New structure in the energy spectrum of reggeon quantum mechanics with quartic couplings

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.

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Dirac Equation in the Presence of Hartmann and Ring-Shaped Oscillator Potentials

Advances in High Energy Physics ◽

10.1155/2018/1940925 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9

Author(s):

Zahra Bakhshi

Keyword(s):

Quantum Mechanics ◽

Energy Spectrum ◽

Dirac Equation ◽

Bound States ◽

Scalar Potential ◽

Solvable Models ◽

Nonrelativistic Quantum ◽

Physical Conditions ◽

Basic Model ◽

Nonrelativistic Quantum Mechanics

The importance of the energy spectrum of bound states and their restrictions in quantum mechanics due to the different methods have been used for calculating and determining the limit of them. Comparison of Schrödinger-like equation obtained by Dirac equation with the nonrelativistic solvable models is the most efficient method. By this technique, the exact relativistic solutions of Dirac equation for Hartmann and Ring-Shaped Oscillator Potentials are accessible, when the scalar potential is equal to the vector potential. Using solvable nonrelativistic quantum mechanics systems as a basic model and considering the physical conditions provide the changes in the restrictions of relativistic parameters based on the nonrelativistic definitions of parameters.

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SPECTRUM DEGENERACY OF GENERAL (p=2)-PARASUPERSYMMETRIC QUANTUM MECHANICS AND PARASUPERSYMMETRIC TOPOLOGICAL INVARIANTS

International Journal of Modern Physics A ◽

10.1142/s0217751x96000493 ◽

1996 ◽

Vol 11 (06) ◽

pp. 1057-1076 ◽

Cited By ~ 15

Author(s):

ALI MOSTAFAZADEH

Keyword(s):

Quantum Mechanics ◽

Energy Spectrum ◽

Topological Invariant ◽

Supersymmetric Quantum Mechanics ◽

Topological Invariants ◽

Witten Index

A thorough analysis of the general features of (p=2)-parasupersymmetric quantum mechanics is presented. It is shown that for both the Rubakov–Spiridonov (RS) formulation and the Beckers–Debergh (BD) formulation of (p=2)-parasupersymmetric quantum mechanics, the degeneracy structure of the energy spectrum can be derived using the defining parasuperalgebras. Thus the results of this article are independent of the details of the Hamiltonian. In fact, they are valid for arbitrary systems based on arbitrary-dimensional coordinate manifolds. In particular, the RS and BD systems possess identical degeneracy structures. For a subclass of RS (alternatively, BD) systems, a new topological invariant is introduced. This is a counterpart of the Witten index of supersymmetric quantum mechanics.

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Quantum mechanics of electromagnetically bounded spin-1/2 particles in expanding universes: II. Energy spectrum of the hydrogen atom

General Relativity and Gravitation ◽

10.1007/bf00759543 ◽

1978 ◽

Vol 9 (6) ◽

pp. 489-500 ◽

Cited By ~ 39

Author(s):

J. Audretsch ◽

G. Sch�fer

Keyword(s):

Quantum Mechanics ◽

Energy Spectrum ◽

Hydrogen Atom

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Bound states of the D-dimensional Schrödinger equation for the generalized Woods–Saxon potential

Modern Physics Letters A ◽

10.1142/s0217732319501074 ◽

2019 ◽

Vol 34 (14) ◽

pp. 1950107 ◽

Cited By ~ 1

Author(s):

V. H. Badalov ◽

B. Baris ◽

K. Uzun

Keyword(s):

Quantum Mechanics ◽

Energy Spectrum ◽

Schrödinger Equation ◽

Schrodinger Equation ◽

Wave Functions ◽

Basis Set ◽

Saxon Potential ◽

Radial Wave ◽

Energy Eigenvalues ◽

Approximate Analytical Solutions

The formal framework for quantum mechanics is an infinite number of dimensional space. Hereby, in any analytical calculation of the quantum system, the energy eigenvalues and corresponding wave functions can be represented easily in a finite-dimensional basis set. In this work, the approximate analytical solutions of the hyper-radial Schrödinger equation are obtained for the generalized Wood–Saxon potential by implementing the Pekeris approximation to surmount the centrifugal term. The energy eigenvalues and corresponding hyper-radial wave functions are derived for any angular momentum case by means of state-of-the-art Nikiforov–Uvarov and supersymmetric quantum mechanics methods. Hence, the same expressions are obtained for the energy eigenvalues, and the expression of hyper-radial wave functions transforming each other is shown owing to these methods. Furthermore, a finite number energy spectrum depending on the depths of the potential well [Formula: see text] and [Formula: see text], the radial [Formula: see text] and [Formula: see text] orbital quantum numbers and parameters [Formula: see text], [Formula: see text], [Formula: see text] are also identified in detail. Next, the bound state energies and corresponding normalized hyper-radial wave functions for the neutron system of the [Formula: see text]Fe nucleus are calculated in [Formula: see text] and [Formula: see text] as well as the energy spectrum expressions of other higher dimensions are revealed by using the energy spectrum of [Formula: see text] and [Formula: see text].

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The “accidental” degeneracy of the hydrogen atom is no accident

Canadian Journal of Physics ◽

10.1139/cjp-2014-0300 ◽

2015 ◽

Vol 93 (3) ◽

pp. 312-317

Author(s):

P.C. Deshmukh ◽

Aarthi Ganesan ◽

N. Shanthi ◽

Blake Jones ◽

James Nicholson ◽

...

Keyword(s):

Quantum Mechanics ◽

Energy Spectrum ◽

Hydrogen Atom ◽

Schrödinger Equation ◽

Atomic Physics ◽

Schrodinger Equation ◽

Spin States ◽

Interesting Aspect ◽

Accidental Degeneracy ◽

Pedagogical Analysis

The Schrödinger equation does not account for the 2n2degeneracy of the hydrogen atom, which it dismisses as an “accidental” degeneracy. The factor of “2” in the 2n2degeneracy is well-accounted-for in the relativistic formulation by the two spin states of the electron. The n2degeneracy is nevertheless not quite an “accident”; it is due to the SO(4), rather than SO(3), symmetry of the hydrogen atom. This result is well known, but is inadequately commented upon in most courses in quantum mechanics and atomic physics, leaving the student wondering about the origins of the n2degeneracy of the hydrogen atom. A pedagogical analysis of this interesting aspect, which highlights the fundamental principles of quantum mechanics, is presented in this article. While doing so, not only is the n2degeneracy of the hydrogen atom explained, but its energy spectrum and eigenfunctions are obtained without even using the Schrödinger equation, employing only the fundamental principles of quantum mechanics rather than the Schrödinger equation.

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Time Observables in a Timeless Universe

Quantum ◽

10.22331/q-2020-10-29-354 ◽

2020 ◽

Vol 4 ◽

pp. 354

Author(s):

Tommaso Favalli ◽

Augusto Smerzi

Keyword(s):

Quantum Mechanics ◽

Energy Spectrum ◽

Hermitian Operator ◽

Dynamical Evolution ◽

Emergent Property ◽

Time Operator ◽

Static Universe ◽

Unequally Spaced ◽

Energy Ratios ◽

Time In Quantum Mechanics

Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially undistinguishables.

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Confined systems with a linear energy spectrum

Modern Physics Letters A ◽

10.1142/s0217732321500644 ◽

2021 ◽

Vol 36 (10) ◽

pp. 2150064

Author(s):

A. D. Alhaidari ◽

T. J. Taiwo

Keyword(s):

Quantum Mechanics ◽

Energy Spectrum ◽

Orthogonal Polynomials ◽

Bound States ◽

Quantum Systems ◽

Potential Functions ◽

Physical Parameters ◽

Confined Systems

Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we study quantum systems totally confined in space and with a linearly spaced energy spectrum. We present several examples of such systems, derive their corresponding potential functions, and plot some of their bound states.

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