If an ensemble of particles moving parallel to the x axis in the positive direction impinges on a piecewise continuous potential confined to the interval [−a,a] (a>0) it will divide into a transmitted ensemble and a reflected ensemble. It is shown that the classical results for the means, variances, and covariance of position and velocity of the transmitted and reflected ensembles hold in quantum mechanics if the incident state is assumed to be pure and defined by a C∞ wave function whose Fourier transform has bounded, positive support, on which the modulus and argument of the transmission and reflection coefficients are C∞.
If an ensemble of particles moving parallel to the x axis in the positive direction impinges on a piecewise continuous potential confined to the interval [−a,a] (a>0) it will divide into a transmitted ensemble and a reflected ensemble. It is shown that the classical results for the means, variances, and covariance of position and velocity of the transmitted and reflected ensembles hold in quantum mechanics if the incident state is assumed to be pure and defined by a C∞ wave function whose Fourier transform has bounded, positive support, on which the modulus and argument of the transmission and reflection coefficients are C∞.
If an ensemble of particles moving parallel to the x axis in the positive direction impinges on a piecewise continuous potential confined to the interval [−a,a] (a>0) it will divide into a transmitted ensemble and a reflected ensemble. It is shown that the classical results for the means, variances, and covariance of position and velocity of the transmitted and reflected ensembles hold in quantum mechanics if the position and momentum probability densities of the incident ensemble are assumed to be sufficiently localized, both in momentum and position. It is also assumed that the support of the probability density of momentum is bounded and positive, and on it the moduli and arguments of the transmission and reflection coefficients are sufficiently well behaved.
In one-dimensional scattering, the portion of particles that is transmitted in general vanishes as the kinetic energy of the incident particles approaches zero. Based upon published results from the literature, the present work shows that for a potential consisting of two Dirac δ functions of arbitrary strength, a finite portion of the incident particles is transmitted at threshold for certain choices of the set of parameters defining the potential. The underlying reasons are shown to be related to the structure of the bound levels of the potential. It is shown that for a potential with a finite range but of general shape the reflection coefficient in general tends to unity near threshold, which is what is usually observed. However, in the presence of a bound level at the onset of the continuum, a finite portion of the incident particles is transmitted at threshold for a potential with a general shape and, furthermore, it can be shown that if the potential V is symmetric with V(x)=V(−x) then resonance transmission is observed, i.e., the reflection coefficient vanishes at threshold.
It is shown that in the case of the one-particle one-dimensional scattering problem for a given time-independent potential, for each state of the whole quantum ensemble of identically prepared particles, there is an unique pair of (subensemble's) solutions to the Schrödinger equation, which, as we postulate, describe separately transmission and reflection: in the case of nonstationary states, for any instant of time, these functions are orthogonal and their sum describes the state of all particles; evolving with constant norms, one of them approaches at late times the transmitted wave packet and another approaches the reflected packet. Both for transmission and reflection, 1) well before and after the scattering event, the average kinetic energy of particles is the same, 2) the average starting point differs, in the general case, from that for all particles. It is shown that for reflection, in the case of symmetric potential barriers, the domain of the motion of particles is bounded by the midpoint of the barrier region. We define (exact and asymptotic) transmission and reflection times and show that the basic results of our formalism can be, in principle, checked experimentally.
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x) exceeds the value of the total energy E, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=a at about the time at which the incident packet reaches the point x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.
Abstract We consider a quantum particle with energy E incident upon a one-dimensional potential. We show that the probabilities of transmission and reflection are the same for incidence upon a general potential from either side (from ‘the left’ or ‘the right’). This equality holds true for any potential which goes to constant values as x → ± ∞ and is finite for all x . We present a remarkably simple proof that the probabilities are equal. The simplicity of our proof is the most important pedagogical result of our paper, and will be easily understood by undergraduate students in second to fourth year. We discuss several cases, including the step potential and the potential barrier.
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.