Generalized Electromagnetic Fields Associated with the Hydrogen-Like Atom Problem

2018 ◽  
Vol 74 (1) ◽  
pp. 43-50 ◽  
Author(s):  
S.A. Bruce ◽  
J.F. Diaz-Valdes
Keyword(s):  
Quantum Mechanics ◽  
Electromagnetic Coupling ◽  
Relativistic Quantum ◽  
Scalar Potential ◽  
Symmetry Algebra ◽  
Dynamical Symmetry ◽  
Canonical Commutation Relation ◽  
Quantum Systems ◽  
Minimal Coupling ◽  
Exactly Solvable

AbstractIt is known that the principle of minimal coupling in quantum mechanics determines a unique interaction form for a charged particle. By properly redefining the canonical commutation relation between (canonical) conjugate components of position and momentum of the particle, e.g. an electron, we restate the Dirac equation for the hydrogen-like atom problem incorporating a generalized minimal electromagnetic coupling. The corresponding interaction keeps the $1/\left|\mathbf{q}\right|$ dependence in both the scalar potential $V\left({\left|\mathbf{q}\right|}\right)$ and the vector potential $\mathbf{A}\left(\mathbf{q}\right)$ ($\left|{\mathbf{A}\left(\mathbf{q}\right)}\right|\sim 1/\left|\mathbf{q}\right|$). This problem turns out to be exactly solvable; moreover, the eigenstates and eigenvalues can be obtained in an elementary fashion. Some feasible models within this approach are discussed. Then we make a few remarks about the breaking of supersymmetry. Finally, we briefly comment on the possible Lie algebra (dynamical symmetry algebra) of these relativistic quantum systems.

Ordering Problems in Fermionic Relativistic Quantum Mechanics

1997 ◽  
Vol 12 (01) ◽  
pp. 243-248 ◽  
Author(s):  
Rodolfo P. Martínez Y Romero ◽  
Antonio Del Sol Mesa
Keyword(s):  
Quantum Mechanics ◽  
Degrees Of Freedom ◽  
Relativistic Quantum ◽  
Typical Case ◽  
Relativistic Quantum Mechanics ◽  
Dirac Oscillator ◽  
Minimal Coupling ◽  
Relativistic Systems

We discuss the existence of ambiguities, or anomalies of fermionic nature as we call them, in the quantization of relativistic systems with odd Grassmann degrees of freedom. We propose in this work a way of avoiding such ambiguities in the case of relativistic quantum mechanics, by including the odd degrees of freedom into a generalization of the momentum, in a similar manner to the minimal coupling. We illustrate our results with some examples, including the Dirac oscillator as a typical case of the problems we are dealing with.

scholarly journals Exactly and Quasi-Exactly Solvable Models on the Basis of OSP(2|1)

1997 ◽  
Vol 12 (22) ◽  
pp. 1655-1661
Author(s):  
A. Shafiekhani ◽  
M. Khorrami
Keyword(s):  
Symmetry Algebra ◽  
Dynamical Symmetry ◽  
One Dimension ◽  
Exactly Solvable Models ◽  
Solvable Models ◽  
Exactly Solvable ◽  
Solvable Problems

The exactly and quasi-exactly solvable problems for spin one-half in one dimension on the basis of a hidden dynamical symmetry algebra of Hamiltonian are discussed. We take the supergroup, OSP(2|1), as such a symmetry. A number of exactly solvable examples are considered and their spectrum are evaluated explicitly. Also, a class of quasi-exactly solvable problems on the basis of such a symmetry has been obtained.

A new theoretical study of the deformed unequal scalar and vector Hellmann plus modified Kratzer potentials within the deformed Klein–Gordon equation in RNCQM symmetries

2021 ◽  
Author(s):  
Abdelmadjid Maireche
Keyword(s):  
Quantum Mechanics ◽  
Yukawa Potential ◽  
Relativistic Quantum ◽  
Scalar Potential ◽  
Relativistic Quantum Mechanics ◽  
Energy Eigenvalues ◽  
Quantum Numbers ◽  
Noncommutative Quantum Mechanics ◽  
Hellmann Potential ◽  
Klein Gordon

In this paper, within the framework of relativistic quantum mechanics and using the improved approximation scheme to the centrifugal term for any [Formula: see text]states via Bopp’s shift method and standard perturbation theory, we have obtained the modified energy eigenvalues of a newly proposed modified unequal vector and scalar Hellmann plus modified Kratzer potentials (DUVSHMK-Ps) for some diatomic N2, I2, CO, NO, O2 and HCl molecules. This study includes corrections of the first-order in noncommutativity parameters [Formula: see text]. This potential is a superposition of the attractive Coulomb Yukawa potential plus the Kratzer potential and new central terms appear as a result of the effects of noncommutativity properties of space–space. The obtained energy eigenvalues appear as a function of noncommutativity parameters, the strength parameters [Formula: see text] and [Formula: see text] of the (scalar vector) Hellmann potential, the screening range parameter [Formula: see text], the dissociation energy of the vector, and scalar potential [Formula: see text], the equilibrium inter-nuclear distance [Formula: see text] in addition to the atomic quantum numbers [Formula: see text]. Furthermore, we obtained the corresponding modified energy of DUVSHMK-Ps in the symmetries of non-relativistic noncommutative quantum mechanics (NRNCQM). In both relativistic and non-relativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM.

scholarly journals Exactly and quasi-exactly solvable ‘discrete’ quantum mechanics

2011 ◽  
Vol 369 (1939) ◽  
pp. 1301-1318 ◽  
Author(s):  
Ryu Sasaki
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Essential Role ◽  
Dynamical Symmetry ◽  
Oscillator Algebra ◽  
Discrete Variable ◽  
Shape Invariance ◽  
One Dimensional ◽  
Exactly Solvable ◽  
Creation Operators

A brief introduction to discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators and dynamical symmetry algebras, including the q -oscillator algebra and the Askey–Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable (QES) Hamiltonians in one-dimensional ‘discrete’ quantum mechanics is presented. It reproduces all the known Hamiltonians whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and QES Hamiltonians are constructed. The sinusoidal coordinate plays an essential role.

scholarly journals MINIMAL BOSONIZATION OF SUPERSYMMETRY

1996 ◽  
Vol 11 (05) ◽  
pp. 397-408 ◽  
Author(s):  
MIKHAIL S. PLYUSHCHAY
Keyword(s):  
Quantum Mechanics ◽  
Symmetry Algebra ◽  
Dynamical Symmetry ◽  
Degree Of Freedom ◽  
Supersymmetric Quantum Mechanics ◽  
Supersymmetric Extension ◽  
Calogero Model ◽  
Parity Operator ◽  
Relationship Of ◽  
Bosonic Oscillator

The minimal bosonization of supersymmetry in terms of one bosonic degree of freedom is considered. A nontrivial relationship of the construction to the Witten supersymmetric quantum mechanics is illustrated with the help of the simplest N=2 SUSY system realized on the basis of the ordinary (undeformed) bosonic oscillator. It is shown that the generalization of such a construction to the case of Vasiliev deformed bosonic oscillator gives a supersymmetric extension of the two-body Calogero model in the phase of exact or spontaneously broken N=2 SUSY. The construction admits an extension to the case of the OSp(2|2) supersymmetry, and, as a consequence, osp(2|2) superalgebra is revealed as a dynamical symmetry algebra for the bosonized supersymmetric Calogero model. Realizing the Klein operator as a parity operator, we construct the bosonized Witten supersymmetric quantum mechanics. Here the general case of the corresponding bosonized N=2 SUSY is given by an odd function being a superpotential.

Quantum mechanics

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Transition Amplitude ◽  
External Source ◽  
Quantum Field ◽  
Damping Term ◽  
Relativistic Quantum Theory ◽  
Generating Functional

After a brief review of the operator approach to quantum mechanics, Feynmans path integral, which expresses a transition amplitude as a sum over all paths, is derived. Adding a linear coupling to an external source J and a damping term to the Lagrangian, the ground-state persistence amplitude is obtained. This quantity serves as the generating functional Z[J] for n-point Green functions which are the main target when studying quantum field theory. Then the harmonic oscillator as an example for a one-dimensional quantum field theory is discussed and the reason why a relativistic quantum theory should be based on quantum fields is explained.

scholarly journals The internal conversion of gamma-rays.—Part II

1928 ◽  
Vol 121 (787) ◽  
pp. 447-456
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Gamma Rays ◽  
Internal Conversion ◽  
Scalar Potential ◽  
X Ray ◽  
Electro Magnetic Field ◽  
Γ Rays ◽  
Electro Magnetic

In a previous paper the absorption of γ-rays in the K-X-ray levels of the atom in which they are emitted was calculated according to the Quantum Mechanics, supposing the γ-rays to be emitted from a doublet of moment f ( t ) at the centre of the atom. The non-relativity wave equation derived from the relativity wave equation for an electron of charge — ε moving in an electro-magnetic field of vector potential K and scalar potential V is h 2 ∇ 2 ϕ + 2μ ( ih ∂/∂ t + εV + ih ε/μ c (K. grad)) ϕ = 0. (1) Suppose, however, that K involves the space co-ordinates. Then, (K. grad) ϕ ≠ (grad . K) ϕ , and the expression (K . grad) ϕ is not Hermitic. Equation (1) cannot therefore be the correct non-relativity wave equation for a single electron in an electron agnetic field, and we must substitute h 2 ∇ 2 ϕ + 2μ ( ih ∂/∂ t + εV) ϕ + ih ε/ c ((K. grad) ϕ + (grad. K) ϕ ) = 0. (2)

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