Abstract A new approach to Quantum Scattering using the Theory of Sources and Absorbers is developed in these chapters. Existing theories of quantum scattering usually calculate the scattered wavefunction and the scattering cross-section by obtaining the solution to Schrödinger’s equation and ignoring the possibility of representing the problem in terms of the principle of stationary action. The action integral approach has several advantages, including quadratic convergence to the accurate results in the variational approach; also, the symmetry of the Lagrangian is readily manifest. Furthermore, the “near-field” wavefunctions and evanescent wave contributions are invariably ignored. Computational methods using discretization of space require the equivalent of “radiation boundary conditions” at large distances, and these are not cleanly implementable.
Abstract A new approach to Quantum Scattering using the Theory of Sources and Absorbers is developed in these chapters. Existing theories of quantum scattering usually calculate the scattered wavefunction and the scattering cross-section by obtaining the solution to Schrödinger’s equation and ignoring the possibility of representing the problem in terms of the principle of stationary action. The action integral approach has several advantages, including quadratic convergence to the accurate results in the variational approach; also, the symmetry of the Lagrangian is readily manifest. Furthermore, the “near-field” wavefunctions and evanescent wave contributions are invariably ignored. Computational methods using discretization of space require the equivalent of “radiation boundary conditions” at large distances, and these are not cleanly implementable.
Abstract The author has developed a new approach to treat quantum scattering using the method of sources and absorbers. Textbook discussions of quantum scattering use the radiation, or asymptotic boundary conditions first developed by Lord Rayleigh. These treatments do not allow the prediction and study of effects associated with evanescent modes since these modes do not survive at large distances. The author has developed the theory of quantum scattering with sources and absorbers to show that evanescent modes are always present in quantum waveguides and in multi-component wavefunctions, such as those occurring in energy band structure analysis and in relativistic wave equations.
Abstract This chapter introduces Schrödinger’s equation and the action integral from which the differential equation is derived by the principle of stationary action. This principle is central to almost all of physics. The discretization of the integral with finite elements and interpolation with polynomials scaled by interpolation coefficients at the ends of the elements (nodes) allows us to integrate out the spatial dependence leaving the action as a binomial expression in the coefficients. Now the action is varied with respect to these coefficients to obtain a discretized matrix version of the Schrödinger’s equation. This typically leads to matrix equation or a generalized eigenvalue problem to solve. In simple terms, this is the essence of the finite element method. The boundary conditions for the wavefunction and the derivative of the wavefunction can be very easily imposed at the nodes prior to the solution of the matrix equations. The probability current has to be continuous at interfaces in the interior of the physical domain. The form of the current is obtained by a gauge-variational method in an unambiguous manner. Going from the differential equation to the action integral, and vice versa, are treated in great detail so that the reader can gain confidence in working with variational principles.
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.
Due to the space and time dependence of the wavefunction in the time-dependent Schrödinger equation, different boundary conditions are possible. The equation is usually solved as an `initial-value problem', by fixing the value of the wavefunction in all space at a given instant. We compare this standard approach to `source boundary conditions' that fix the wave at all times in a given region, in particular at a point in one dimension. In contrast to the well known physical interpretation of the initial-value-problem approach, the interpretation of the source approach has remained unclear, since it introduces negative energy components, even for `free motion', and a time-dependent norm. This work provides physical meaning to the source method by finding the link with equivalent initial-value problems.