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Boundary conditions in tunneling via quantum hydrodynamics

Antonio Boulhosa Nassar

Published Jan 1, 1993• 4 citations
OpenAlex

Abstract

Via the hydrodynamical formulation of quantum mechanics, an approach to the problem of tunneling through sharp-edged potential barriers is developed. Above all, it is shown how more general boundary conditions follow from the continuity of mass, momentum, and energy.

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The process of dynamical tunneling between local mode states is analyzed within the Bohmian approach to quantum mechanics. A number of quantum trajectories are followed for both a coupled Morse potential and the Barbanis potential. The superposition of nearly degenerate eigenstates leads to the formation of numerous permanent nodes concentrated near the potential valley. Around these nodes, vortices are evident on the flux map. Quantum trajectories can be attracted by a vortex and their velocity and kinetic energy will be significantly changed as they spiral around the vortex. However, trajectories starting remotely from vortices reflect more realistically the transition between a pair of localized states. These quantum trajectories can go through one or more regions where the potential energy is higher than the total energy. Hence, barrier tunneling accompanies dynamical tunneling.

The Journal of Chemical Physics•Sep 18, 2003•15 citations
Via the hydrodynamical formulation of quantum mechanics, we propose new boundary conditions to the problem of tunneling through sharp-edged potential barriers with and without dissipation. Above all, these boundary conditions can be matched without assuming continuity of the wave function at a surface σ where the potential undergoes a finite jump. Our approach yields a general expression for transmission probability from which tunneling with and without dissipation can be directly correlated to standard discussions found in textbooks.
Abstract A distinguishing feature of quantum mechanics, as compared with classical mechanics, is the phenomenon of matter exhibiting wave-like properties. A direct consequence of this wave property of particles is the tunneling of particles through potential energy barriers.-3 The classical particle is forbidden from penetrating the barrier region. In this chapter, the problem of one-dimensional (1D) quantum mechanical tunneling of particles through potential barriers of arbitrary shape is treated. We show that tunneling is a mixed boundary value problem, and obtain the action integral that leads to Schrodinger’s equation and at the same time gives the equations defining the boundary conditions (BCs).
The quantum hydrodynamic approach is based on the de Broglie-Bohm formulation of quantum mechanics. The resulting fluid-like equations of motion describe the flow of probability and an accurate solution to these equations is equivalent to solving the time-dependent Schroedinger equation. Furthermore, the hydrodynamic approach provides new insight into the mechanisms as well as an alternative computational approach for treating tunneling phenomena. New concepts include well-defined 'quantum trajectories', 'quantum potential', and 'quantum force' all of which have classical analogues. The quantum potential and its associated force give rise to all quantum mechanical effects such as zero point energy, tunneling, and interference. A new numerical approach called the Iterative Finite Difference Method (IFDM) will be discussed. The IFDM is used to solve the set of non-linear coupled hydrodynamic equations. It is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a 'smart' Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is also capable of treating anharmonic potentials. Example calculations using the IFDM will be presented which include: a one-dimensional Gaussian wave packet tunneling through an Eckart barrier, a one-dimensional bound-state Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. Approximate treatments of the quantum hydrodynamic equations will also be discussed which could allow scaling of the calculations to hundreds of degrees of freedom which is important for treating tunneling phenomena in condensed phase systems.
This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.
The process of dynamical tunneling between local mode states is analyzed within the Bohmian approach to quantum mechanics. A number of quantum trajectories are followed for both a coupled Morse potential and the Barbanis potential. The superposition of nearly degenerate eigenstates leads to the formation of numerous permanent nodes concentrated near the potential valley. Around these nodes, vortices are evident on the flux map. Quantum trajectories can be attracted by a vortex and their velocity and kinetic energy will be significantly changed as they spiral around the vortex. However, trajectories starting remotely from vortices reflect more realistically the transition between a pair of localized states. These quantum trajectories can go through one or more regions where the potential energy is higher than the total energy. Hence, barrier tunneling accompanies dynamical tunneling.