34 research outputs found
Auxiliary field method and analytical solutions of the Schr\"{o}dinger equation with exponential potentials
The auxiliary field method is a new and efficient way to compute approximate
analytical eigenenergies and eigenvectors of the Schr\"{o}dinger equation. This
method has already been successfully applied to the case of central potentials
of power-law and logarithmic forms. In the present work, we show that the
Schr\"{o}dinger equation with exponential potentials of the form can also be analytically solved by using the
auxiliary field method. Formulae giving the critical heights and the energy
levels of these potentials are presented. Special attention is drawn on the
Yukawa potential and the pure exponential one
Duality relations in the auxiliary field method
The eigenenergies of a system of
identical particles with a mass are functions of the various radial quantum
numbers and orbital quantum numbers . Approximations
of these eigenenergies, depending on a principal quantum number
, can be obtained in the framework of the auxiliary field
method. We demonstrate the existence of numerous exact duality relations
linking quantities and for various forms of the
potentials (independent of and ) and for both nonrelativistic and
semirelativistic kinematics. As the approximations computed with the auxiliary
field method can be very close to the exact results, we show with several
examples that these duality relations still hold, with sometimes a good
accuracy, for the exact eigenenergies
The quantum N-body problem and the auxiliary field method
Approximate analytical energy formulas for N-body relativistic Hamiltonians
with one- and two-body interactions are obtained within the framework of the
auxiliary field method. This method has already been proved to be a powerful
technique in the case of two-body problems. A general procedure is given and
applied to various Hamiltonians of interest, in atomic and hadronic physics in
particular. A test of formulas is performed for baryons described as a
three-quark system.Comment: References adde
Extensions of the auxiliary field method to solve Schr\"{o}dinger equations
It has recently been shown that the auxiliary field method is an interesting
tool to compute approximate analytical solutions of the Schr\"{o}dinger
equation. This technique can generate the spectrum associated with an arbitrary
potential starting from the analytically known spectrum of a particular
potential . In the present work, general important properties of the
auxiliary field method are proved, such as scaling laws and independence of the
results on the choice of . The method is extended in order to find
accurate analytical energy formulae for radial potentials of the form , and several explicit examples are studied. Connections existing
between the perturbation theory and the auxiliary field method are also
discussed
Some equivalences between the auxiliary field method and the envelope theory
The auxiliary field method has been recently proposed as an efficient
technique to compute analytical approximate solutions of eigenequations in
quantum mechanics. We show that the auxiliary field method is completely
equivalent to the envelope theory, which is another well-known procedure to
analytically solve eigenequations, although relying on different principles
\textit{a priori}. This equivalence leads to a deeper understanding of both
frameworks.Comment: v2 to appear in J. Math. Phy
The few-body problem in terms of correlated gaussians
In their textbook, Suzuki and Varga [Y. Suzuki and K. Varga, {\em Stochastic
Variational Approach to Quantum-Mechanical Few-Body Problems} (Springer,
Berlin, 1998)] present the stochastic variational method in a very exhaustive
way. In this framework, the so-called correlated gaussian bases are often
employed. General formulae for the matrix elements of various operators can be
found in the textbook. However the Fourier transform of correlated gaussians
and their application to the management of a relativistic kinetic energy
operator are missing and cannot be found in the literature. In this paper we
present these interesting formulae. We give also a derivation for new
formulations concerning central potentials; the corresponding formulae are more
efficient numerically than those presented in the textbook.Comment: 10 page
Semirelativistic potential model for low-lying three-gluon glueballs
The three-gluon glueball states are studied with the generalization of a
semirelativistic potential model giving good results for two-gluon glueballs.
The Hamiltonian depends only on 3 parameters fixed on two-gluon glueball
spectra: the strong coupling constant, the string tension, and a gluon size
which removes singularities in the potential. The Casimir scaling determines
the structure of the confinement. Low-lying states are computed and
compared with recent lattice calculations. A good agreement is found for
and states, but our model predicts a state much
higher in energy than the lattice result. The mass is also computed.Comment: 2 figure
Semirelativistic Hamiltonians and the auxiliary field method
Approximate analytical closed energy formulas for semirelativistic
Hamiltonians of the form are obtained within
the framework of the auxiliary field method. This method, which is equivalent
to the envelope theory, has been recently proposed as a powerful tool to get
approximate analytical solutions of the Schr\"odinger equation. Various shapes
for the potential are investigated: power-law, funnel, square root, and
Yukawa. A comparison with the exact results is discussed in detail
Equation of motion of an interstellar Bussard ramjet with radiation and mass losses
An interstellar Bussard ramjet is a spaceship using the protons of the
interstellar medium in a fusion engine to produce thrust. In recent papers, it
was shown that the relativistic equation of motion of an ideal ramjet and of a
ramjet with radiation loss are analytical. When a mass loss appears, the limit
speed of the ramjet is more strongly reduced. But, the parametric equations, in
terms of the ramjet's speed, for the position of the ramjet in the inertial
frame of the interstellar medium, the time in this frame, and the proper time
indicated by the clocks on board the spaceship, can still be obtained in an
analytical form. The non-relativistic motion and the motion near the limit
speed are studied.Comment: 4 figure