4,384 research outputs found
Two charges on plane in a magnetic field I. "Quasi-equal" charges and neutral quantum system at rest cases
Low-lying bound states for the problem of two Coulomb charges of finite
masses on a plane subject to a constant magnetic field perpendicular to the
plane are considered. Major emphasis is given to two systems: two charges with
the equal charge-to-mass ratio (quasi-equal charges) and neutral systems with
concrete results for the Hydrogen atom and two electrons (quantum dot). It is
shown that for these two cases, but when a neutral system is at rest (the
center-of-mass momentum is zero), some outstanding properties occur: in double
polar coordinates in CMS and relative coordinate
systems (i) the eigenfunctions are factorizable, all factors except for
-dependent are found analytically, they have definite relative angular
momentum, (ii) dynamics in -direction is the same for both systems being
described by a funnel-type potential; (iii) at some discrete values of
dimensionless magnetic fields the system becomes {\it
quasi-exactly-solvable} and a finite number of eigenfunctions in are
polynomials. The variational method is employed. Trial functions are based on
combining for the phase of a wavefunction (a) the WKB expansion at large
distances, (b) the perturbation theory at small distances (c) with a form of
the known analytically (quasi-exactly-solvable) eigenfunctions. For the lowest
states with relative magnetic quantum numbers this approximation
gives not less than 7 s.d., 8 s.d., 9 s.d., respectively, for the total energy
for magnetic fields
(Hydrogen atom) and (two electrons).Comment: 38 pages, 8 figures, 11 table
Two charges on plane in a magnetic field: II. Moving neutral quantum system across a magnetic field
The moving neutral system of two Coulomb charges on a plane subject to a
constant magnetic field perpendicular to the plane is considered. It is
shown that the composite system of finite total mass is bound for any
center-of-mass momentum and magnetic field strength; the energy of the
ground state is calculated accurately using a variational approach. Their
accuracy is cross-checked in a Lagrange-mesh method for a.u. and in a
perturbation theory at small and . The constructed trial function has
the property of being a uniform approximation of the exact eigenfunction. For a
Hydrogen atom and a Positronium a double perturbation theory in and is
developed and the first corrections are found algebraically. A phenomenon of a
sharp change of energy behavior for a certain center-of-mass momentum and a
fixed magnetic field is indicated.Comment: 24 pages, 5 figures, 13 tables (6 in main body and 7 moved in a
supplementary material), several clarifying sentences and 3 extra references
added, Chapter II rectified, typos fixed, Annals of Physics (to be published
Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions
The quantum mechanics of two Coulomb charges on a plane and
subject to a constant magnetic field perpendicular to the
plane is considered. Four integrals of motion are explicitly indicated. It is
shown that for two physically-important particular cases, namely that of two
particles of equal Larmor frequencies, (e.g. two electrons) and one of a neutral
system (e.g. the electron - positron pair, Hydrogen atom) at rest (the
center-of-mass momentum is zero) some outstanding properties occur. They are
the most visible in double polar coordinates in CMS and relative
coordinate systems: (i) eigenfunctions are factorizable, all
factors except one with the explicit -dependence are found analytically,
they have definite relative angular momentum, (ii) dynamics in -direction
is the same for both systems, it corresponds to a funnel-type potential and it
has hidden algebra; at some discrete values of dimensionless magnetic
fields , (iii) particular integral(s) occur, (iv) the hidden
algebra emerges in finite-dimensional representation, thus, the system becomes
{\it quasi-exactly-solvable} and (v) a finite number of polynomial
eigenfunctions in appear. Nine families of eigenfunctions are presented
explicitly.Comment: 20 page
Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces
2nd-order conformal superintegrable systems in dimensions are Laplace
equations on a manifold with an added scalar potential and independent
2nd order conformal symmetry operators. They encode all the information about
Helmholtz (eigenvalue) superintegrable systems in an efficient manner: there is
a 1-1 correspondence between Laplace superintegrable systems and Stackel
equivalence classes of Helmholtz superintegrable systems. In this paper we
focus on superintegrable systems in two dimensions, , where there are 44
Helmholtz systems, corresponding to 12 Laplace systems. For each Laplace
equation we determine the possible 2-variate polynomial subspaces that are
invariant under the action of the Laplace operator, thus leading to families of
polynomial eigenfunctions. We also study the behavior of the polynomial
invariant subspaces under a Stackel transform. The principal new results are
the details of the polynomial variables and the conditions on parameters of the
potential corresponding to polynomial solutions. The hidden gl_3-algebraic
structure is exhibited for the exact and quasi-exact systems. For physically
meaningful solutions, the orthogonality properties and normalizability of the
polynomials are presented as well. Finally, for all Helmholtz superintegrable
solvable systems we give a unified construction of 1D and 2D quasi-exactly
solvable potentials possessing polynomial solutions, and a construction of new
2D PT-symmetric potentials is established.Comment: 28 page
The spin representations via sigma models
We establish and analyze a new relationship between the matrices describing
an arbitrary component of a spin , where , and the
matrices of two-dimensional Euclidean sigma models. The spin
matrices are constructed from the rank-1 Hermitian projectors of the sigma
models or from the antihermitian immersion functions of their soliton surfaces
in the algebra. For the spin matrices which can be
represented as a linear combination of the generalized Pauli matrices, we find
the dynamics equation satisfied by its coefficients. The equation proves to be
identical to the stationary equation of a two-dimensional Heisenberg model. We
show that the same holds for the matrices congruent to the generalized Pauli
ones by any coordinate-independent unitary linear transformation. These
properties open the possibility for new interpretations of the spins and also
for application of the methods known from the theory of sigma models to the
situations described by the Heisenberg model, from statistical mechanics to
quantum computing.Comment: 17 page
Three-loop Correction to the Instanton Density. II. The Sine-Gordon potential
In this second paper on quantum fluctuations near the classical instanton
configurations, see {\em Phys. Rev. D \bf 92}, 025046 (2015) and
arXiv:1501.03993, we focus on another well studied quantum-mechanical problem,
the one-dimensional Sine-Gordon potential (the Mathieu potential). Using only
the tools from quantum field theory, the Feynman diagrams in the instanton
background, we calculate the tunneling amplitude (the instanton density) to the
three-loop order. The result confirms (to seven significant figures) the one
recently recalculated by G. V. Dunne and M. \"{U}nsal, {\it Phys. Rev. \bf D
89}, 105009 (2014) from the resurgence perspective. As in the double well
potential case, we found that the largest contribution is given by the diagrams
originating from the Jacobian. We again observe that in the three-loop case
individual Feynman diagrams contain irrational contributions, while their sum
does not.Comment: 14 pages, 3 figures, 1 table, already published at Phys.Rev.D: a post
published version with tadpole diagrams redrawn to mark clearly the Jacobian
source, Note Added about significance of a tadpole diagram in 3-,4-,5-loops
contribution
Fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II
This is the second paper on semiclassical approach based on the density
matrix given by the Euclidean time path integral with fixed coinciding
endpoints. The classical path, interpolating between this point and the
classical vacuum, called "flucton", plus systematic one- and two-loop
corrections, has been calculated in the first paper \cite{Escobar-Ruiz:2016aqv}
for double-well potential and now extended for a number of quantum-mechanical
problems (anharmonic oscillator, sine-Gordon potential). The method is based on
systematic expansion in Feynman diagrams and thus can be extended to QFTs. We
show that the loop expansion in QM reminds the leading log-approximations in
QFT. In this sequel we present complete set of results obtained using this
method in unified way. Alternatively, starting from the Schr\"{o}dinger
equation we derive a {\it generalized} Bloch equation which semiclassical-like,
iterative solution generates the loop expansion. We re-derive two loop
expansions for all three above potentials and now extend it to three loops,
which has not yet been done via Feynman diagrams. All results for both methods
are fully consistent with each other. Asymmetric (tilted) double-well potential
(non-degenerate minima) is also studied using the second method
Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential
This paper deals with quantum fluctuations near the classical instanton
configuration. Feynman diagrams in the instanton background are used for the
calculation of the tunneling amplitude (the instanton density) in the
three-loop order for quartic double-well potential. The result for the
three-loop contribution coincides in six significant figures with one given
long ago by J.~Zinn-Justin. Unlike the two-loop contribution where all involved
Feynman integrals are rational numbers, in the three-loop case Feynman diagrams
can contain irrational contributions.Comment: 15 pages, 3 figures, 1 table, already published at Phys.Rev.D92
(2015) 025046, 089902(erratum), a post published version with tadpole
diagrams redrawn to clearly mark the Jacobian source and misprint on 3-loop
tadpole contribution on p.8 fixe
Quantum and thermal fluctuations in quantum mechanics and field theories from a new version of semiclassical theory
We develop a new semiclassical approach, which starts with the density matrix
given by the Euclidean time path integral with fixed coinciding endpoints, and
proceed by identifying classical (minimal Euclidean action) path, to be
referred to as {\it flucton}, which passes through this endpoint. Fluctuations
around flucton path are included, by standard Feynman diagrams, previously
developed for instantons. We calculate the Green function and evaluate the one
loop determinant both by direct diagonalization of the fluctuation equation,
and also via the trick with the Green functions. The two-loop corrections are
evaluated by explicit Feynman diagrams, and some curious cancellation of
logarithmic and polylog terms is observed. The results are fully consistent
with large-distance asymptotics obtained in quantum mechanics. Two classic
examples -- quartic double-well and sine-Gordon potentials -- are discussed in
detail, while power-like potential and quartic anharmonic oscillator are
discussed in brief. Unlike other semiclassical methods, like WKB, we do not use
the Schr\"{o}dinger equation, and all the steps generalize to multi-dimensional
or quantum fields cases straightforwardly.Comment: Title slightly changed, typos corrected, Eq.(50) rectified, two
references added, to appear at Phys Rev D9
Three charges on a plane in a magnetic field: Special trajectories
As a generalization and extension of JMP 54 (2013) 022901, the classical
dynamics of three non-relativistic Coulomb charges ,
and on the plane placed in a constant magnetic field perpendicular
to the plane is considered. Special trajectories for which the distances
between the charges remain unchanged are presented and their corresponding
integrals of motion are indicated. For these special trajectories the number of
integrals of motion is larger than the dimension of the configuration space and
hence they can be called \emph{particularly superintegrable}. Three physically
relevant cases are analyzed in detail, namely that of three electrons, a
neutral system and a Helium-like system. The -body case is discussed as
well.Comment: 27 pages, 5 figure
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