We consider the scattering of nonrelativistic particles in three dimensions by a contact potential Ωℏ 2 δ( r )/2µ r α , which is defined as the a →0 limit of Ωℏ 2 δ( r - a )/2µ r α . It is surprising that it gives a nonvanishing cross section when α = 1 and Ω = -1. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter Ω that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.
We consider the scattering of nonrelativistic particles in three dimensions by a contact potential Ωℏ 2 δ( r )/2µ r α , which is defined as the a →0 limit of Ωℏ 2 δ( r - a )/2µ r α . It is surprising that it gives a nonvanishing cross section when α = 1 and Ω = -1. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter Ω that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.
We consider the scattering of nonrelativistic particles in three dimensions by a contact potential $\\Omega\\hbar^2\\delta(r)/ 2\\mu r^\\alpha$ which is defined as the ato0 limit of $\\Omega\\hbar^2\\delta(r-a)/2\\mu r^\\alpha$. It is surprising that it gives a nonvanishing cross section when alpha=1 and Omega=−1. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter Omega that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.
Abstract In classical mechanics (c.m.), and near the semi-classical limit h→0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ0/ρ∞ are found for scattering by attractive central potentials U(r); here ρ0,∞ (and v0,∞) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials (U(0) > — ∞), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high v∞ ), α1 = v∞/v0, α2 = 1, α3 = v0/v∞ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U(r→0) ~ ─ C/rq, with 0 < q < 1 in ID (where r ≡ |x|), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., α1(c.m.) = 0, α3(c.m.) = ∞, and α2(c.m.) = (1 —½q) N, where N = [integer part of (1 ─½q)-1]is the number of trajectories through any point (r, θ) in the limit r → 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low v∞) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r; for small v∞/u it yields the leading terms α1(s.c.l.) = Λ1(q)v∞/u, α2(s.c.I) = (1 ─½q)-1, α3(s.c.l.)= Λ3(q)u/v∞, where u ≡ (C/hqm1-q)1/(2-q) is a generalized Bohr velocity. Here Λ1,3 are functions of q alone, given in the text; as q →0 the α (s.c.l.) agree with the α quoted above for finite potentials. Even in the limit h = 0, α2 (s.c.l.) and α2 (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential ─ C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter α(e.g. U = ─ Cexp (─r/a)/rq), and if, near the s.c.l., v∞/u is regarded as a function not of h but more realistically of v∞, then the expressions α (s.c.l.) above apply only in an intermediate range 1/a ≪ mv∞/h ≪ (mC/h2)1/(2-q) which exists only if a ≫ (h2/mC)1/(2-q)).
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 exact solution is presented of the scattering problem of three equal particles interacting in one-dimension via two- and/or three-body inverse-square potentials. Both the classical and the quantal problems are treated. It is shown that the outcome of this scattering problem is an extremely simple relation between initial and final momenta, the latter being univocally determined by the former even in the quantal case. The solvability of the problem, and the simple results just mentioned, are peculiar to the equal particle case.
We consider the resonant Fermi gas, that is, two-component fermions in three dimensions interacting by a short-range potential of large scattering length. We introduce a quantity, the three-body contact, that determines several observables. Within the zero-range model, the number of nearby fermion triplets, the large-momentum tail of the center-of-mass momentum distribution of nearby fermion pairs, as well as the large-momentum tail of the two-particle momentum distribution, are expressed in terms of the three-body contact. For a small finite interaction range, the formation rate of deeply bound dimers by three-body recombination, as well as the three-body contribution to the finite-range correction to the energy, are expressed in terms of the three-body contact and of a three-body parameter. This three-body parameter, which vanishes in the zero-range limit, is defined through the asymptotic behavior of the zero-energy scattering state at distances intermediate between the range and the two-body scattering length. In general, the three-body contact has different contributions labeled by spin and angular momentum indices, and the three-body parameter can depend on those indices. We also include the generalization to unequal masses for ↑ and ↓ particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.