1,308 research outputs found
Exchange Interactions and Principle of Minimal Energy in Relativistic Schroedinger Theory
The principle of minimal energy, which has been set up in the preceding
papers for systems of non-identical particles (e.g. positronium), is now
generalized to include also identical particles. Since the latter kind of
particles feels also the exchange forces (besides the usual electromagnetic
forces), one has to deal with non-zero exchange potentials which render the
theory nonlinear, according to the non-Abelian character of Relativistic
Schroedinger Theory (RST). However, the present extension of the variational
principle refers only to the linearized version of RST in order to keep the
calculations sufficiently simple. It is also demonstrated that in RST the Dirac
particles can occur in fermionic and bosonic quantum states; and the
mathematical and physical consistency of the variational principle is validated
for both types of states (concretely the fermionic hydrogen state 2p_3/2 and
the bosonic positronium state 2^1P_1).Comment: 110 pages, 1 figur
Principle of Minimal Energy in Relativistic Schroedinger Theory
The Hamilton-Lagrange action principle for Relativistic Schr\"odinger Theory
(RST) is converted to a variational principle (with constraints) for the
stationary bound states. The groundstate energy is the minimally possible value
of the corresponding energy functional and the relativistic energy eigenvalue
equations do appear as the corresponding variational equations. The matter part
of these eigenvalue equations is a relativistic generalization of the
well-known Ritz principle in non-relativistic quantum mechanics which however
disregards the dynamical character of the particle interactions. If the latter
are included in the proposed principle of minimal energy for the bound states,
one obtains a closed dynamical system for both matter and gauge fields. The new
variational principle enables the development of variational techniques for
solving approximately the energy eigenvalue equations. As a demonstration, the
positronium groundstate is treated in great detail. Here a simple exponential
trial function is sufficient in order to reproduce the (exact) result of
conventional quantum mechanics where the relativistic and spin effects are
neglected.Comment: 65 page
Spherically Symmetric Approximation (and beyond) in Relativistic Schroedinger Theory
The energy eigenvalue problem of non-relativistic positronium is considered
within the framework of Relativistic Schroedinger Theory (RST), and the results
are compared to those of the conventional quantum theory. For the range of
princi- pal quantum numbers n = 2;3;::: ;30, the RST predictions for the
non-relativistic positronium energies deviate now from the corresponding
predictions of the conven- tional quantum theory at an average of (roughly) 3%.
These results suggest that the deviations will be further diminished in the
higher orders of approximation.Comment: 150 pages, 10 figures and 4 table
Non-Relativistic Positronium Spectrum in Relativistic Schroedinger Theory
The lowest energy levels of positronium are studied in the non-relativistic
approximation within the framework of Relativistic Schr\"odinger Theory (RST).
Since it is very difficult to find the exact solutions of the RST field
equations (even in the non-relativistic limit), an approximation scheme is set
up on the basis of the hydrogen-like wave functions (i.e. polynomial times
exponential). For any approximation order \NN (\NN=0,1,2,3,...) there arises
a spectrum of approximate RST solutions with the associated energies, quite
similarly to the conventional treatment of positronium in the standard quantum
theory (Appendix). For the lowest approximation order (\NN=0) the RST
prediction for the \emph{groundstate} energy exactly agrees with the
conventional prediction of the standard theory. However for the higher
approximation orders (\NN=1,2,3), the corresponding RST prediction differs
from the conventional result by (roughly) which confirms the
previous estimate of the error being due to the use of the spherically
symmetric approximation. The excited states require the application of
higher-order approximations (\NN>>3) and are therefore not adequately
described by the present orders (\NN\le 3).Comment: 67 pages and 3 figure
Magnetic Interactions in Relativistic Two Particle Systems
The magnetic interactions of the two electrons in helium-like ions are
studied in detail within the framework of Relativistic Schroedinger Theory
(RST). The general results are used to compute the ground-state interaction
energy of some highly-ionized atoms ranging from germanium (Z=32) up to bismuth
(Z=83). When the magnetic interaction energy is added to its electric
counterpart resulting from the electrostatic approximation, the present RST
predictions reach a similar degree of precision (relative to the experimental
data) as the other theoretical approaches known in the literature. However
since the RST magnetism is then treated only in lowest-order approximation,
further improvements of the RST predictions seem possible.Comment: 60 pages and 1 figur
Positive and Negative Charges in Relativistic Schroedinger Theory
Relativistic Schroedinger Theory (RST), as a general gauge theory for the
description of relativistic N-particle systems, is shown to be a mathematically
consistent and physically reasonable framework for an arbitrary assemblage of
positive and negative charges. The electromagnetic plus exchange interactions
within the subset of {\it identical} particles are accounted for in a
consistent way, whereas {\it different} particles can undergo only the
electromagnetic interactions. The origin of this different interaction
mechanism for the subsets of identical and non-identical particles is traced
back to the fundamental conservation laws for charge and energy-momentum: in
order that these conservation laws can hold also for different particles, the
structure group of the fibre bundles must be reduced to its
maximal Abelian subgroup , which eliminates the exchange part of the bundle connection.
The persisting Abelian gauge symmetry adopts the meaning of the proper gauge
group for the electromagnetic interactions which apply to the identical and
non-identical particles in the same way. Thus in RST there is an intrinsic
dynamical foundation of the emergence of exchange effects for identical
particles, whereas the conventional theory is invaded by the exchange
phenomenon via a purely kinematical postulate, namely the antisymmetrization
postulate for the wave functions due to Pauli's exclusion principle. As a
concrete demonstration, a three-particle system is considered which consists of
a positively charged particle of arbitrary rest mass and of two negatively
charged particles of equal spin, mass and charge (e.g. electrons).Comment: 57 page
Geometry and Topology of Relativistic Two-Particle Quantum Mixtures
Within the framework of Relativistic Schroedinger Theory (an alternative form
of quantum mechanics for relativistic many-particle systems) it is shown that a
general N-particle system must occur in one of two forms: either as a
``positive'' or as a ``negative'' mixture, in analogy to the fermion-boson
dichotomy of matter in the conventional theory. The pure states represent a
limiting case between the two types of mixtures which themselves are considered
as the RST counterparts of the entangled (fermionic or bosonic) states of the
conventional quantum theory. Both kinds of mixtures are kept separated from
dynamical as well as from topological reasons. The 2-particle configurations
(N=2) are studied in great detail with respect to their geometric and
topological properties which are described in terms of the Euler class of an
appropriate bundle connection. If the underlying space-time manifold (as the
base space of the fibre bundles applied) is parallelisable, the 2-particle
configurations can be thought to be generated geometrically by an appropriate
(2+2) splitting of the local tangent space.Comment: 35 pages and 1 figur
Positronium Groundstate in Relativistic Schroedinger Theory
The usefulness of the Relativistic Schr\"odinger Theory (RST) is studied in
the field of atomic physics. As a concrete demonstration, the positronium
groundstate is considered in great detail; especially the groundstate energy
is worked out in the non-relativistic approximation and under neglection
of the magnetic interactions between the positron and the electron. The
corresponding RST prediction misses the analogous
conventional Schr\"odinger result but is closer to the
latter than the corresponding Hartree approximation . The missing
binding energy of can be attributed to the approximative
use of an SO(3) symmetric interaction potential which in RST, however, is
actually only SO(2) invariant against rotations around the z-axis. It is
expected that, with the correct use of an anisotropic interaction potential due
to the SO(2) symmetry, the RST predictions will come even closer to the
conventional Schr\"odinger result, where however the mathematical structure of
RST relies on exotic (i.e. double-valued) wave functions and on the
corresponding unconventional interaction potentials (e.g. Struve-Neumann
potential).Comment: 80 pages and 2 figure
Relativistic Energy Levels of Para-Helium
The practical usefulness of Relativistic Schr\"odinger Theory (RST) is tested
by calculating approximately the energy difference between the excited singlet
state and the ground state of the helium-like
ions with arbitrary charge number . The
results are compared to the corresponding predictions of other theoretical
approaches in the literature and to the experimental data. Since the exact
solutions of the RST energy eigenvalue problem are unknown, one has to resort
to approximative methods. However the crudest approximation (``spherically
symmetric approximation'') yields relatively accurate results so that it seems
worth while to develop more powerful approximation techniquesComment: 45 pages, 2 figure
Helium Multiplet Structure in Relativistic Schr\"odinger Theory
The emergence of a multiplet structure of the helium-like ions is studied
within Relativistic Schr\"odinger Theory (RST), a fluid-dynamic approach to the
relativistic quantum theory of the many-particle systems. The fluid-dynamic
character of RST demands to specify the electronic current densities \jmu for
any -particle configuration which is exemplified here by considering the
helium singlet () and triplet () states in great detail.
Since the use of densities in RST is based upon the concept of wave functions,
the new theory appears as a certain kind of (relativistic) unification of the
conventional wave function formalism and the density functional theory, which
both are the most prominent theoretical tools in atomic and molecular physics.
As a demonstration of the practical usefulness of RST, the energy difference
of the helium singlet states and
is calculated for a large range of nuclear charge numbers
(), whereas the corresponding experimental
values are available only up to (molybdenum). The deviations of
these RST results from the observational data is less than $0,3Comment: 84 pages, 2 figure
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