The study of the semiclassical limit of quantum
systems with non-integrable classical
analogues has been object of intense research
in the past 30 years. At the quantum level, systems
that are classicaly chaotic behave qualitatively
different from systems that are classicaly
integrable or regular. The most striking difference
between the two is perhaps the statistical
properties of their energy levels. Wave
functions also display distinct behavior and 'feel',
in the semiclassical limit, the nature of the
underlying classical dynamics.
An important tool for describing the semiclassical
physics of chaotic or mixed systems
is provided by the coherent states. These states
allow for a quantum mechanical representantion
in terms of phase space variables, similar to
the Wigner representation. Coherent states
are minimum uncertainty Gaussian states, |qp>,
centered at the phase space point
(q,p). In a way, they are the best possible representation
of a classical particle in quantum
mechanics, and we may call them 'quantum points'
in phase space, following the nomenclature
introduced by Michel Baranger. The semiclassical
limit of the evolution operator in this representation
brings up information about the dynamics of these
'quantum points' , particularly in situations
involving tunneling and chaos.
Here is a list of topics we have been working on in the last years:
The semiclassical propagator and Green's function
in the coherent states representation
Complex trajectories and their role in semiclassical
tunneling
Tunneling in the presence of chaos
Calculation of Periodic orbits in smooth systems
and billiards
The plots below show an example of the type of calculation that can be
performed. They show
the time evolution of a Gaussian wave-packet
reflecting off the repulsive barrier V(x)= 1/x².
The continuous line
shows the semiclassical approximation and the dotted line the exact
(numerical)
quantum evolution. The not so good agreement in the last figure is due to phase
space caustics,
which can
be corrected with the use of improved (third order) approximations. See
more details in
A Regular Semiclassical Approximation for the Propagation of Wave
Packets with Complex Trajectories
Fernando Parisio and M.A.M. de Aguiar, J. Phys. A: Math. Gen. 38 (2005) 9317.
Recent publications (click here for the full list)
A Uniform Approximation for the Coherent States Propagator
using a Conjugate Application of the Bargmann Representation
A. D. Ribeiro, M.Novaes and M.A.M. de Aguiar,
Phys. Rev. Lett. 95 (2005) 050405.
Semiclassical Propagation of Wavepackets with Complex Trajectories
M.A.M. de Aguiar, M. Baranger, L. Jaubert, Fernando
Parisio and A.D. Ribeiro, J. Phys. A38 (2005) 4645.
Semiclassical
Approximations Based on Complex Trajectories
A.D. Ribeiro, M.A.M. de Aguiar and M. Baranger,
Phys. Rev. E69 (2004) 66204.
A
Semiclassical Coherent-State Propagator via Path Integrals with
Intermediate
States of Variable Width F. Parisio and M.A.M. de Aguiar,
Phys. Rev. A68 (2003) 062112.
Reply
to`Comment
on ``Semiclassical Approximations in Phase Space with Coherent States'' '
M. Baranger, M.A.M. de Aguiar, F. Keck, H.J. Korsch and B. Schellaas
J. Phys. A35 (2002) 9493
Semiclassical approximations in phase-space with coherent states
M. Baranger, M.A.M. de Aguiar, F. Keck, H.J.
Korsch and B. Schellaas
J. Phys. A34 (2001) 7227-7286 (462 Kb)
Scars of the Wigner function
F. Toscano, M.A.M. de Aguiar and Alfredo M. Ozorio
de Almeida
Phys. Rev. Lett 86 (2001) 59-62 (730 Kb)
Phase-Space
Approach to the Tunnel Effect: a New Semiclassical Traversal Time
A.L. Xavier Jr. e M.A.M. de Aguiar
Phys. Rev. Lett. 79 (1997) 3323 (400 Kb)