17 research outputs found
Semiclassical Approximations Based On Complex Trajectories.
The semiclassical limit of the coherent state propagator involves complex classical trajectories of the Hamiltonian H(u,v) = satisfying u(0) = z' and v(T) = z*. In this work we study mostly the case z' = z. The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as Planck's over 2 pi goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a phenomenon closely related to Stokes lines. When the contributions of these trajectories are filtered out, the semiclassical results show excellent agreement with the exact calculations.6906620
Semiclassical Tunneling of Wavepackets with Real Trajectories
Semiclassical approximations for tunneling processes usually involve complex
trajectories or complex times. In this paper we use a previously derived
approximation involving only real trajectories propagating in real time to
describe the scattering of a Gaussian wavepacket by a finite square potential
barrier. We show that the approximation describes both tunneling and
interferences very accurately in the limit of small Plank's constant. We use
these results to estimate the tunneling time of the wavepacket and find that,
for high energies, the barrier slows down the wavepacket but that it speeds it
up at energies comparable to the barrier height.Comment: 23 pages, 7 figures Revised text and figure
Semiclassical Propagation of Wavepackets with Real and Complex Trajectories
We consider a semiclassical approximation for the time evolution of an
originally gaussian wave packet in terms of complex trajectories. We also
derive additional approximations replacing the complex trajectories by real
ones. These yield three different semiclassical formulae involving different
real trajectories. One of these formulae is Heller's thawed gaussian
approximation. The other approximations are non-gaussian and may involve
several trajectories determined by mixed initial-final conditions. These
different formulae are tested for the cases of scattering by a hard wall,
scattering by an attractive gaussian potential, and bound motion in a quartic
oscillator. The formula with complex trajectories gives good results in all
cases. The non-gaussian approximations with real trajectories work well in some
cases, whereas the thawed gaussian works only in very simple situations.Comment: revised text, 24 pages, 6 figure
Semiclassical coherent state propagator for systems with spin
We derive the semiclassical limit of the coherent state propagator for
systems with two degrees of freedom of which one degree of freedom is canonical
and the other a spin. Systems in this category include those involving
spin-orbit interactions and the Jaynes-Cummings model in which a single
electromagnetic mode interacts with many independent two-level atoms. We
construct a path integral representation for the propagator of such systems and
derive its semiclassical limit. As special cases we consider separable systems,
the limit of very large spins and the case of spin 1/2.Comment: 19 pages, no figure
Coherent State Path Integrals in the Weyl Representation
We construct a representation of the coherent state path integral using the
Weyl symbol of the Hamiltonian operator. This representation is very different
from the usual path integral forms suggested by Klauder and Skagerstan in
\cite{Klau85}, which involve the normal or the antinormal ordering of the
Hamiltonian. These different representations, although equivalent quantum
mechanically, lead to different semiclassical limits. We show that the
semiclassical limit of the coherent state propagator in Weyl representation is
involves classical trajectories that are independent on the coherent states
width. This propagator is also free from the phase corrections found in
\cite{Bar01} for the two Klauder forms and provides an explicit connection
between the Wigner and the Husimi representations of the evolution operator.Comment: 23 page
Semiclassical mechanics of a non-integrable spin cluster
We study detailed classical-quantum correspondence for a cluster system of
three spins with single-axis anisotropic exchange coupling. With autoregressive
spectral estimation, we find oscillating terms in the quantum density of states
caused by classical periodic orbits: in the slowly varying part of the density
of states we see signs of nontrivial topology changes happening to the energy
surface as the energy is varied. Also, we can explain the hierarchy of quantum
energy levels near the ferromagnetic and antiferromagnetic states with EKB
quantization to explain large structures and tunneling to explain small
structures.Comment: 9 pages. For related works see
"http://www.msc.cornell.edu/~clh/clh.html
Semiclassical Approximations in Phase Space with Coherent States
We present a complete derivation of the semiclassical limit of the coherent
state propagator in one dimension, starting from path integrals in phase space.
We show that the arbitrariness in the path integral representation, which
follows from the overcompleteness of the coherent states, results in many
different semiclassical limits. We explicitly derive two possible semiclassical
formulae for the propagator, we suggest a third one, and we discuss their
relationships. We also derive an initial value representation for the
semiclassical propagator, based on an initial gaussian wavepacket. It turns out
to be related to, but different from, Heller's thawed gaussian approximation.
It is very different from the Herman--Kluk formula, which is not a correct
semiclassical limit. We point out errors in two derivations of the latter.
Finally we show how the semiclassical coherent state propagators lead to
WKB-type quantization rules and to approximations for the Husimi distributions
of stationary states.Comment: 80 pages, 4 figure