Quantum Scattering

Abstract

Scattering is easier than gathering.-Irish proverb (A. Wirzba, P. Cvitanovi´cCvitanovi´c and N. Whelan) S o far the trace formulas have been derived assuming that the system under consideration is bound. As we shall now see, we are in luck-the semiclas-sics of bound systems is all we need to understand the semiclassics for open, scattering systems as well. We start by a brief review of the quantum theory of elastic scattering of a point particle from a (repulsive) potential, and then develop the connection to the standard Gutzwiller theory for bound systems. We do this in two steps-first, a heuristic derivation which helps us understand in what sense density of states is "density," and then we sketch a general derivation of the central result of the spectral theory of quantum scattering, the Krein-Friedel-Lloyd formula. The end result is that we establish a connection between the scattering resonances (both positions and widths) of an open quantum system and the poles of the trace of the Green's function, which we learned to analyze in earlier chapters. 39.1 Density of states For a scattering problem the density of states (35.16) appear ill defined since formulas such as (38.6) involve integration over infinite spatial extent. What we will now show is that a quantity that makes sense physically is the difference of two densities-the first with the scatterer present and the second with the scatterer absent. In non-relativistic dynamics the relative motion can be separated from the center-of-mass motion. Therefore the elastic scattering of two particles can be treated as the scattering of one particle from a static potential V(q). We will study 700 CHAPTER 39. QUANTUM SCATTERING 701 the scattering of a point-particle of (reduced) mass m by a short-range potential V(q), excluding inter alia the Coulomb potential. (The Coulomb potential decays slowly as a function of q so that various asymptotic approximations which apply to general potentials fail for it.) Although we can choose the spatial coordinate frame freely, it is advisable to place its origin somewhere near the geometrical center of the potential. The scattering problem is solved, if a scattering solution to the time-independent Schrödinger equation (36.2) − 2 2m ∂ 2 ∂q 2 + V(q) φ k (q) = Eφ k (q) (39.1) can be constructed. Here E is the energy, p = k the initial momentum of the particle, and k the corresponding wave vector. When the argument r = |q| of the wave function is large compared to the typical size a of the scattering region, the Schrödinger equation effectively becomes a free particle equation because of the short-range nature of the potential. In the asymptotic domain r ≫ a, the solution φ k (q) of (39.1) can be written as superpo-sition of ingoing and outgoing solutions of the free particle Schrödinger equation for fixed angular momentum: φ(q) = Aφ (−) (q) + Bφ (+) (q) , (+ boundary conditions) , where in 1-dimensional problems φ (−) (q), φ (+) (q) are the "left," "right" moving plane waves, and in higher-dimensional scattering problems the "incoming," "out-going" radial waves, with the constant matrices A, B fixed by the boundary conditions. What are the boundary conditions? The scatterer can modify only the outgoing waves (see figure 39.1), since the incoming ones, by definition, have yet to encounter the scattering region. This defines the quantum mechanical scattering matrix, or the S matrix φ m (r) = φ (−) m (r) + S mm ′ φ (+) m ′ (r). (39.2) All scattering effects are incorporated in the deviation of S from the unit matrix, the transition matrix T S = 1 − iT. (39.3) For concreteness, we have specialized to two dimensions, although the final formula is true for arbitrary dimensions. The indices m and m ′ are the angular mo-menta quantum numbers for the incoming and outgoing state of the scattering wave function, labeling the S-matrix elements S mm ′. More generally, given a set of quantum numbers β, γ, the S matrix is a collection S βγ of transition amplitudes β → γ normalized such that |S βγ | 2 is the probability of the β → γ transition. The total probability that the ingoing state β ends up in some outgoing state must add up to unity γ |S βγ | 2 = 1 , (39.4) scattering-29dec2004 ChaosBook.org version15.7, Apr 8 2015