237 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
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
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
The H molecular ion: a solution
Combining the WKB expansion at large distances and Perturbation Theory at
small distances it is constructed a compact uniform approximation for
eigenfunctions. For lowest states 1s\si_{g} and 2p\si_{u} this
approximation provides the relative accuracy (5 s.d.) for
any real in eigenfunctions and for total energy it gives 10-11 s.d.
for internuclear distances . Corrections to proposed
approximations are evaluated. Separation constants and the oscillator strength
for the transition 1s\si_{g} \rar 2p\si_{u} are calculated and compared with
existing data.Comment: 16 pages, 4 figures, 6 tables, typos are corrected and small
additions are inserted, to be published at JPB (fast track comm
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
The H ion in a strong magnetic field. Lowest excited states
As a continuation of our previous work ({\it Phys. Rev. A68, 012504 (2003)})
an accurate study of the lowest 1\si_g and the low-lying excited 1\si_u,
2\si_g, , 1\de_{g,u} electronic states of the molecular ion
is made. Since the parallel configuration where the molecular axis
coincides with the magnetic field direction is optimal, this is the only
configuration which is considered. The variational method is applied and the
{\it same} trial function is used for different magnetic fields. The magnetic
field ranges from to where non-relativistic
considerations are justified. Particular attention is paid to the 1\si_u
state which was studied for an arbitrary inclination. For this state a
one-parameter vector potential is used which is then variationally optimized.Comment: 25 pages, 2 figure
Hydrogen atom in a magnetic field: The quadrupole moment
The quadrupole moment of a hydrogen atom in a magnetic field for field
strengths from 0 to 4.414e13 G is calculated by two different methods. The
first method is variational, and based on a single trial function. The second
method deals with a solution of the Schroedinger equation in the form of a
linear combination of Landau orbitals.Comment: 4 pages, 1 figure, 1 table; RevTeX. Final (proofs-stage) version of
the text; corrected numbers in Table 1 and in Eq.(15
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