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

Mathematical Physics Seminar

19th June 2018, 2:00 pm – 3:00 pm

Howard House, 4th Floor Seminar Room

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

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