) ◽  
pp. 1650003 ◽  
Author(s):  
Aleksandar Demić ◽  
Vitomir Milanović ◽  
Jelena Radovanović ◽  
Milenko Musić
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Wave Functions ◽  
Wkb Method ◽  
Symmetric Potential ◽  
Overlap Integral ◽  
Entire Spectrum ◽  
One Dimensional ◽  
Model Based ◽  
Exactly Solvable

Bound states degenerated in energy (and differing in parity) may form in one-dimensional quantum mechanics if the potential is unbounded from below. We focus on symmetric potential and present quasi-exactly solvable (QES) model based on WKB method. The application of this method is limited on slow-changing potentials. We consider the overlap integral of WKB wave functions [Formula: see text] and [Formula: see text] which correspond to energies [Formula: see text] and [Formula: see text], and by setting [Formula: see text], we determine the type of spectrum depending on parameter [Formula: see text] which arises from this method. For finite value [Formula: see text], we show that the entire spectrum will consist of degenerated bound states.

scholarly journals ALTERNATIVE CONFORMAL QUANTUM MECHANICS

2011 ◽  
Vol 26 (16) ◽  
pp. 2735-2742 ◽  
Author(s):  
S.-H. HO
Keyword(s):  
Quantum Mechanics ◽  
Conformal Invariance ◽  
Invariant Theory ◽  
Mechanical Model ◽  
Conformal Transformation ◽  
Quantum Mechanical ◽  
Scale Invariant ◽  
One Dimensional ◽  
Quantum Mechanical Model ◽  
Conformal Quantum Mechanics

We investigate a one-dimensional quantum mechanical model, which is invariant under translations and dilations but does not respect the conventional conformal invariance. We describe the possibility of modifying the conventional conformal transformation such that a scale invariant theory is also invariant under this new conformal transformation.

scholarly journals Quantum SU(2|1) supersymmetric ℂN Smorodinsky-Winternitz system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Evgeny Ivanov ◽  
Armen Nersessian ◽  
Stepan Sidorov
Keyword(s):  
Quantum Mechanics ◽  
Energy Levels ◽  
Wave Functions ◽  
Supersymmetric Extension ◽  
Quantum Properties ◽  
Hidden Symmetry ◽  
Euclidian Space ◽  
Symmetry Operators ◽  
Symmetry Generators ◽  
Equivalent Description

Abstract We study quantum properties of SU(2|1) supersymmetric (deformed $$ \mathcal{N} $$ N = 4, d = 1 supersymmetric) extension of the superintegrable Smorodinsky-Winternitz system on a complex Euclidian space ℂN. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU(2|1) supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the “even-odd” splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU(2|1) multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU(2|1, 1) quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU(2|1, 1) ones is found.

Complete determination of the conical Mueller matrix for one-dimensional rough metallic surfaces

2006 ◽  
Vol 257 (1) ◽  
pp. 62-71 ◽  
Author(s):  
Rafael Espinosa-Luna ◽  
Gelacio Atondo-Rubio ◽  
Alberto Mendoza-Suárez
Keyword(s):  
Mueller Matrix ◽  
Metallic Surfaces ◽  
One Dimensional ◽  
Complete Determination

Upper bounds to the overlap between approximate and exact wave functions

1970 ◽  
Vol 48 (14) ◽  
pp. 1681-1686 ◽  
Author(s):  
Maurice Cohen ◽  
Tova Feldmann
Keyword(s):  
Trial Function ◽  
Practical Method ◽  
Hamiltonian Operator ◽  
Upper Bounds ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Lower And Upper Bounds ◽  
Present Procedure ◽  
Classical Procedure

Rigorous lower and upper bounds to the eigenvalues E of a quantum-mechanical system with Hamiltonian operator H are derived from the Gramian determinant of a set of suitably chosen functions. This procedure, which also yields a rigorous upper bound to the overlap between a given trial function and the corresponding (unknown) exact eigenfunction, is shown to be equivalent to a generalization of the classical procedure of Weinstein. In the absence of a rigorous lower bound to the overlap, the present procedure provides a practical method of assessing the influence of the ground state on a given trial function for an excited state.

Probability in quantum mechanics

1978 ◽  
Vol 10 (4) ◽  
pp. 725-729 ◽  
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

Probability in quantum mechanics

1978 ◽  
Vol 10 (04) ◽  
pp. 725-729
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

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