Abstract A familiar classroom heuristic for hard-sphere scattering is that the low-energy cross section ‘equals the surface area’ because a long-wavelength particle can ‘feel’ the sphere from all directions. More precisely, this heuristic means that, when the wavelength is large compared with the size of the scatterer, the wave is sensitive to the global boundary condition imposed by the whole hard obstacle; in the zero-energy limit this global response is represented by the monopole term of an exterior Laplace problem. While useful as intuition, the heuristic prompts a natural question for students: is there a general connection between surface area and the threshold cross section of a non-spherical hard target? This pedagogical note, aimed at advanced undergraduates and beginning graduate students, shows how to answer that question systematically. For a perfectly hard obstacle the zero-energy Schrödinger equation reduces to an exterior Laplace problem with Dirichlet boundary conditions. Comparing its far-field monopole term with the corresponding electrostatic problem yields a direct mapping between the scattering length and the electrostatic capacitance of the same shape. The mapping provides a simple route to threshold cross sections for bodies of arbitrary shape, without relying on separability. Using published capacitances for several examples, we compare the threshold cross section with surface area and show that the equality for a sphere is not a general surface-area law.
Abstract A familiar classroom heuristic for hard-sphere scattering is that the low-energy cross section ‘equals the surface area’ because a long-wavelength particle can ‘feel’ the sphere from all directions. More precisely, this heuristic means that, when the wavelength is large compared with the size of the scatterer, the wave is sensitive to the global boundary condition imposed by the whole hard obstacle; in the zero-energy limit this global response is represented by the monopole term of an exterior Laplace problem. While useful as intuition, the heuristic prompts a natural question for students: is there a general connection between surface area and the threshold cross section of a non-spherical hard target? This pedagogical note, aimed at advanced undergraduates and beginning graduate students, shows how to answer that question systematically. For a perfectly hard obstacle the zero-energy Schrödinger equation reduces to an exterior Laplace problem with Dirichlet boundary conditions. Comparing its far-field monopole term with the corresponding electrostatic problem yields a direct mapping between the scattering length and the electrostatic capacitance of the same shape. The mapping provides a simple route to threshold cross sections for bodies of arbitrary shape, without relying on separability. Using published capacitances for several examples, we compare the threshold cross section with surface area and show that the equality for a sphere is not a general surface-area law.
The most important parameters in the study of low-energy scattering are the s-wave and p-wave scattering lengths and the s-wave effective range. We solve the scattering problem and find two useful formulas for the scattering length and the effective range for any angular momentum, as long as the Wigner threshold law holds. Using that formalism, we obtain a set of useful formulas for the angular-momentum scattering parameters of four different model potentials: hard-sphere, soft-sphere, spherical well, and well-barrier potentials. The behavior of the scattering parameters close to Feshbach resonances is also analyzed. Our derivations can be useful as hands-on activities for learning scattering theory.
Introduction.We give three applications of probability to scattering theory.The first question we consider is that of the "scattering length."We obtain a probabilistic expression for scattering length and use this expression to prove that in the case of an impenetrable obstacle ("hard" potential) the scattering length T is the same as the electrostatic capacity.Next, we briefly touch upon a problem which is inherently probabilistic-namely, scattering in a medium with random boundary.We find that if a large number of small hard spheres ("dust particles") are distributed at random in the medium, the effect is to shift all the eigenvalues by a constant.Our final topic is the so-called "inverse problem" of scattering theory: given the scattering operator, to compute the potential.For certain spherically symmetric potentials, this problem was solved in a famous paper of GelTand and Levitan.Here we show that their result can be included in the circle of potential-theoretic problems which can be solved by means of Brownian motion and Wiener integrals.We quickly sketch some of the physical background.We are concerned with the Schrodinger equation of non-relativistic quantum mechanics.In its stationary or time-independent form, this equation isThe "wave function" I/J depends on both the position vector fiEfi 3 and the wave number k.The real parameter k 2 is proportional to the energy.The function q(R) is called the "potential."It represents the interaction of the quantum-mechanical "particle" with some force field.The "scattering problem" is, roughly speaking, to determine the effect of this interaction asymptotically for large R.More precisely, we seek a solution of (1) which at |fl| = » is of the asymptotic form e iV2kz + fk(e)e iV ^k> ^l\\R\\ where e is the unit vector in the direction of R. The first term, e iV2kz , is an incident wave in the direction of the zaxis; the second term /fc(e)e iV ^I^H/||R||, is the "scattered wave."
Abstract The advances in cold atom experiments have allowed construction of confining traps in the form of curved surfaces. This opens up the possibility of studying quantum gases in curved manifolds. On closed surfaces, many fundamental processes are affected by the local and global properties, i.e. the curvature and the topology of the surface. In this paper, we study the problem of potential scattering on a spherical surface and discuss its difference with that on a 2D plane. For bound states with angular momentum m , their energies ( E m ) on a sphere are related to those on a 2D plane ( − ∣ E m , o ∣ ) as E m = − ∣ E m , o ∣ + E R m 2 − 1 3 + O r o 2 R 2 , where E R = ℏ 2 / ( 2 MR 2 ) , and R is the radius of the sphere. Due to the finite extent of the manifold, the phase shifts on a sphere at energies E ∼ E R differ significantly from those on a 2D plane. As energy E approaches zero, the phase shift in the planar case approaches 0, whereas in the spherical case it reaches a constant that connects the microscopic length scale to the largest length scale R .
We are concerned with the justification of the statement, commonly (explicitly or implicitly) used in quantum scattering theory, that for a free non-relativistic quantum particle with initial wave function Ψ0(x), surrounded by detectors along a sphere of large radius R, the probability distribution of the detection time and place has asymptotic density (i.e., scattering cross section) σ(x,t)=m3ℏ−3Rt−4∣Ψ0(mx/ℏt)∣2 with Ψ0 the Fourier transform of Ψ0. We give two derivations of this formula, based on different macroscopic models of the detection process. The first one consists of a negative imaginary potential of strength λ>0 in the detector volume (i.e., outside the sphere of radius R) in the limit R→∞,λ→0,Rλ→∞. The second one consists of repeated nearly-projective measurements of (approximately) the observable 1∣x∣>R at times T,2T,3T,… in the limit R→∞,T→∞,T/R→0; this setup is similar to that of the quantum Zeno effect, except that there one considers T→0 instead of T→∞. We also provide a comparison to Bohmian mechanics: while in the absence of detectors, the arrival times and places of the Bohmian trajectories on the sphere of radius R have asymptotic distribution density given by the same formula as σ, their deviation from the detection times and places is not necessarily small, although it is small compared to R, so the effect of the presence of detectors on the particle can be neglected in the far-field regime. We also cover the generalization to surfaces with non-spherical shape, to the case of N non-interacting particles, to time-dependent surfaces, and to the Dirac equation.
Transmission and scattering problems involving complex potentials are important in physics, in particular in describing nuclear collisions. We describe many pedagogical features of transmission in one dimension and scattering and absorption cross sections in three dimensions for a rectangular absorptive potential and compare the results with scattering from real barriers and a hard sphere. For a given energy in one dimension we show that the absorption reaches a maximum for a critical value of the absorption strength and then monotonically decreases with an increase in the absorption strength. An infinitely absorptive well becomes almost fully reflective and thus is similar to an infinitely high barrier. Similar results are found in three dimensions. We show that the absorption cross section in three dimensions diverges at a threshold energy, which is of critical importance for exothermic reactions.
The scattering of a monoenergetic beam of atoms by a hard hemisphere on a hard flat surface is formulated quantum mechanically. Differential and integral cross sections are calculated as a function of incident beam energy and incident angle and compared with the familiar results of scattering from a free hard sphere. Two types of interference processes are identified. In addition to the previously discussed off-specular reflection symmetry interferences, Rayleigh oscillations are found in the integral cross sections. They result from the interference of specular scattering from the hemisphere and from the flat surface. Their phases are found to be independent of the angle of incidence when plotted versus the normal component of the incident wave vector.