28 research outputs found
Pseudo-Hermitian Operators in a Description of Physical Systems
We present some basic features of pseudo-hermitian quantum mechanics and illustrate the use of pseudo-hermitian Hamiltonians in a description of physical systems.
Aharonov-Bohm Effect and the Supersymmetry of Identical Anyons
We briefly review the relation between the Aharonov-Bohm effect and the dynamical realization of anyons. We show how the particular symmetries of the Aharonov-Bohm model give rise to the (nonlinear) supersymmetry of the two-body system of identical anyons
Comment on `Solution of the Dirac equation for the Woods-Saxon potential with spin and pseudospin symmetry' [J. Y. Guo and Z-Q. Sheng, Phys. Lett. A 338 (2005) 90]
Out of the four bound-state solutions presented in loc. cit., only one (viz.,
the spin-symmetric one, in the low-mass regime) is shown compatible with the
physical boundary conditions. We clarify the problem, correct the method and
offer another, "missing" (viz., pseudospin-symmetric) new solution with certain
counterintuitive "repulsion-generated" property.Comment: 6 p
Klein tunneling in carbon nanostructures: a free particle dynamics in disguise
The absence of backscattering in metallic nanotubes as well as perfect Klein
tunneling in potential barriers in graphene are the prominent electronic
characteristics of carbon nanostructures. We show that the phenomena can be
explained by a peculiar supersymmetry generated by a first order Hamiltonian
and zero order supercharge operators. Like the supersymmetry associated with
second order reflectionless finite-gap systems, it relates here the low-energy
behavior of the charge carriers with the free particle dynamics.Comment: 4 pages, 1 fig., typos correcte
Supersymmetric quantum mechanics living on topologically nontrivial Riemann surfaces
Supersymmetric quantum mechanics is constructed in a new non-Hermitian
representation. Firstly, the map between the partner operators is
chosen antilinear. Secondly, both these components of a super-Hamiltonian
are defined along certain topologically nontrivial complex curves
which spread over several Riemann sheets of the wave function.
The non-uniqueness of our choice of the map between "tobogganic"
partner curves and is emphasized.Comment: 14p
Quadratic pseudosupersymmetry in two-level systems
Using the intertwining relation we construct a pseudosuperpartner for a
(non-Hermitian) Dirac-like Hamiltonian describing a two-level system
interacting in the rotating wave approximation with the electric component of
an electromagnetic field. The two pseudosuperpartners and pseudosupersymmetry
generators close a quadratic pseudosuperalgebra. A class of time dependent
electric fields for which the equation of motion for a two level system placed
in this field can be solved exactly is obtained. New interesting phenomenon is
observed. There exists such a time-dependent detuning of the field frequency
from the resonance value that the probability to populate the excited level
ceases to oscillate and becomes a monotonically growing function of time
tending to 3/4. It is shown that near this fixed excitation regime the
probability exhibits two kinds of oscillations. The oscillations with a small
amplitude and a frequency close to the Rabi frequency (fast oscillations) take
place at the background of the ones with a big amplitude and a small frequency
(slow oscillations). During the period of slow oscillations the minimal value
of the probability to populate the excited level may exceed 1/2 suggesting for
an ensemble of such two-level atoms the possibility to acquire the inverse
population and exhibit lasing properties.Comment: 5 figure
Crypto-unitary forms of quantum evolution operators
For the description of quantum evolution, the use of a manifestly
time-dependent quantum Hamiltonian
is shown equivalent to the work with its simplified, time-independent
alternative . A tradeoff analysis is performed recommending
the latter option. The physical unitarity requirement is shown fulfilled in a
suitable ad hoc representation of Hilbert space.Comment: 15 p
PT-symmetric regularizations in supersymmetric quantum mechanics
Supersymmetry offers one of the deepest insights in the concept of
solvability in quantum mechanics. This insight is, paradoxically, restricted by
one of the most serious formal drawbacks of the standard Witten's formulation
of supersymmetric quantum mechanics which lies in the Jevicki-Rodrigues'
postulate of absence of poles in superpotentials W(x) over all the real axis of
coordinates x. In our review we emphasize that this obstacle is artificial and
that it disappears immediately after a suitable (say, constant) shift of the
axis of x into complex plane. Detailed attention is paid to a close
relationship between this common trick and the recent not quite expected
increase of interest in non-Hermitian (a. k. a. PT-symmetric or
pseudo-Hermitian) Hamiltonians. We show that the resulting PT-SUSY
regularization recipe proves both easy and universal. An insight into its
mathematics is mediated by the complex harmonic oscillator with a
centrifugal-like spike. An exhaustive discussion of the role of the strength of
this spike is offered. In addition we recollect the possibility of a
re-formulation of the recipe in the second-order SUSY language. Finally we list
a few promising directions of applicability of our PT-SUSY regularization
prescription to a few more complicated nonrelativistic models (superintegrable
Hamiltonians of the Smorodinsky-Winternitz and of the Calogero-Sutherland type)
and to the relativistic Klein-Gordon equation (as well as to all of its
unphysical higher-order analogues).Comment: 17 pages, based on the talk during SUSY QM conference in Valladolid
in the summer in 2003, to appear in J. Phys. A: Math. Gen. (spec. issue
Pseudo-Hermitian Representation of Quantum Mechanics
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used
to define a unitary quantum system, if one modifies the inner product of the
Hilbert space properly. We give a comprehensive and essentially self-contained
review of the basic ideas and techniques responsible for the recent
developments in this subject. We provide a critical assessment of the role of
the geometry of the Hilbert space in conventional quantum mechanics to reveal
the basic physical principle motivating our study. We then offer a survey of
the necessary mathematical tools and elaborate on a number of relevant issues
of fundamental importance. In particular, we discuss the role of the antilinear
symmetries such as PT, the true meaning and significance of the charge
operators C and the CPT-inner products, the nature of the physical observables,
the equivalent description of such models using ordinary Hermitian quantum
mechanics, the pertaining duality between local-non-Hermitian versus
nonlocal-Hermitian descriptions of their dynamics, the corresponding classical
systems, the pseudo-Hermitian canonical quantization scheme, various methods of
calculating the (pseudo-) metric operators, subtleties of dealing with
time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation
of the theory, and the structure of the state space and its ramifications for
the quantum Brachistochrone problem. We also explore some concrete physical
applications of the abstract concepts and tools that have been developed in the
course of this investigation. These include applications in nuclear physics,
condensed matter physics, relativistic quantum mechanics and quantum field
theory, quantum cosmology, electromagnetic wave propagation, open quantum
systems, magnetohydrodynamics, quantum chaos, and biophysics.Comment: 76 pages, 2 figures, 243 references, published as Int. J. Geom. Meth.
Mod. Phys. 7, 1191-1306 (2010
PT symmetric models in more dimensions and solvable square-well versions of their angular Schroedinger equations
For any central potential V in D dimensions, the angular Schroedinger
equation remains the same and defines the so called hyperspherical harmonics.
For non-central models, the situation is more complicated. We contemplate two
examples in the plane: (1) the partial differential Calogero's three-body model
(without centre of mass and with an impenetrable core in the two-body
interaction), and (2) the Smorodinsky-Winternitz' superintegrable harmonic
oscillator (with one or two impenetrable barriers). These examples are solvable
due to the presence of the barriers. We contemplate a small complex shift of
the angle. This creates a problem: the barriers become "translucent" and the
angular potentials cease to be solvable, having the sextuple-well form for
Calogero model and the quadruple or double well form otherwise. We mimic the
effect of these potentials on the spectrum by the multiple, purely imaginary
square wells and tabulate and discuss the result in the first nontrivial
double-well case.Comment: 21 pages, 5 figures (see version 1), amendment (a single comment
added on p. 7