Thomas Dittrich

Systems in confined spaces: challenge for semiclassical propagation in phase space

A particle confined to a potential box with infinitely high walls is maybe the most elementary textbook
example of a bound quantum system. Yet it gives rise to fascinating effects such as "quantum
carpets". Cast in terms of a phase-space representation, e.g., the Wigner function, it turns out
that the time evolution induced by this potential is identical to that of a classical free particle, i.e.,
it reduces to a shear of phase space. The reason for this apparent disappearance of quantum effects
lies in the discrete translation invariance achieved if the potential box is repeated periodically in
space, resulting in a chain of delta potentials. Obviously, semiclassical approximations are useless
in this case.

In the present project, we deliberately give up this helpful symmetry, take the concept of a box
seriously, and consider a system confined between two "massive" walls which require the
wavefunction to vanish everywhere outside the box. This system constitutes a hybrid between two
extremes, a free particle inside and a highly nonlinear potential at the walls, considering them as
the limit n to infinity of V(x) ~ x^2n. For this system we find the exact propagator of the Wigner
function, which now does exhibit strong dynamical quantum effects, and compare it to the Wigner propagator for smooth confining potentials.

In constructing a semiclassical approximation, we have to discard approximation schemes developed
earlier for the Wigner propagator in smooth anharmonic potentials. Instead, we find that the decisive
mechanism to include dynamical coherence effects is the superposition of topologically distinct
classical trajectories, as they arise through multiple reflections between the walls of the box.

As a side task, we conceive coherent states for the potential box as minimum-uncertainty Gaussians
superposed such as to comply with the boundary conditions for this system. Our results for the
potential box are complemented by corresponding calculations for a system confined to a half space,
following the same strategy of replacing a “thin delta” by a “massive Heaviside” function representing
the wall. Besides illuminating mathematical subtleties, we expect this project to be of interest for
solid-state physics where quantum dots cannot be replaced by delta-chain potentials as in optical
grids.

T. Dittrich, O. Rodríguez, C. Viviescas, Y. Zuleta