Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Possible Characterization of Short-Range Interactions

J. D. Finley

Published Nov 1, 1969• 1 citation
Publisher

Abstract

In the preceding paper [J. Math. Phys. 10, 2047 (1969)], we investigated possible characterizations of ``short-range'' interactions by means of conditions on the total transition probability ∥Tψ∥2 which were valid only for certain types of sequences of wavepackets. In this paper we generalize the assumption on the transition probability so that it will apply to an arbitrary wavepacket. We do this by utilizing a measure G[R, ψ] of how much of a wavepacket ψ is within a sphere of radius R about the scattering center at all times. We then characterize a ``short-range'' interaction by the existence of a positive constant A and positive functions B(p) and C(p) such that for all R ≥ 0, ‖Tψ‖2≤AG[R,ψ]+〈C(p)e−RB(p)〉ψ.We then show that for such interactions the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse the size of which depends upon B(p). We compare our results with the similar results obtained by the method in the preceding paper and show that these new results much more closely approximate the behavior of the Yukawa interaction.

DOI
10.1063/1.1664802

Related papers

Short-Range Interactions and Analyticity in Momentum Transfer

J. D. Finley

We investigate possible characterizations of ``short-range'' interactions by means of conditions on the total transition probability ∥Tψ∥2. We desire to require that the total transition probability decrease exponentially as the displacement of the wavepacket ψ increases. It is shown that this can only be done for certain types of sequences of wavepackets, and a criterion is developed for selecting these ``proper'' wavepackets. We then show for all such interactions that what is essentially the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse which always includes the physical region. We then compare these results with the Schrödinger potential theory and thereby relate the type of the exponential decrease of a potential to the type of the exponential decrease of the corresponding transition probability. Finally, our results are compared with similar results previously obtained by others working on this problem.

Journal of Mathematical Physics•Nov 1, 1969•1 citation

Analytical calculations of scattering lengths in atomic physics

Radosław Szmytkowski

We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e - -Xe and Cs-Cs systems.

Journal of Physics A Mathematical and General•Dec 21, 1995•41 citations

Integral Approximations in Potential Scattering

Philip B. Burt, Forrest James Woods

A development of the scattering amplitude in potential scattering which is based upon replacing a smooth potential by a sequence of potential steps is given. The Schrödinger equation is solved exactly in each interval over which the potential is constant and by matching boundary conditions an expression relating the coefficients of two linearly independent solutions at small radius to those at large radius is found. From this the scattering amplitude is calculated. The limits of validity are found to be dependent upon the value of the expression (r/2)ddrln(2mℏ2[E−V]). Limiting cases of the amplitude are shown to be a modified WKB approximation and the Born approximation. A particular application to the Yukawa potential is discussed. The discussion is restricted to attractive potentials.

Journal of Mathematical Physics•Nov 1, 1971•1 citation

Analytical calculations of scattering lengths for a class of long-range potentials of interest for atomic physics

Radosław Szmytkowski

We derive two equivalent analytical expressions for an lth partial-wave scattering length al for central potentials with long-range tails of the form V(r)=−ℏ22mBrn−4(rn−2 + Rn−2)2−ℏ22mCr2(rn−2 + Rn−2), (r ⩾ rs, R > 0). For C = 0, this family of potentials reduces to the Lenz potentials discussed in a similar context in our earlier works [R. Szmytkowski, Acta Phys. Pol. A 79, 613 (1991); J. Phys. A: Math. Gen. 28, 7333 (1995)]. The formulas for al that we provide in this paper depend on the parameters B, C, and R characterizing the tail of the potential, on the core radius rs, as well as on the short-range scattering length als, the latter being due to the core part of the potential. The procedure, which may be viewed as an analytical extrapolation from als to al, is relied on the fact that the general solution to the zero-energy radial Schrödinger equation with the potential given above may be expressed analytically in terms of the generalized associated Legendre functions.

Journal of Mathematical Physics•Jan 1, 2020•1 citation

Introduction to the scattering theory

The asymptotic condition: A qualitative look Suppose that we have prepared an initial quantum state |ψ at t = 0. We may view |ψ as a square integrable wavepacket ψ(0, x) = x|ψ .Assume that this wavepacket does not populate any bound states of the studied problem.According to the central assumption of the scattering theory, our wavepacket behaves for t = 0 as follows: As t increases from 0 to +∞, the wavepacket ψ(t, x) = x|ψ(t) tends to leave the interaction region (= region of x ∈ (a, b) with nonzero potential) and becomes thus freely evolving at very large positive times.[ Note that some portion of ψ(0, x) would generally escape to the right (x → +∞) and some portion to the left (x → -∞).] Similarly, as t decreases from 0 to -∞, the wavepacket ψ(t, x) = x|ψ(t) tends to leave the interaction region and becomes thus freely evolving at very large negative times.What we just described is the so called Asymptotic Condition (AC).It has crucial importance for all our subsequent elaborations.

Advanced series in mathematical physics•Aug 1, 1996•11 citations

Measurability of Coulomb wavepacket scattering effects

Scott E. Hoffmann

A previous paper [J. Phys. B: At. Mol. Opt. Phys. 50, 215302 (2017)] showed that partial wave analysis becomes applicable to nonrelativistic Coulomb scattering if wavepackets are used. The scattering geometry considered was special: that of a head-on collision between the wavepacket and the centre of the potential. Our results predicted, in this case, a shadow zone of low probability for small angles around the forward direction for the description of alpha scattering from a gold foil. In this paper we generalize the results to the case of a nonzero impact parameter, a displacement of the wavepacket centre perpendicular to the average momentum direction. We predict a large flux in the forward direction from events with large impact parameters. We find a significant probability of scattering into the deviation region for impact parameters of order the spatial width of the wavepacket. Averaging over impact parameters produces predictions in excellent agreement with the Rutherford formula down to lower angles than for the zero impact parameter prediction. We consider issues that would arise in a real experiment and discuss the possibility of measuring a deviation from the Rutherford formula.

arXiv:2002.02047•Feb 6, 2020•0 citations
In the preceding paper [J. Math. Phys. 10, 2047 (1969)], we investigated possible characterizations of short-range'' interactions by means of conditions on the total transition probability ∥Tψ∥2 which were valid only for certain types of sequences of wavepackets. In this paper we generalize the assumption on the transition probability so that it will apply to an arbitrary wavepacket. We do this by utilizing a measure G[R, ψ] of how much of a wavepacket ψ is within a sphere of radius R about the scattering center at all times. We then characterize a short-range'' interaction by the existence of a positive constant A and positive functions B(p) and C(p) such that for all R ≥ 0, ‖Tψ‖2≤AG[R,ψ]+〈C(p)e−RB(p)〉ψ.We then show that for such interactions the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse the size of which depends upon B(p). We compare our results with the similar results obtained by the method in the preceding paper and show that these new results much more closely approximate the behavior of the Yukawa interaction.
We investigate possible characterizations of ``short-range'' interactions by means of conditions on the total transition probability ∥Tψ∥2. We desire to require that the total transition probability decrease exponentially as the displacement of the wavepacket ψ increases. It is shown that this can only be done for certain types of sequences of wavepackets, and a criterion is developed for selecting these ``proper'' wavepackets. We then show for all such interactions that what is essentially the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse which always includes the physical region. We then compare these results with the Schrödinger potential theory and thereby relate the type of the exponential decrease of a potential to the type of the exponential decrease of the corresponding transition probability. Finally, our results are compared with similar results previously obtained by others working on this problem.
We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e - -Xe and Cs-Cs systems.
A development of the scattering amplitude in potential scattering which is based upon replacing a smooth potential by a sequence of potential steps is given. The Schrödinger equation is solved exactly in each interval over which the potential is constant and by matching boundary conditions an expression relating the coefficients of two linearly independent solutions at small radius to those at large radius is found. From this the scattering amplitude is calculated. The limits of validity are found to be dependent upon the value of the expression (r/2)ddrln(2mℏ2[E−V]). Limiting cases of the amplitude are shown to be a modified WKB approximation and the Born approximation. A particular application to the Yukawa potential is discussed. The discussion is restricted to attractive potentials.
We derive two equivalent analytical expressions for an lth partial-wave scattering length al for central potentials with long-range tails of the form V(r)=−ℏ22mBrn−4(rn−2 + Rn−2)2−ℏ22mCr2(rn−2 + Rn−2), (r ⩾ rs, R > 0). For C = 0, this family of potentials reduces to the Lenz potentials discussed in a similar context in our earlier works [R. Szmytkowski, Acta Phys. Pol. A 79, 613 (1991); J. Phys. A: Math. Gen. 28, 7333 (1995)]. The formulas for al that we provide in this paper depend on the parameters B, C, and R characterizing the tail of the potential, on the core radius rs, as well as on the short-range scattering length als, the latter being due to the core part of the potential. The procedure, which may be viewed as an analytical extrapolation from als to al, is relied on the fact that the general solution to the zero-energy radial Schrödinger equation with the potential given above may be expressed analytically in terms of the generalized associated Legendre functions.
The asymptotic condition: A qualitative look Suppose that we have prepared an initial quantum state |ψ at t = 0. We may view |ψ as a square integrable wavepacket ψ(0, x) = x|ψ .Assume that this wavepacket does not populate any bound states of the studied problem.According to the central assumption of the scattering theory, our wavepacket behaves for t = 0 as follows: As t increases from 0 to +∞, the wavepacket ψ(t, x) = x|ψ(t) tends to leave the interaction region (= region of x ∈ (a, b) with nonzero potential) and becomes thus freely evolving at very large positive times.[ Note that some portion of ψ(0, x) would generally escape to the right (x → +∞) and some portion to the left (x → -∞).] Similarly, as t decreases from 0 to -∞, the wavepacket ψ(t, x) = x|ψ(t) tends to leave the interaction region and becomes thus freely evolving at very large negative times.What we just described is the so called Asymptotic Condition (AC).It has crucial importance for all our subsequent elaborations.
A previous paper [J. Phys. B: At. Mol. Opt. Phys. 50, 215302 (2017)] showed that partial wave analysis becomes applicable to nonrelativistic Coulomb scattering if wavepackets are used. The scattering geometry considered was special: that of a head-on collision between the wavepacket and the centre of the potential. Our results predicted, in this case, a shadow zone of low probability for small angles around the forward direction for the description of alpha scattering from a gold foil. In this paper we generalize the results to the case of a nonzero impact parameter, a displacement of the wavepacket centre perpendicular to the average momentum direction. We predict a large flux in the forward direction from events with large impact parameters. We find a significant probability of scattering into the deviation region for impact parameters of order the spatial width of the wavepacket. Averaging over impact parameters produces predictions in excellent agreement with the Rutherford formula down to lower angles than for the zero impact parameter prediction. We consider issues that would arise in a real experiment and discuss the possibility of measuring a deviation from the Rutherford formula.