The Schrödinger equation for the scattered particle is converted into an integral equation, in which the kernel is obtained by the use of the condition that the events described by the solutions of the equation must be causally related, i.e., that causes must always precede the effects which they produce. The integral equation is iterated once, and it is found that its solution corresponds to the usual description of a scattering process in terms of an undeflected wave traversing the system plus a radially scattered wave moving outwards. In the second part, it is shown that a simple relation exists between solutions of Schrödinger's equation of the sort just described and the solution of the initial-value problem associated with the equation of motion for a free particle. By this means a solution of the scattering problem is set up which describes how the wave function develops as the wave which it describes propagates through the system. The series which results was first given by Feynman. As an example, the first Born approximation for the scattering problem is developed by each approach.
The Schrödinger equation for the scattered particle is converted into an integral equation, in which the kernel is obtained by the use of the condition that the events described by the solutions of the equation must be causally related, i.e., that causes must always precede the effects which they produce. The integral equation is iterated once, and it is found that its solution corresponds to the usual description of a scattering process in terms of an undeflected wave traversing the system plus a radially scattered wave moving outwards. In the second part, it is shown that a simple relation exists between solutions of Schrödinger's equation of the sort just described and the solution of the initial-value problem associated with the equation of motion for a free particle. By this means a solution of the scattering problem is set up which describes how the wave function develops as the wave which it describes propagates through the system. The series which results was first given by Feynman. As an example, the first Born approximation for the scattering problem is developed by each approach.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. e 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. e 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. e chapter concludes with a consideration of identical particle scattering and an application to the protonproton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
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.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.