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American Journal of Physics

Quantum scattering in two dimensions

Sadhan K. Adhikari

Published Apr 1, 1986• 249 citations
Publisher

Abstract

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.

DOI
10.1119/1.14623

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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.
A self-contained discussion of integral equations of scattering is presented in the case of centrally symmetric potentials in one dimension, which will facilitate the understanding of more complex scattering integral equations in two and three dimensions. The present discussion illustrates in a simple fashion the concept of partial-wave decomposition, Green’s function, Lippmann–Schwinger integral equations of scattering for wave function and transition operator, optical theorem, and unitarity relation. We illustrate the present approach with a Dirac delta potential.
A self-contained discussion of nonrelativistic quantum scattering is presented in the case of central potentials in one space dimension, which will facilitate the understanding of the more complex scattering theory in two and three dimensions. The present discussion illustrates in a simple way the concept of partial-wave decomposition, phase shift, optical theorem and There has been great interest in studying tunneling phenomena in a finite superlattice [1], which essentially deals with quantum scattering in one dimension. It has also been emphasized [2] that a study of quantum scattering in one dimension is important for the study of Levinson’s theorem and virial coefficient. Moreover, one-dimensional quantum
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.
After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.
The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain (Asymptotic Condition, in- and out- states, scattering operator S^\hat{S}S^). The aim of Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain. This amounts to introduce the Lippmann-Schwinger equation for the stationary scattering states (denoted as ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩), to discuss fundamental properties of ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩, and subsequently to construct S^\hat{S}S^ and T^\hat{T}T^ operators in terms of ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩. Physical contents of the S^\hat{S}S^ and T^\hat{T}T^ operators is then illuminated by deriving explicit formulas for the probability of transmission/reflection of our quantum particle through/from the interaction region of the potential. An illustrative numerical example is given, which also highlights an existence of scattering resonances. Finally, Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism), which offers an extremely powerful tool suitable for clear cut understanding of the resonance phenomena.