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Physics of the Solid State

Stationary motion of a quantum particle in an arbitrary one-dimensional potential

А. Ж. Хачатрян, Д. М. Седракян, V. A. Khoetsyan

Published Jul 1, 2010• 0 citations
Publisher

Abstract

A consistent approach to the description of a stationary motion of a quantum particle in an arbitrary one-dimensional potential has been developed. It has been proved that the wave function of an infinite motion can be expressed accurate to up two arbitrary constants with the use of one particular solution to the system of first-order linear differential equations. It has been shown that many well-known methods, such as the integral equation method, the transfer matrix method, the embedding method, and the method of combination of scattering parameters, are based on a general property of the solutions to the Schrödinger equation. Within the proposed approach, the relation between these methods becomes more transparent and their description can be well within a unified context.

DOI
10.1134/s1063783410070279

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A consistent approach to the description of a stationary motion of a quantum particle in an arbitrary one-dimensional potential has been developed. It has been proved that the wave function of an infinite motion can be expressed accurate to up two arbitrary constants with the use of one particular solution to the system of first-order linear differential equations. It has been shown that many well-known methods, such as the integral equation method, the transfer matrix method, the embedding method, and the method of combination of scattering parameters, are based on a general property of the solutions to the Schrödinger equation. Within the proposed approach, the relation between these methods becomes more transparent and their description can be well within a unified context.
A new approach to the problem of description of the stationary motion of a quantum particle in the field of a one-dimensional arbitrary potential is developed. It is shown that the wave function of infinite motion, with the accuracy to two arbitrary constants, can be expressed by means of an arbitrary single solution for some set of linear differential equations of the first order. It is shown that one general property of the Schrödinger equation solutions lies in the basis of many known methods of the problem consideration, such as the method of integral equations, transfer-matrix method, imbedding method, and method of combination of scattering parameters. Within the framework of the proposed approach, the connection between the above-mentioned methods becomes more transparent.
The exact solution to the one-dimensional Schr\"odinger equation which describes the motion of a particle scattering off a finite set of $\ensuremath{\delta}$-function potentials of arbitrary strengths and positions is found. The behavior of the TTT matrix both on and off the energy shell is also precisely given. Localization of states is discussed.
The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.
Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]
We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a $2\ifmmode\times\else\texttimes\fi{}2$ matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. We show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices, each related to the time-evolution operator for an effective nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for $\ensuremath{\delta}$-function potentials in 2D and 3D and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.
A simple exact solution of a one-dimensional Schr\"odinger equation which describes the motion of a particle in a continuous potential V(x)V(x)V(x) and scattering from a finite set of $\ensuremath{\delta}$-function potentials of arbitrary positions and strengths is found.