15 research outputs found
Inner products of resonance solutions in 1-D quantum barriers
The properties of a prescription for the inner products of the resonance
(Gamow states), scattering (Dirac kets), and bound states for 1-dimensional
quantum barriers are worked out. The divergent asypmtotic behaviour of the
Gamow states is regularized using a Gaussian convergence factor first
introduced by Zel'dovich. With this prescription, most of these states (with
discrete complex energies) are found to be orthogonal to each other, to the
bound states, and to the Dirac kets, except when they are neighbors, in which
case the inner product is divergent. Therefore, as it happens for the continuum
scattering states, the norm of the resonant ones remains non-calculable. Thus,
they exhibit properties half way between the (continuum real) Dirac-delta
orthogonality and the (discrete real) Kronecker-delta orthogonality of the
bound states.Comment: 13 pages, 2 figure
Tunnelling of plane waves through a square barrier
The time evolution of plane waves in the presence of a 1-dimensional square
quantum barrier is considered. Comparison is made between the cases of an
infinite and a cut-off (shutter) initial plane wave. The difference is relevant
when the results are applied to the analysis of the tunnelling regime. This
work is focused on the analytical calculation of the time-evolved solution and
highlights the contribution of the resonant (Gamow) states.
PACS numbers: 11.10.Ef, 11.10.Lm, 04.60Comment: 16 page
Time of arrival in the presence of interactions
We introduce a formalism for the calculation of the time of arrival t at a
space point for particles traveling through interacting media. We develop a
general formulation that employs quantum canonical transformations from the
free to the interacting cases to construct t in the context of the Positive
Operator Valued Measures. We then compute the probability distribution in the
times of arrival at a point for particles that have undergone reflection,
transmission or tunneling off finite potential barriers. For narrow Gaussian
initial wave packets we obtain multimodal time distributions of the reflected
packets and a combination of the Hartman effect with unexpected retardation in
tunneling. We also employ explicitly our formalism to deal with arrivals in the
interaction region for the step and linear potentials.Comment: 20 pages including 5 eps figure