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 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 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.
The problem of the quantization of evanescent waves, which appear in the angular spectrum representation of the electromagnetic field in a half-space, is discussed. Although evanescent waves are associated with material sources, scatterers, etc., we are able to treat the electromagnetic field, including the evanescent waves, effectively as a free field, by making use of the idea of the refractive index of a passive, macroscopically continuous medium. We consider a space which is filled with a homogeneous dielectric to the left of the plane z=0, and is empty to the right of the plane. Triplets of incident, reflected, and transmitted waves at the interface form the fundamental orthogonal modes of the space. By expanding the field in terms of these triplet modes, we show that the field Hamiltonian reduces to the sum of independent harmonic-oscillator Hamiltonians. The quantization is therefore straightforward. We introduce the creation and annihilation operators for the triplet wave modes, and encounter Fock states, coherent states, etc., for a field having evanescent wave components. The field commutator at two space-time points in the right half-space is shown to have an explicit contribution from evanescent waves, characterized by an exponential decay to the right and a propagation parallel to the interface. We also examine the problem of atomic excitation by quantized evanescent waves, and show that the results are of the form given by semiclassical treatments.
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.
In a collision problem, the energy of the incident particle is specified in advance, unlike the bound state case where the boundary conditions on the wavefunctions at large distances give rise to the quantization of energy into discrete energy levels of the system. In the scattering problems the behaviour of the wavefunction at large distances is obtained in terms of the energy of the incident particle. We discuss some of the simpler theories in this chapter, deferring some of the formal theories for the next chapter.
A self-contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in coordinate and momentum space, Lippmann–Schwinger integral equations of scattering for the scattering wavefunction and the transition operator, optical theorem, and the unitarity relation for the transition operator. The present definition of the scattering amplitude in terms of the asymptotic wavefunction differs from the usual definition. The present definition has certain advantages, for example, in writing the optical theorem and in studying the analytical properties of the scattering amplitude.
The evanescent states are common in the electro-magnetic environments,it's basic feature is the field strength decrease according to the exponential law from orginal position to far away.The electro-magnetic waves propagate through the reactive medium are common situations,corresponding wave propagation in the guided wave modes,dispersion mediums,and the ionic gases.In 1897,Lord Rayleigh's calculation predicated the evanescent states appear to propagate in rectangular metal waveguides.In order to analyze the phenomenon of waveguide below cut-off(WBCO),it seems that the evanescent states have a imaginary wave vector(a imaginary wave number).And so,we can say it is like a standing wave,the variation of field occurs in every position simultaneously,this means that they do not experience a phase shift in propagate direction.But the behaviour of WBCO was a bit strange,sometimes the phase constant β is negative,then in THz experiments the pulse delay is negative.Although the phase velocity of waveguide to be superluminal is well-known,the theoretical discussion does not explain why the phase velocity and the phase refractive index becomes negative.In quantum mechanics(QM),in barrier the evanescent states also have a imaginary wave parame-ter.This state can be explained by the Schrdinger equation,it is analogy with the Helmholtz equation in classical electro-magnetic theory.In practice,quantum tunnelling is the most popular physical process,it can be observed in many cases: the propagation of the graded index optical fiber,the two glass prisms of Bose experiment,etc.In scientist's opinion,to defy the light-speed limit is possible;for example,photons tunnel across a gap between two prisms yet arrive at same time as reflected photons that travelled a shorter distance,that means an ultra-fast transit between the two prisms:so much faster than the speed of light.As well-known,we are used the virtual particles in microscopic interaction processes since R.Feynman,but now we perhaps demonstrated here an example in the macroscopic range of a meter since G.Nimtz's experiment.