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Physical Review A

Mechanics of particles with nonmonotonic dispersion laws

Leonid D. Shvartsman, Dmitri Romanov, J. E. Golub

Published Sep 1, 1994• 9 citations
Publisher

Abstract

Motivated by the behavior of carriers in various condensed systems, we formulate and discuss the mechanics of a particle described by a nonmonotonic dispersion law. For illustration, we consider the scattering of a particle with a @sw-shaped dispersion law by a one-dimensional potential barrier. The negative mass feature of the dispersion gives rise to a number of unusual phenomena, including barrier penetration by a classical particle, particle trapping by the barrier, and scattering resonances in both transmission and reflection.

DOI
10.1103/physreva.50.r1969

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Motivated by the behavior of carriers in various condensed systems, we formulate and discuss the mechanics of a particle described by a nonmonotonic dispersion law. For illustration, we consider the scattering of a particle with a @sw-shaped dispersion law by a one-dimensional potential barrier. The negative mass feature of the dispersion gives rise to a number of unusual phenomena, including barrier penetration by a classical particle, particle trapping by the barrier, and scattering resonances in both transmission and reflection.
The interaction between incident and target particles is inferred from the numbers and angular distribution of scattered particles and from their energy distribution. It is important to recognize that a scattering region, as represented for example by a rectangular barrier, is a region where, in classical terms, forces are applied to the incident particles. A full analysis of scattering events must, in principle, embody the description of a localized free particle that travels toward a center of force and is scattered by it. Quantum-mechanically, that implies a theoretical analysis in terms of packet states, with an explicit account of their change of position with time. One of the most striking effects described by quantum mechanics is the transmission of incident particles across a potential barrier whose height is greater than the particle energy. Classically, these particles lack the energy to get over the barrier, yet for quantum-mechanical systems some fraction of incident particles are transmitted.
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).
Abstract Having examined free particles and particles that are confined in space by a potential energy term, we now consider the impact of a disturbance in the flat energy landscape for a free particle. By disturbance we means some kind of fixed “obstacle” which is either a positive (repelling) or negative (attractive) potential. We are interested in determining the impact on the free particle. Continuing to work mostly in one dimension, the particle described by a plane wave corresponding to momentum moving in the positive direction (a positive k−vector in the x−direction), we study elastic scattering. In one dimension, this means that we determine the probability that the particle is transmitted (continuing in the forward direction) or reflected (now moving in the backward direction.) We will also determine the nature of the solution inside the potential and in the case that the potential energy maximum is greater than the kinetic energy of the particle, we will show that the particle tunnels through the barrier. Interestingly, when we have two barriers, we can find conditions where the probability that the particle is transmitted is unity. This is the result of resonance, a feature of the wave-like nature of the particle’s wave function.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.
This chapter discusses energy levels for more realistic systems. It looks at a different class of problems involving the scattering of free particles from a potential. The chapter uses the free-particle wave functions to describe an important physical process – the scattering of a steady beam of particles from a barrier. To get the first one note that cannot step down for ever, for eventually the ground state will be reached. To construct the excited wave functions one must substitute a full polynomial for F in equation and equate the coefficients of all the powers to zero. The rapidly declining tail on a wave function which penetrates into a classically forbidden region of a barrier could, if the barrier were of finite thickness, emerge from the remote side and become wave-like again, representing the tunnelling of particles through the barrier.