47 research outputs found
Quantum Manifestations of Classical Stochasticity in the Mixed State
We investigate the QMCS in structure of the eigenfunctions, corresponding to
mixed type classical dynamics in smooth potential of the surface quadrupole
oscillations of a charged liquid drop. Regions of different regimes of
classical motion are strictly separated in the configuration space, allowing
direct observation of the correlations between the wave function structure and
type of the classical motion by comparison of the parts of the eigenfunction,
corresponding to different local minima.Comment: 4 pages, 3 figure
Multi-Well Potentials in Quantum Mechanics and Stochastic Processes
Using the formalism of extended N=4 supersymmetric quantum mechanics we
consider the procedure of the construction of multi-well potentials. We
demonstrate the form-invariance of Hamiltonians entering the supermultiplet,
using the presented relation for integrals, which contain fundamental
solutions. The possibility of partial N=4 supersymmetry breaking is determined.
We also obtain exact forms of multi-well potentials, both symmetric and
asymmetric, using the Hamiltonian of harmonic oscillator as initial. The
modification of the shape of potentials due to variation of parameters is also
discussed, as well as application of the obtained results to the study of
tunneling processes. We consider the case of exact, as well as partially broken
N=4 supersymmetry. The distinctive feature of obtained probability densities
and potentials is a parametric freedom, which allows to substantially modify
their shape. We obtain the expressions for probability densities under the
generalization of the Ornstein-Uhlenbeck process
Classification of irreps and invariants of the N-extended Supersymmetric Quantum Mechanics
We present an algorithmic classification of the irreps of the -extended
one-dimensional supersymmetry algebra linearly realized on a finite number of
fields. Our work is based on the 1-to-1 \cite{pt} correspondence between
Weyl-type Clifford algebras (whose irreps are fully classified) and classes of
irreps of the -extended 1D supersymmetry. The complete classification of
irreps is presented up to . The fields of an irrep are accommodated
in different spin states. N=10 is the minimal value admitting length
irreps. The classification of length-4 irreps of the N=12 and {\em real} N=11
extended supersymmetries is also explicitly presented.\par Tensoring irreps
allows us to systematically construct manifestly (-extended) supersymmetric
multi-linear invariants {\em without} introducing a superspace formalism.
Multi-linear invariants can be constructed both for {\em unconstrained} and
{\em multi-linearly constrained} fields. A whole class of off-shell invariant
actions are produced in association with each irreducible representation. The
explicit example of the N=8 off-shell action of the multiplet is
presented.\par Tensoring zero-energy irreps leads us to the notion of the {\em
fusion algebra} of the 1D -extended supersymmetric vacua.Comment: Final version to appear in JHEP. 52 pages. The part with the complete
classification of irreps (and the explicit presentation of length-4 irreps of
N=9,10,11,12 and N=10 length-5 irreps) is unchanged. An extra section has
been added with an entire class of off-shell invariant actions for arbitrary
values N of the 1D extended supersymmetry. A non-trivial N=8 off-shell action
for the (1,8,7) multiplet has been constructed as an example. It is obtained
in terms of the octonionic structure constant
A New Two-Parameter Family of Potentials with a Tunable Ground State
In a previous paper we solved a countably infinite family of one-dimensional
Schr\"odinger equations by showing that they were supersymmetric partner
potentials of the standard quantum harmonic oscillator. In this work we extend
these results to find the complete set of real partner potentials of the
harmonic oscillator, showing that these depend upon two continuous parameters.
Their spectra are identical to that of the harmonic oscillator, except that the
ground state energy becomes a tunable parameter. We finally use these
potentials to analyse the physical problem of Bose-Einstein condensation in an
atomic gas trapped in a dimple potential.Comment: 15 pages, 5 figure