764 research outputs found
Quantum tunneling as a classical anomaly
Classical mechanics is a singular theory in that real-energy classical
particles can never enter classically forbidden regions. However, if one
regulates classical mechanics by allowing the energy E of a particle to be
complex, the particle exhibits quantum-like behavior: Complex-energy classical
particles can travel between classically allowed regions separated by potential
barriers. When Im(E) -> 0, the classical tunneling probabilities persist.
Hence, one can interpret quantum tunneling as an anomaly. A numerical
comparison of complex classical tunneling probabilities with quantum tunneling
probabilities leads to the conjecture that as ReE increases, complex classical
tunneling probabilities approach the corresponding quantum probabilities. Thus,
this work attempts to generalize the Bohr correspondence principle from
classically allowed to classically forbidden regions.Comment: 12 pages, 7 figure
Chaotic systems in complex phase space
This paper examines numerically the complex classical trajectories of the
kicked rotor and the double pendulum. Both of these systems exhibit a
transition to chaos, and this feature is studied in complex phase space.
Additionally, it is shown that the short-time and long-time behaviors of these
two PT-symmetric dynamical models in complex phase space exhibit strong
qualitative similarities.Comment: 22 page, 16 figure
Classical Trajectories for Complex Hamiltonians
It has been found that complex non-Hermitian quantum-mechanical Hamiltonians
may have entirely real spectra and generate unitary time evolution if they
possess an unbroken \cP\cT symmetry. A well-studied class of such
Hamiltonians is (). This paper
examines the underlying classical theory. Specifically, it explores the
possible trajectories of a classical particle that is governed by this class of
Hamiltonians. These trajectories exhibit an extraordinarily rich and elaborate
structure that depends sensitively on the value of the parameter and
on the initial conditions. A system for classifying complex orbits is
presented.Comment: 24 pages, 34 figure
Effects of temperature and carbon source on the isotopic fractionations associated with O_2 respiration for ^(17)O/^(16)O and ^(18)O/^(16)O ratios in E. coli
^(18)O/^(16)O and ^(17)O/^(16)O ratios of atmospheric and dissolved oceanic O_2 are used as biogeochemical tracers of photosynthesis and respiration. Critical to this approach is a quantitative understanding of the isotopic fractionations associated with production, consumption, and transport of O_2 in the ocean both at the surface and at depth. We made measurements of isotopic fractionations associated with O_2 respiration by E. coli. Our study included wild-type strains and mutants with only a single respiratory O_2 reductase in their electron transport chains (either a heme-copper oxygen reductase or a bd oxygen reductase). We tested two common assumptions made in interpretations of O_2 isotope variations and in isotope-enabled models of the O_2 cycle: (i) laboratory-measured respiratory ^(18)O/^(16)O isotopic fractionation factors (^(18)α) of microorganisms are independent of environmental and experimental conditions including temperature, carbon source, and growth rate; And (ii) the respiratory ‘mass law’ exponent, θ, between ^(18)O/^(16)O and ^(17)O/^(16)O, ^(17)α = (^(18)α)^θ, is universal for aerobic respiration. Results demonstrated that experimental temperatures have an effect on both ^(18)α and θ for aerobic respiration. Specifically, lowering temperatures from 37 to 15 °C decreased the absolute magnitude of ^(18)α by 0.0025 (2.5‰), and caused the mass law slope to decrease by 0.005. We propose a possible biochemical basis for these variations using a model of O_2 reduction that incorporates two isotopically discriminating steps: the reversible binding and unbinding of O_2 to a terminal reductase, and the irreversible reduction of that O_2 to water. Finally, we cast our results in a one-dimensional isopycnal reaction-advection-diffusion model, which demonstrates that enigmatic δ^(18)O and Δ^(17)O variations of dissolved O_2 in the dark ocean can be understood by invoking the observed temperature dependence of these isotope effects
Quantum effects in classical systems having complex energy
On the basis of extensive numerical studies it is argued that there are
strong analogies between the probabilistic behavior of quantum systems defined
by Hermitian Hamiltonians and the deterministic behavior of classical
mechanical systems extended into the complex domain. Three models are examined:
the quartic double-well potential , the cubic potential
, and the periodic potential . For the
quartic potential a wave packet that is initially localized in one side of the
double-well can tunnel to the other side. Complex solutions to the classical
equations of motion exhibit a remarkably analogous behavior. Furthermore,
classical solutions come in two varieties, which resemble the even-parity and
odd-parity quantum-mechanical bound states. For the cubic potential, a quantum
wave packet that is initially in the quadratic portion of the potential near
the origin will tunnel through the barrier and give rise to a probability
current that flows out to infinity. The complex solutions to the corresponding
classical equations of motion exhibit strongly analogous behavior. For the
periodic potential a quantum particle whose energy lies between -1 and 1 can
tunnel repeatedly between adjacent classically allowed regions and thus execute
a localized random walk as it hops from region to region. Furthermore, if the
energy of the quantum particle lies in a conduction band, then the particle
delocalizes and drifts freely through the periodic potential. A classical
particle having complex energy executes a qualitatively analogous local random
walk, and there exists a narrow energy band for which the classical particle
becomes delocalized and moves freely through the potential.Comment: 16 pages, 12 figure
Bounding biomass in the Fisher equation
The FKPP equation with a variable growth rate and advection by an
incompressible velocity field is considered as a model for plankton dispersed
by ocean currents. If the average growth rate is negative then the model has a
survival-extinction transition; the location of this transition in the
parameter space is constrained using variational arguments and delimited by
simulations. The statistical steady state reached when the system is in the
survival region of parameter space is characterized by integral constraints and
upper and lower bounds on the biomass and productivity that follow from
variational arguments and direct inequalities. In the limit of
zero-decorrelation time the velocity field is shown to act as Fickian diffusion
with an eddy diffusivity much larger than the molecular diffusivity and this
allows a one-dimensional model to predict the biomass, productivity and
extinction transitions. All results are illustrated with a simple growth and
stirring model.Comment: 32 Pages, 13 Figure
Modified spectrum autointerferometric correlation (MOSAIC) for single-shot pulse characterization
A method for generation of the modified spectrum autointerferometric correlation that allows single-shot pulse characterization is demonstrated. A sensitive graphical representation of the ultrashort pulse phase quality is introduced that delineates the difference between the presence of temporal and spectral phase distortions. Using these schemes, full-field reconstruction of ultrashort laser pulses is obtained in real time using an efficient iterative technique. (a) Single-shot characterization using a combination of fringe-free (noninterferometric) autocorrelation and second-harmonic spectrum (b) A hybrid graphical representation that distinguishes between spectral and temporal phase distortions (c) Real-time full-field reconstruction using the above schemes with an efficient sequential search algorithm Naganuma et al. showed that the pulse spectrum and IAC provide a sufficient dataset to uniquely reconstruct the complex electric field, with only a timedirection ambiguity The increased SNR found on averaged MOSAIC traces extends the utility of all retrieval techniques using the dataset outlined by Naganuma et al. The principle of computing a MOSAIC can be described in the frequency domain as follows: a secondorder IAC waveform with a fringe frequency ⍀ is Fourier transformed to generate a spectrum. Spectral filtering is then performed to remove the ⍀ component and amplify the 2⍀ component by a factor of 2. An inverse Fourier transform generates a new time-domain signal known as a fringe-resolved MOSAIC In the (delay) time-domain analysis, the maximum and minimum envelopes of MOSAIC are given by the intensity autocorrelation, g͑͒ = ͐f͑t͒f͑t + ͒dt, and the difference computation, S min = g͑͒ − ͉g p ͉͑͒, respectivel
Exact Isospectral Pairs of PT-Symmetric Hamiltonians
A technique for constructing an infinite tower of pairs of PT-symmetric
Hamiltonians, and (n=2,3,4,...), that have exactly the
same eigenvalues is described. The eigenvalue problem for the first Hamiltonian
of the pair must be posed in the complex domain, so its
eigenfunctions satisfy a complex differential equation and fulfill homogeneous
boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions
of the second Hamiltonian of the pair obey a real differential
equation and satisfy boundary conditions on the real axis. This equivalence
constitutes a proof that the eigenvalues of both Hamiltonians are real.
Although the eigenvalue differential equation associated with is
real, the Hamiltonian exhibits quantum anomalies (terms
proportional to powers of ). These anomalies are remnants of the complex
nature of the equivalent Hamiltonian . In the classical limit in
which the anomaly terms in are discarded, the pair of Hamiltonians
and have closed classical orbits whose
periods are identical.Comment: 18 pages, 12 figure
- …