arXiv subjects
Publications and source records attributed to Rafael de la Madrid.
We construct the Gamow Golden Rule of multichannel scattering, and use it to obtain the decay distributions, the partial decay constants, the partial decay widths, and the branching fractions of a resonance that has several decay modes. We exemplify the results using two coupled-channel square well potentials.
By using the fact that the Gamow states in the momentum representation are square integrable, we obtain the differential and the total decay width of a two-body, non-relativistic decay. The resulting Gamow Golden Rule is well suited to describe both energy and angular decay distributions, and it becomes the Fermi Golden Rule when the resonance is long-lived and far from the energy threshold. We also show that the correct density of states and phase space factors arise naturally from the Gamow Golden Rule. The upshot is that the Gamow states and the Golden Rule can be combined into a unified description of quantum resonances.
We experimentally show that, unlike the solid-solid frictional force, the solid-liquid retention force is determined by interactions at the triple line rather than over the solid-liquid contact area, as predicted by theory. We have prepared drops whose triple line enclosed a uniform surface, a hydrophobic island, and topographical inhomogeneities, and measured the same retention force for all three cases. We have also studied the retention force on drops whose initial triple line was non-circumferential and measured different retention forces for different shapes, also as predicted by theory. The experiments with non-circumferential drops provide (1) another way to show that the retention force is not proportional to the contact area, and (2) a manner to falsify a recent theory on drop depinning.
We calculate the normal capillary retention force that anchors a drop to a solid surface in the direction perpendicular to the surface, and study the relationship between such force and the Young-Dupre work of adhesion. We also calculate the work necessary to create or destroy a patch of solid-liquid interface by moving the triple line on a solid substrate. We argue that when the capillary number is small and a drop is sliding on a surface at constant speed, the lateral retention force is the major source of energy dissipation, whereas viscous dissipation plays a minor role.
We address the issues raised in [R. Tadmor et al., Langmuir 2020, 36, 475-476]. In particular, we explain why we did not use Tadmor's theory to explain our results.
We compare the lateral retention forces on sessile drops (which are drops that are placed on top of a solid surface), pendant drops (which are drops that are placed on the underside of the surface), and inverted sessile drops (which are drops that are first placed on top and then on the underside of the surface by flipping the surface). We have found experimentally that the retention force on a truly pendant drop is always smaller than that on a sessile drop. However, the retention force on an inverted sessile drop is comparable to, and usually larger than, that on a sessile drop. Thus, the retention force on a drop depends not only on whether it is placed on top or on bottom of a surface, but also on the history of drop deposition, since such history affects the width, the shape and the contact angles of the drop.
We express the resonant energies of the delta-shell potential in terms of the Lambert $W$ function, and we calculate their decay widths and decay constants. The ensuing numerical results strengthen the interpretation of such decay widths and constants as a way to quantify the coupling between a resonance and the continuum. We calculate explicitly the decay energy spectrum of the resonances of the delta-shell potential, and we show numerically that the lineshape of such spectrum is not the same as, and can be very different from, the Breit-Wigner (Lorentzian) distribution. We argue that the standard Golden Rule cannot describe the interference of two resonances, and we show how to describe such interference by way of the decay energy spectrum of two resonant states.
We introduce the differential and the total decay widths of a resonant (Gamow) state decaying into a continuum of stable states. When the resonance has several decay modes, we introduce the corresponding partial decay widths and branching fractions. In the approximation that the resonance is sharp, the expressions for the differential, partial and total decay widths of a resonant state bear a close resemblance with the Golden Rule. In such approximation, the branching fractions of a resonant state are the same as the standard branching fractions obtained by way of the Golden Rule. We also introduce dimensionless decay constants along with their associated differential decay constants, and we express experimentally measurable quantities such as the branching fractions and the energy distributions of decay events in terms of those dimensionless decay constants.
We present a simple experiment that demonstrates how a water drop hanging from a Plexiglas surface (pendant drop) experiences a lateral retention force that is comparable to, and in some cases larger than, the lateral retention force on a drop resting on top of the surface (sessile drop). The experiment also affords a simple demonstration of the Coriolis effect in two dimensions.
We demonstrate that mechanical waves traveling in a torsional, mechanical wave machine exhibit dispersion due to gravity and the discreteness of the medium. We also show that although the dispersion due to discreteness is negligible, the dispersion due to gravity can be easily measured, and can be shown to disappear in a zero-gravity environment.
We present a theoretical analysis of quantum decay in which the survival probability is replaced by a decay rate that is equal to the absolute value squared of the wave function in the time representation. The wave function in the time representation is simply the Fourier transform of the wave function in the energy representation, and it is also the probability amplitude generated by the Positive Operator Valued Measure of a time operator. The present analysis endows time with a dynamical character in quantum decay, and it is applicable only when the unstable system is monitored continuously while it decays. When the analysis is applied to the Gamow state, one recovers the exponential decay law. The analysis allows us to interpret the oscillations in the decay rate of the GSI anomaly, of neutral mesons, and of fluorescence quantum beats as the result of the interference of two resonances in the time representation. In addition, the analysis allows us to show that the time of flight of a resonance coincides with its lifetime.
We use the resonances of the spherical shell potential to present a thorough description of the Gamow (quasinormal) states within the rigged Hilbert space. It will be concluded that the natural setting for the Gamow states is a rigged Hilbert space whose test functions fall off at infinity faster than Gaussians.
We review the way to analytically continue the Lippmann-Schwinger bras and kets into the complex plane. We will see that a naive analytic continuation leads to nonsensical results in resonance theory, and we will explain how the non-obvious but correct analytical continuation is done. We will see that the physical basis for the non-obvious but correct analytic continuation lies in the invariance of the Hamiltonian under anti-unitary symmetries such as time reversal or PT.