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TURKISH JOURNAL OF PHYSICS

General Formulation of the Scattered Matter Waves by a Quantum Shutter

Yusuf Ziya Umul

Published Jan 1, 2009• 5 citations
Publisher

Abstract

The scattering process of matter waves by a quantum shutter is investigated by using the spectrum integral representation. The scattered fields are expressed in terms of the Fresnel function. It is shown that the obtained equation gives the Moshinsky function for a one dimensional problem of the plane wave. Also a general integral representation is derived for two dimensional problems. The scattering of matter waves for some special wave-packets are examined analytically and numerically.

DOI
10.3906/fiz-0808-8

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The scattering process of matter waves by a quantum shutter is investigated by using the spectrum integral representation. The scattered fields are expressed in terms of the Fresnel function. It is shown that the obtained equation gives the Moshinsky function for a one dimensional problem of the plane wave. Also a general integral representation is derived for two dimensional problems. The scattering of matter waves for some special wave-packets are examined analytically and numerically.
In this study, the scattering of matter waves that accompany to the quantum particles by an absorbing half-plane is examined. A quantum shutter, which is opened at an initial time, is proposed to close the semi-infinite complementary aperture of the half-screen. Hence the matter waves, confined in a determined region of space, transmits to the second region by scattering from the semi-infinite aperture. The mathematical expressions of the scattered fields are plotted numerically and interpreted according to the diffraction theory of Young.
In a previous note a dynamical description of resonance scattering was given, and transient terms appeared in the wave function describing the process. To understand the physical significance of these terms, the transient effects that appear when a shutter is opened are discussed in this paper. For a nonrelativistic beam of particles, the transient current has a close mathematical resemblance with the intensity of light in the Fresnel diffraction by a straight edge. This is the reason for calling the transient phenomena by the name of diffraction in time.The shutter problem is discussed for particles whose wave functions satisfy the Schr\"odinger equation, the ordinary wave equation, and the Klein-Gordon equation. Only for the Schr\"odinger time-dependent equation do the transient wave functions resemble the solutions that appear in Sommerfeld's theory of diffraction. The connection of transient phenomena with the time-energy uncertainty relation, and the interpretation of the transient current in a scattering process, are briefly discussed. The relativistic wave functions for the shutter problem may play an important role in the dynamical description of a relativistic scattering process.
We analytically compute the time evolution of an initial infinite plane wave in the presence of a 1-dimensional square quantum barrier. This calculation generalizes the analysis of the shutter problem and sets the basis for the calculation of the transmission of general wave packets, aiming to work out the explicit contribution of the resonant (Gamow) states. The method relies mainly on the analytical properties of the Green function. The role of separate boundary conditions on the Green function and on the evolution equation is highlighted. As in previous works on related problems, only the determination of the resonant momenta requires numerical methods.
Abstract The diffraction process of an entangled non-relativistic quantum particle in time domain is investigated. The scenario consists of two entangled spin-0 particles and a quantum shutter, which is opened at an initial time. The particles are traveling in two opposite sides and expressed with a single wave function. The integral solution of the Schrödinger equation is considered for two particles case. The initial condition is expressed in terms of an entangled wave function. The time-diffracted matter wave is obtained in terms of the Fresnel integral. The effect of the phenomenon of diffraction in time is examined numerically for the entangled system of two particles.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. The integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. The relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. The chapter concludes with a consideration of identical particle scattering and an application to the proton–proton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
Abstract In quantum mechanics textbooks, a single-particle scattering theory is introduced. In the present work, a generalized scattering theory is presented, which can be in principle applied to the scattering problems of arbitrary number of particle. In laboratory frame, a generalized Lippmann-Schwinger scattering equation is derived. We emphasized that the derivation is rigorous, even for treating infinitesimals. No manual operation such as analytical continuation is allowed. In the case that before scattering N particles are plane waves and after the scattering they are new plane waves, the transition amplitude and transition probability are given and the generalized S matrix is presented. It is proved that the transition probability from a set of plane waves to a new set of plane waves of the N particles equal to that of the reciprocal process. The generalized theory is applied to the cases of one- and two-particle scattering as two examples. When applied to single-particle scattering problems, our generalized formalism degrades to that usually seen in the literature. When our generalized theory is applied to two-particle scattering problems, the formula of the transition probability of two-particle collision is given. It is shown that the transition probability of the scattering of two free particles is identical to that of the reciprocal process. This transition probability and the identity are needed in deriving Boltzmann transport equation in statistical mechanics. The case of identical particles is also discussed.