1,592 research outputs found
On the exactness of the Semi-Classical Approximation for Non-Relativistic One Dimensional Propagators
For one dimensional non-relativistic quantum mechanical problems, we
investigate the conditions for all the position dependence of the propagator to
be in its phase, that is, the semi-classical approximation to be exact. For
velocity independent potentials we find that:
(i) the potential must be quadratic in space, but can have arbitrary time
dependence.
(ii) the phase may be made proportional to the classical action, and the
magnitude (``fluctuation factor'') can also be found from the classical
solution.
(iii) for the driven harmonic oscillator the fluctuation factor is
independent of the driving term.Comment: 7 pages, latex, no figures, published in journal of physics
Analysis of a three-component model phase diagram by Catastrophe Theory
We analyze the thermodynamical potential of a lattice gas model with three
components and five parameters using the methods of Catastrophe Theory. We find
the highest singularity, which has codimension five, and establish its
transversality. Hence the corresponding seven-degree Landau potential, the
canonical form Wigwam or , constitutes the adequate starting point to
study the overall phase diagram of this model.Comment: 16 pages, Latex file, submitted to Phys. Rev.
Exact propagators on the lattice with applications to diffractive effects
The propagator of the discrete Schr\"odinger equation is computed and its
properties are revealed through a Feynman path summation in discrete space.
Initial data problems such as diffraction in discrete space and continuous time
are studied analytically by the application of the new propagator. In the
second part of this paper, the analogy between time propagation and 2D
scattering by 1D obstacles is explored. New results are given in the context of
diffraction by edges within a periodic medium. A connection with tight-binding
arrays and photonic crystals is indicated.Comment: Final version with two appendices. Published in J. Phys. A: Math.
Theo
Comment on "Why quantum mechanics cannot be formulated as a Markov process"
In the paper with the above title, D. T. Gillespie [Phys. Rev. A 49, 1607,
(1994)] claims that the theory of Markov stochastic processes cannot provide an
adequate mathematical framework for quantum mechanics. In conjunction with the
specific quantum dynamics considered there, we give a general analysis of the
associated dichotomic jump processes. If we assume that Gillespie's
"measurement probabilities" \it are \rm the transition probabilities of a
stochastic process, then the process must have an invariant (time independent)
probability measure. Alternatively, if we demand the probability measure of the
process to follow the quantally implemented (via the Born statistical
postulate) evolution, then we arrive at the jump process which \it can \rm be
interpreted as a Markov process if restricted to a suitable duration time.
However, there is no corresponding Markov process consistent with the
event space assumption, if we require its existence for all times .Comment: Latex file, resubm. to Phys. Rev.
Three manifestations of the pulsed harmonic potential
We consider, in turn, three systems being acted upon by a regularly pulsed
harmonic potential (PHP). These are i) a classical particle, ii) a quantum
particle, and iii) a directed line. We contrast the mechanics of the first two
systems by parameterizing their bands of stability and periodicity. Interesting
differences due to quantum fluctuations are examined in detail. The
fluctuations of the directed line are calculated in the two cases of a binding
PHP, and an unbinding PHP. In the latter case there is a finite maximum line
length for a given potential strength.Comment: 34 Revtex pages, with 5 attached figure
Spin relaxation dynamics of quasiclassical electrons in ballistic quantum dots with strong spin-orbit coupling
We performed path integral simulations of spin evolution controlled by the
Rashba spin-orbit interaction in the semiclassical regime for chaotic and
regular quantum dots. The spin polarization dynamics have been found to be
strikingly different from the D'yakonov-Perel' (DP) spin relaxation in bulk
systems. Also an important distinction have been found between long time spin
evolutions in classically chaotic and regular systems. In the former case the
spin polarization relaxes to zero within relaxation time much larger than the
DP relaxation, while in the latter case it evolves to a time independent
residual value. The quantum mechanical analysis of the spin evolution based on
the exact solution of the Schroedinger equation with Rashba SOI has confirmed
the results of the classical simulations for the circular dot, which is
expected to be valid in general regular systems. In contrast, the spin
relaxation down to zero in chaotic dots contradicts to what have to be expected
from quantum mechanics. This signals on importance at long time of the
mesoscopic echo effect missed in the semiclassical simulations.Comment: 14 pages, 9 figure
Space-Time Evolution of the Oscillator, Rapidly moving in a random media
We study the quantum-mechanical evolution of the nonrelativistic oscillator,
rapidly moving in the media with the random vector fields. We calculate the
evolution of the level probability distribution as a function of time, and
obtain rapid level diffusion over the energy levels. Our results imply a new
mechanism of charmonium dissociation in QCD media.Comment: 32 pages, 13 figure
Path Integral Approach for Spaces of Non-constant Curvature in Three Dimensions
In this contribution I show that it is possible to construct
three-dimensional spaces of non-constant curvature, i.e. three-dimensional
Darboux-spaces. Two-dimensional Darboux spaces have been introduced by Kalnins
et al., with a path integral approach by the present author. In comparison to
two dimensions, in three dimensions it is necessary to add a curvature term in
the Lagrangian in order that the quantum motion can be properly defined. Once
this is done, it turns out that in the two three-dimensional Darboux spaces,
which are discussed in this paper, the quantum motion is similar to the
two-dimensional case. In \threedDI we find seven coordinate systems which
separate the Schr\"odinger equation. For the second space, \threedDII, all
coordinate systems of flat three-dimensional Euclidean space which separate the
Schr\"odinger equation also separate the Schr\"odinger equation in
\threedDII. I solve the path integral on \threedDI in the -system,
and on \threedDII in the -system and in spherical coordinates
Quantum creep and variable range hopping of one-dimensional interacting electrons
The variable range hopping results for noninteracting electrons of Mott and
Shklovskii are generalized to 1D disordered charge density waves and Luttinger
liquids using an instanton approach. Following a recent paper by Nattermann,
Giamarchi and Le Doussal [Phys. Rev. Lett. {\bf 91}, 56603 (2003)] we calculate
the quantum creep of charges at zero temperature and the linear conductivity at
finite temperatures for these systems. The hopping conductivity for the short
range interacting electrons acquires the same form as for noninteracting
particles if the one-particle density of states is replaced by the
compressibility. In the present paper we extend the calculation to dissipative
systems and give a discussion of the physics after the particles materialize
behind the tunneling barrier. It turns out that dissipation is crucial for
tunneling to happen. Contrary to pure systems the new metastable state does not
propagate through the system but is restricted to a region of the size of the
tunneling region. This corresponds to the hopping of an integer number of
charges over a finite distance. A global current results only if tunneling
events fill the whole sample. We argue that rare events of extra low tunneling
probability are not relevant for realistic systems of finite length. Finally we
show that an additional Coulomb interaction only leads to small logarithmic
corrections.Comment: 15 pages, 3 figures; references adde
Brownian Motion and Polymer Statistics on Certain Curved Manifolds
We have calculated the probability distribution function G(R,L|R',0) of the
end-to-end vector R-R' and the mean-square end-to-end distance (R-R')^2 of a
Gaussian polymer chain embedded on a sphere S^(D-1) in D dimensions and on a
cylinder, a cone and a curved torus in 3-D.
We showed that: surface curvature induces a geometrical localization area; at
short length the polymer is locally "flat" and (R-R')^2 = L l in all cases; at
large scales, (R-R')^2 is constant for the sphere, it is linear in L for the
cylinder and reaches different constant values for the torus. The cone vertex
induces (function of opening angle and R') contraction of the chain for all
lengths. Explicit crossover formulas are derived.Comment: 9 pages, 4 figures, RevTex, uses amssymb.sty and multicol.sty, to
appear in Phys. Rev
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