32,546 research outputs found
Is there a prescribed parameter's space for the adiabatic geometric phase?
The Aharonov-Anandan and Berry phases are determined for the cyclic motions
of a non-relativistic charged spinless particle evolving in the superposition
of the fields produced by a Penning trap and a rotating magnetic field.
Discussion about the selection of the parameter's space and the relationship
between the Berry phase and the symmetry of the binding potential is given.Comment: 7 pages, 2 figure
Harmonic Oscillator SUSY Partners and Evolution Loops
Supersymmetric quantum mechanics is a powerful tool for generating exactly
solvable potentials departing from a given initial one. If applied to the
harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg
algebras is obtained. In this paper it will be shown that the SUSY partner
Hamiltonians of the harmonic oscillator can produce evolution loops. The
corresponding geometric phases will be as well studied
Trends in Supersymmetric Quantum Mechanics
Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for studying solvable quantum potentials. It is the simplest method to build Hamiltonians with prescribed spectra in the spectral design. The key is to pair two Hamiltonians through a finite order differential operator. Some related subjects can be simply analyzed, as the algebras ruling both Hamiltonians and the associated coherent states. The technique has been applied also to periodic potentials, where the spectra consist of allowed and forbidden energy bands. In addition, a link with non-linear second-order differential equations, and the possibility of generating some solutions, can be explored. Recent applications concern the study of Dirac electrons in graphene placed either in electric or magnetic fields, and the analysis of optical systems whose relevant equations are the same as those of SUSY QM. These issues will be reviewed briefly in this paper, trying to identify the most important subjects explored currently in the literature
Supersymmetric Quantum Mechanics and Painlev\'e IV Equation
As it has been proven, the determination of general one-dimensional
Schr\"odinger Hamiltonians having third-order differential ladder operators
requires to solve the Painlev\'e IV equation. In this work, it will be shown
that some specific subsets of the higher-order supersymmetric partners of the
harmonic oscillator possess third-order differential ladder operators. This
allows us to introduce a simple technique for generating solutions of the
Painlev\'e IV equation. Finally, we classify these solutions into three
relevant hierarchies.Comment: Proceedings of the Workshop 'Supersymmetric Quantum Mechanics and
Spectral Design' (July 18-30, 2010, Benasque, Spain
Supersymmetric quantum mechanics and Painleve equations
In these lecture notes we shall study first the supersymmetric quantum
mechanics (SUSY QM), specially when applied to the harmonic and radial
oscillators. In addition, we will define the polynomial Heisenberg algebras
(PHA), and we will study the general systems ruled by them: for zero and first
order we obtain the harmonic and radial oscillators, respectively; for second
and third order PHA the potential is determined by solutions to Painleve IV
(PIV) and Painleve V (PV) equations. Taking advantage of this connection, later
on we will find solutions to PIV and PV equations expressed in terms of
confluent hypergeometric functions. Furthermore, we will classify them into
several solution hierarchies, according to the specific special functions they
are connected with.Comment: 38 pages, 20 figures. Lecture presented at the XLIII Latin American
School of Physics: ELAF 2013 in Mexico Cit
Geometric Phases and Mielnik's Evolution Loops
The cyclic evolutions and associated geometric phases induced by
time-independent Hamiltonians are studied for the case when the evolution
operator becomes the identity (those processes are called {\it evolution
loops}). We make a detailed treatment of systems having equally-spaced energy
levels. Special emphasis is made on the potentials which have the same spectrum
as the harmonic oscillator potential (the generalized oscillator potentials)
and on their recently found coherent states.Comment: 11 pages, harvmac, 2 figures available upon request; CINVESTAV-FIS
GFMR 11/9
Semi-classical spectrum of the Homogeneous sine-Gordon theories
The semi-classical spectrum of the Homogeneous sine-Gordon theories
associated with an arbitrary compact simple Lie group G is obtained and shown
to be entirely given by solitons. These theories describe quantum integrable
massive perturbations of Gepner's G-parafermions whose classical
equations-of-motion are non-abelian affine Toda equations. One-soliton
solutions are constructed by embeddings of the SU(2) complex sine-Gordon
soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits
both stable and unstable particles, which is a peculiar feature shared with the
spectrum of monopoles and dyons in N=2 and N=4 supersymmetric gauge theories.Comment: 28 pages, plain TeX, no figure
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