314 research outputs found
Canonical Discretization. I. Discrete faces of (an)harmonic oscillator
A certain notion of canonical equivalence in quantum mechanics is proposed.
It is used to relate quantal systems with discrete ones. Discrete systems
canonically equivalent to the celebrated harmonic oscillator as well as the
quartic and the quasi-exactly-solvable anharmonic oscillators are found. They
can be viewed as a translation-covariant discretization of the (an)harmonic
oscillator preserving isospectrality. The notion of the deformation of the
canonical equivalence leading to a dilatation-covariant discretization
preserving polynomiality of eigenfunctions is also presented.Comment: 29 pages, LaTe
On polynomial solutions of differential equations
A general method of obtaining linear differential equations having polynomial
solutions is proposed. The method is based on an equivalence of the spectral
problem for an element of the universal enveloping algebra of some Lie algebra
in the "projectivized" representation possessing an invariant subspace and the
spectral problem for a certain linear differential operator with variable
coefficients. It is shown in general that polynomial solutions of partial
differential equations occur; in the case of Lie superalgebras there are
polynomial solutions of some matrix differential equations, quantum algebras
give rise to polynomial solutions of finite--difference equations.
Particularly, known classical orthogonal polynomials will appear when
considering acting on . As examples, some
polynomials connected to projectivized representations of ,
, and are briefly discussed.Comment: 12p
One-electron atomic-molecular ions containing Lithium in a strong magnetic field
The one-electron Li-containing Coulomb systems of atomic type and
molecular type , and are studied in
the presence of a strong magnetic field a.u. in the
non-relativistic framework. They are considered at the Born-Oppenheimer
approximation of zero order (infinitely massive centers) within the parallel
configuration (molecular axis parallel to the magnetic field). The variational
and Lagrange-mesh methods are employed in complement to each other. It is
demonstrated that the molecular systems , and
can exist for sufficiently strong magnetic fields a.u. and that can even be stable at
magnetic fields typical of magnetars.Comment: 22 pages, 9 figures, 4 table
Particular Integrability and (Quasi)-exact-solvability
A notion of a particular integrability is introduced when two operators
commute on a subspace of the space where they act. Particular integrals for
one-dimensional (quasi)-exactly-solvable Schroedinger operators and
Calogero-Sutherland Hamiltonians for all roots are found. In the classical case
some special trajectories for which the corresponding particular constants of
motion appear are indicated.Comment: 13 pages, typos correcte
The molecular ion in a magnetic field
A detailed study of the low-lying electronic states
{}^1\Si,{}^3\Si,{}^3\Pi,{}^3\De of the molecular ion in parallel
to a magnetic field configuration (when \al-particle and proton are situated
on the same magnetic line) is carried out for G in
the Born-Oppenheimer approximation. The variational method is employed using a
physically adequate trial function. It is shown that the parallel configuration
is stable with respect to small deviations for \Si-states. The quantum
numbers of the ground state depend on the magnetic field strength. The ground
state evolves from the spin-singlet {}^1\Si state for small magnetic fields
a.u. to the spin-triplet {}^3\Si unbound state for
intermediate fields and to the spin-triplet strongly bound state for a.u. When the molecular ion exists, it is stable with
respect to a dissociation.Comment: 13 pages, 5 figures, 4 table
A note about the ground state of the hydrogen molecular ion
Three simple parametric trial functions for the molecular ion are presented. Each of them provides subsequently the
most accurate approximation for the Born-Oppenheimer ground state energy among
several-parametric trial functions. These trial functions are chosen following
a criterion of physical adequacy and includes the electronic correlation in the
exponential form , where is a variational
parameter. The Born-Oppenheimer energy is found to be \,a.u., respectively, for optimal equilateral triangular
configuration of protons with the equilibrium interproton distance
\,a.u. The variational energy agrees in three significant digits (s.d.)
with most accurate results available at present as well as for major
expectation values.Comment: 12 pages, 1 figure, 3 table
Two electrons in an external oscillator potential: hidden algebraic structure
It is shown that the Coulomb correlation problem for a system of two
electrons (two charged particles) in an external oscillator potential possesses
a hidden -algebraic structure being one of recently-discovered
quasi-exactly-solvable problems. The origin of existing exact solutions to this
problem, recently discovered by several authors, is explained. A degeneracy of
energies in electron-electron and electron-positron correlation problems is
found. It manifests the first appearence of hidden -algebraic structure
in atomic physics.Comment: 7 pages (plus one figure avaliable via direct request), LaTeX,
Preprint IFUNAM FT 94-4
molecular ions can exist in strong magnetic fields
Using the variational method it is shown that for magnetic fields G there can exist a molecular ion .Comment: LaTeX, 7 pp, 1 table, 4 figures. Title modified. Consideration of the
longitudinal size of the system was adde
Energy Reflection Symmetry of Lie-Algebraic Problems: Where the Quasiclassical and Weak Coupling Expansions Meet
We construct a class of one-dimensional Lie-algebraic problems based on sl(2)
where the spectrum in the algebraic sector has a dynamical symmetry E -> - E.
All 2j+1 eigenfunctions in the algebraic sector are paired, and inside each
pair are related to each other by simple analytic continuation x -> ix, except
the zero mode appearing if j is integer. At j-> infinity the energy of the
highest level in the algebraic sector can be calculated by virtue of the
quasiclassical expansion, while the energy of the ground state can be
calculated as a weak coupling expansion. The both series coincide identically.Comment: Latex, 16 pages, 3 figures. Minor styllistic changes made, typos
corrected, a remark on the energy-reflection symmetry in the
quantum-algebraic Hamiltonians emerging in finite-difference problems added.
Final version, to be published in Physical Review
Charged Hydrogenic, Helium and Helium-Hydrogenic Molecular Chains in a Strong Magnetic Field
A non-relativistic classification of charged molecular hydrogenic, helium and
mixed helium-hydrogenic chains with one or two electrons which can exist in a
strong magnetic field G is given. It is shown that for
both cases at the strongest studied magnetic fields the longest
hydrogenic chain contains at most five protons indicating to the existence of
the and ions, respectively. In the case of the
helium chains the longest chains can exist at the strongest studied magnetic
fields with three and four \al-particles for cases, respectively. For
mixed helium-hydrogenic chains the number of heavy centers can reach five for
highest magnetic fields studied. In general, for a fixed magnetic field
two-electron chains are more bound than one-electron ones.Comment: 32 pages, 2 figures, 9 table
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