Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.1036

Transfer matrices for piecewise constant potentials

G. F. Torres del Castillo·I. Rubalcava Garcia
View original source ↗Read source document ↗Export metadata ↓

Abstract

By expressing the time-independent Schrodinger equation in one dimension as a system of two first-order differential equations, the transfer matrix for a rectangular potential barrier is obtained making use of the matrix exponential. It is shown that the transfer matrix allows one to find the bound states and the quasinormal modes. A similar treatment for the one-dimensional propagation of electromagnetic waves in a homogeneous medium is also presented.

Explore related subjects

physics.class-phphysics.gen-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

G. F. Torres del Castillo, I. Rubalcava Garcia. 2007-06-07. Transfer matrices for piecewise constant potentials. https://arxiv.org/abs/0706.1036

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Automated Sailboat Navigation using Controllabilty-Aware Nonlinear Model Predicitive Control under Stochastic Winds

This paper presents a systematic approach to plan and execute a time-efficient sailboat trajectory under the challenging stochastic wind conditions for automated sailboat. Unlike deterministic scenarios, stochastic wind disturbances introduce challenges such as gust unpredictability, directional shifts, and fluctuating apparent wind speeds. These characteristics render traditional planning methods, such as Line-of-Sight (LoS) and waypoint approaches, to be ineffective and not applicable, as they often rely on static or deterministic environmental models. This paper proposed a new path planner and controller based on nonlinear model predictive control method. It is compared with a baseline planner and controller. A Lie-algebraic analysis of the sailboat dynamics is carried out to identify the operating conditions under which the vessel loses first-order control authority in surge, and these conditions are embedded as constraints in the NMPC. Finally, simulation results are provided to validate the proposed framework.

physics.class-ph↗
arXiv2026

Recovering long-range cumulative response to geometric frustration in quasi-1d systems, mediated by constitutive softness

Cumulative geometric frustration can drive self-limited assembly and morphology selection through size-dependent energetic costs. However, the slenderness of quasi-one-dimensional systems generally suppresses the formation of long-range longitudinal gradients. We show that the suppression of longitudinal gradients can be overcome by tuning the ratio between the longitudinal and transverse (shear) moduli. We demonstrate the recovery of cumulative frustration across distinct quasi-one-dimensional systems, each frustrated through a different mechanism, by the introduction of a soft response mode. The resulting saturation length scale is universal and is dictated by the local constitutive information of the system.

physics.class-ph↗
arXiv2026

Generalized reverberation theory in diffuse sound fields: Introducing mean residual free path for macroscopic and microscopic unification

Sabine's foundational theory established the cornerstone of modern architectural acoustics. However, it fails to predict zero reverberation time in perfectly absorptive rooms. Eyring subsequently addressed this issue by proposing a new formula, which Knudsen later extended to incorporate air absorption. Nevertheless, Eyring's underlying approach contains fundamental theoretical contradictions. To resolve these flaws, the author previously introduced macroscopic and microscopic models for a revised reverberation theory; however, their mathematical unification, rigorous derivation from differential equations, and the mechanism behind Eyring's underestimation remained unclarified. This paper presents a comprehensive generalization and mathematical foundation of the revised theory. The macroscopic model is derived directly from fundamental energy differential equations, incorporating air absorption and a generalized mean residual free path. By reformulating the microscopic model as a sequential convolution process, its fundamental consistency with the macroscopic model is mathematically demonstrated. Crucially, it is shown that Eyring's underestimation can be interpreted as stemming universally from the structural omission of temporal variance expansion (σ_{n}^{2}=nσ^{2}) inherent to multiple reflections, regardless of the assumed probability distribution. Finally, ray-tracing simulations validate the scale-invariant accuracy of the proposed theory. Ultimately, this mathematically consistent framework establishes the theoretical limit for future generalized theories in non-diffuse sound fields.

physics.class-ph↗