Rutherford scattering of quantum and classical fields

Pijnenburg, Martin; Cusin, Giulia; Pitrou, Cyril; Uzan, Jean-Philippe

doi:10.1119/5.0175025

Quantum Rutherford scattering and the scattering of classical waves by black holes have similar formal structures and can be studied using the same mathematical techniques. In both contexts, the long-range nature of the interaction leads to a divergent total cross section, which has been interpreted and regularized in various ways in the past literature. We review in detail the origin of this divergence, in both real and multipole spaces, and show that it arises from the incorrect use of approximations outside their domain of validity. We also stress that although black hole and quantum Rutherford scattering share the same formalism, the natures of the associated physical observables differ. We comment on the role of interference: while interference may be safely neglected in the context of quantum Rutherford scattering (due to the fact that the observable quantity is a flux, and the incoming flux is collimated), it should not be neglected in the context of classical waves scattered by a black hole, where one expects to see a superposition of transmitted and scattered waves in a broad region downstream from the target and a cross section is not connected to any physically observable quantity.

I. INTRODUCTION

Scattering phenomena are ubiquitous in physics, and range from quantum mechanics systems (scattering in the Schrödinger framework) to relativity problems (light or gravitational waves scattering off astrophysical sources). In all of these contexts, one encounters scattering equations which have a structure similar to the Schrödinger equation. The time-independent Schrödinger equation for a monochromatic wavefunction associated with a particle of mass m, frequency ω, and energy

E = ℏ ω

scattering off a potential V is

(- \frac{ℏ^{2}}{2 m} \nabla^{2} + V) ψ = E ψ .

(1)

In most cases, one makes a “large distance” approximation in which the solution behaves as the superposition of a spherical (scattered) wave and incident plane wave. Indeed, the definition of the differential cross section,

d σ / d Ω

⁠, relies on the assumption that it is possible to split the wave solution of Eq. into the sum of incoming and outgoing waves as

ψ = ψ_{in} + ψ_{scat} .

(2)

Then one defines the differential cross section in a direction θ relative to the incoming wave direction as the flux of the current of the scattered waves in a surface

d S

normalized to the initial flux,

J_{scat} \cdot d S e_{r} \equiv | J_{in} | d σ (θ),

(3)

where

d S = r^{2} d Ω

⁠, and r is the radial distance from the scattering potential.

In many physical situations the potential in Eq. (1) has poles (r = 0 in the case of the Coulomb/Newtonian potential). As a consequence, the first derivative of the solution cannot be continuous at the pole location but the solution may nonetheless be well-defined and continuous. Also, when the potential does not fall off rapidly enough, as is the case for the Coulomb potential on which we focus hereafter, the cross section diverges in the forward direction. Exact solutions for such Rutherford scattering¹ problems do actually exist and have been known for a long time.^2–9 Since these exact solutions are well-defined everywhere in space, they allow one to track back the origin of the divergence of approximate solutions.

We discuss some usual methods used in the literature in order to regularize or justify Rutherford-like $θ \to 0$ divergences. Those methods become, strictly speaking, unnecessary once the exact solution is considered, and they misleadingly suggest that $r^{- 1}$ scattering problems are ill-defined. We provide a pedagogical example stressing the importance of correctly stating the regime of validity of an approximate solution in order to correctly draw a physical interpretation from it. We also show that great care has to be taken in the inversion of limits to avoid misinterpreting anomalous behaviors of results that are in fact simply a consequence of an incorrect extrapolation of the approximate solution itself.

This paper is structured as follows. First, in Sec. II, we briefly review standard concepts in scattering problems and apply the usual procedure to compute the scattering cross section of Rutherford scattering. We show that the amplitude of the resulting scattered wave is divergent in the forward direction (hence the cross section has the same type of divergence). In Sec. III A, we share the exact solution to the Rutherford scattering equation and discuss its behavior, noting that it is well-defined (not divergent) everywhere in space. For an incoming wave function with wave vector $k$ ⁠, we then identify a variable $k r (1 - cos θ)$ that allows us to split the total solution in terms of incoming and outgoing waves as in Eq. (2) thanks to the properties of the exact solution. The “large-r” limit appears as an approximation valid only for large values of this dimensionless variable. As a consequence, for any given radial distance r, one can define a small θ region around the forward direction inside which the expansion of Eq. (2) is not valid. Hence, the Rutherford formula for the scattering cross section only holds outside this region. The θ = 0 divergence of the usually reported Rutherford result arises from the extrapolation of the approximate solution in a regime where it never holds. The exact solution of Rutherford scattering can also be obtained using a multipolar decomposition, and this is introduced in Sec. IV. Finally, Sec. V turns to a different problem: scattering of classical waves off Coulomb-like potentials. Mathematically the equation governing this scattering problem is very similar to the Rutherford equation describing quantum scattering,¹ but we stress some caveats of this formal analogy. In particular, we show that a definition of a classical cross section in this context does not correspond to any observable quantity and can be misleading when trying to interpret physical effects in this classical framework.

II. REVIEW OF RUTHERFORD SCATTERING

The starting point of the traditional approach to this scattering problem is the Schrödinger equation, where the solution ψ contains an incident plane wave incoming along the e_z-axis,

ψ_{in} = e^{i k z} .

(4)

We use spherical coordinates

(r, θ, φ)

so that

z = r cos θ

and

e_{z} = cos θ e_{r} - sin θ e_{θ}

⁠. The spherically symmetric Coulomb potential for the Rutherford problem is of the form

V = \frac{A}{r},

(5)

for a constant parameter A with value

A = Q_{1} Q_{2} / 4 π ϵ_{0}

in the Rutherford case of a particle of electric charge Q₁ and target charge Q₂, ϵ₀ being the vacuum permittivity. Defining

ℏ k = \sqrt{2 m E}, γ = \frac{m A}{ℏ^{2} k}, ρ = k r,

(6)

the Schrödinger equation reads

(\nabla^{2} + k^{2} - 2 \frac{γ k}{r}) ψ = 0,

(7)

or equivalently

(\partial_{ρ}^{2} + \frac{2}{ρ} \partial_{ρ} + \frac{1}{ρ^{2}} Δ_{2} + 1 - 2 \frac{γ}{ρ}) ψ = 0,

(8)

where the angular part of the Laplacian is

Δ_{2} ψ = \frac{1}{sin θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial ψ}{\partial θ}) + \frac{1}{{sin}^{2} θ} \frac{\partial^{2} ψ}{\partial φ^{2}} .

(9)

In the standard analysis, one considers the problem in a regime where the waveform can be split into an incoming plane wave and an outgoing spherical wave as in Eq. (2). This regime is commonly referred to as the $r \to + \infty$ limit (however, as we will show in Sec. III, this definition induces misconceptions when considering the infinitesimal scattering angle limit).

In the standard scattering literature, the incoming wave is superposed on the scattered wave, parameterized as

ψ_{scat} = f (θ, φ) \frac{e^{i k r}}{r} .

(10)

The scattering amplitude f is obtained by solving Eq. , with proper boundary conditions. Since the potential in Eq. is spherically symmetric, and having chosen that the incoming wave travels along the azimuthal direction z, it is clear by symmetry that f depends only on the angle θ, i.e.,

f = f (θ)

⁠.

So far, the mathematical description of Rutherford scattering relies on an (infinitely extended) plane wave, scattering off a Coulomb potential. The fundamental object of this description is a wavefunction, which is related to a probability density and a probability current. In general, the current associated with a given waveform ψ is given by

J [ψ] = \frac{ℏ}{m} Im [ψ^{*} \nabla ψ],

(11)

such that the outgoing current (using

ψ = ψ_{scat}

in Eq. ) is

J_{scat} = \frac{ℏ k}{m} \frac{| f |^{2}}{r^{2}} e_{r} + O (1 / r^{3}) .

(12)

Similarly,

| J_{in} | = ℏ k / m

⁠. From Eq. , the differential cross section is related to f by

\frac{d σ}{d Ω} (θ) \equiv \frac{J_{scat} . e_{r} ρ^{2}}{k^{2} | J_{in} |} = | f (θ) |^{2} .

(13)

Note that we have taken the large-r limit on the wave solution and extracted the scattered wave from the total wave before computing the current. Indeed

J [ψ] - J [ψ_{in}] \neq J [ψ_{scat}]

⁠, where the difference arises from the presence of interference terms. The relevance of these terms will be discussed at the end of Sec. III E.

There are different methods to compute

f (θ)

and the scattered wave in the so-called large-r limit. Here, we provide the result based on the Born approximation. We define

k

as the initial wavevector and

k'

as the wavevector of the scattered wave. The scattered wave is spherical by assumption, so

k' = | k' | e_{r}

⁠. We assume that the scattering potential does not absorb momentum (elastic collision, fixed target) so that

| k | = | k' | = k

⁠. Defining

q \equiv k - k'

⁠, the Born approximation yields^10–12

\begin{matrix} f (θ) = - \frac{2 m}{4 π ℏ^{2}} \int e^{- i q \cdot r} V (r) d^{3} r \\ = - \frac{2 m}{ℏ^{2}} \frac{1}{q} \int_{0}^{\infty} V (r) r sin (q r) d r, \end{matrix}

(14)

where

q \equiv | q | = 2 k sin (θ / 2)

and θ is the angle between

k

and

k'

⁠. For the

r^{- 1}

Coulomb potential of our problem, it is obvious that the integral in Eq. is, strictly speaking, ill-defined. This fact should not be interpreted as an ill definition of wave scattering problems in

r^{- 1}

potentials, but only as the inapplicability of the Born approximation to the latter case.

It is, however, standard practice to substitute the initial Coulomb potential from Eq. with a Yukawa-like exponentially suppressed potential, for which the integral of Eq. possesses a closed form for

μ > 0

only,

V (r) = \frac{A}{r} e^{- μ r} \Rightarrow f (θ) = \frac{- 2 k γ}{μ^{2} + q^{2}}, μ > 0 .

(15)

Setting μ = 0 in the latter result, one obtains

f (θ) = - \frac{γ}{2 k {sin}^{2} (θ / 2)}, \frac{d σ}{d Ω} = \frac{γ^{2}}{4 k^{2} {sin}^{4} (θ / 2)} .

(16)

The differential cross section is exactly the one found for the scattering of a massive particle off a Coulomb potential in classical mechanics, a textbook result known as classical Rutherford scattering cross section, which is usually obtained from the conservation of energy, angular momentum, and Runge–Lenz vectors.

This cross section is divergent for $θ \to 0$ ⁠. In the literature, different explanations have been provided to justify the presence of such a divergence. This divergence is often claimed to be artificial due to the unphysicality of the Coulomb potential itself because of screening.^11,13,14 For example, in situations with many charges at finite temperature, the potential of a point charge is suppressed due to Debye–Hückel screening¹⁵ and a Yukawa potential is one possible screening parametrization. Equation (15) is then regular as $θ \to 0$ for $μ > 0$ ⁠. While being physically motivated, this argument relies on a particular screening model and misleadingly suggests that the pure Coulomb potential model leads to divergences, a statement which Sec. III A shows not to be true. It is important to emphasize this when dealing with formally similar $r^{- 1}$ classical wave scattering (off black holes) for which there is no immediate analogous regularization.

We also stress that in the classical particle computation of Rutherford scattering, the divergence is present and has been addressed in the literature. In this case, there exists a well-defined map between deflection angle and impact parameter b, and considering $θ_{defl} \to 0$ means considering $b \to \infty$ ⁠.^16,17 In other words, only particles passing infinitely far away from the potential are undeflected, and precisely those give divergent contributions to the cross section. Thus, in this case, the divergence issue is alleviated by arguing that in a concrete experimental setup, $b \to \infty$ never happens, hence neither does $θ_{defl} \to 0$ ⁠. Moreover, typically the classical situation is such that there are multiple scattering centers (e.g., in a classical gold foil model), and there is a maximum value for b, given by half of the inter atomic distance. This provides a lower bound on $θ_{defl}$ ⁠.¹⁷ However, one cannot rely on this classical analogy when considering scattering of waves, as a definition of impact parameter is ambiguous in this case.

In Sec. III, we show that the problem of Rutherford scattering for waves is actually inherently free of divergence: the divergence is just an artifact of the fact that we take the $θ \to 0$ limit after having taken the large distance limit (and the two limits do not commute). It follows that all the arguments mentioned above presented to justify or deal with the Rutherford divergence are rather unnecessary, if not formally artificial, when discussing scattering by a pure Coulomb potential.

III. FROM EXACT SOLUTION TO THE RUTHERFORD SOLUTION

A. Exact solution

The Schrödinger equation of Eq. can be solved in terms of the confluent hypergeometric function

_{1} F_{1}

as^2–9

ψ = e^{i k . r} e^{- π γ / 2} Γ {(1 + i γ)}_{1} F_{1} [- i γ, 1; i (k r - k \cdot r)] .

(17)

This exact solution and its normalization have been obtained by imposing that, in the absence of interactions, it reduces to an incoming plane wave

e^{i k . z}

⁠, in such a way that it matches the desired asymptotic behavior from Eq. . The derivation of Eq. may be found in Sec. A of the supplementary material.

It is convenient to introduce the notation

s \equiv 2 {sin}^{2} \frac{θ}{2} = 1 - cos θ .

(18)

The exact solution then takes the following compact form:

ψ = e^{i ρ (1 - s)} e^{- π γ / 2} Γ {(1 + i γ)}_{1} F_{1} [- i γ, 1; i ρ s] .

(19)

Note that the first exponential prefactor is simply

e^{i k z}

since

ρ (1 - s) = k z

⁠.

B. Approximate solution

To find the Rutherford solution, one usually expands the function $_{1} F_{1}$ for large values of its argument, i.e., $ρ s ≫ 1$ ⁠. In other words, for a fixed geometry and wave frequency (r and k), we identify an angular region $s ≪ 1 / (r k)$ and compute the approximate solution valid asymptotically outside this region.

Explicitly, one makes the following expansion valid for asymptotically large values of q:

\begin{matrix} _{1} F_{1} (a, b; q) \sim e^{q} q^{a - b} \frac{Γ (b)}{Γ (a)} \sum_{k = 0}^{\infty} \frac{{(b - a)}_{k} {(1 - a)}_{k}}{k! q^{k}} \\ + e^{\pm i π a} q^{- a} \frac{Γ (b)}{Γ (b - a)} \sum_{k = 0}^{\infty} \frac{{(a)}_{k} {(a - b + 1)}_{k}}{k! {(- q)}^{k}}, \end{matrix}

(20)

where the sign of the complex phase depends on the argument of q.^18,19 In our case,

q = i s

⁠, implying that the positive sign has to be chosen.¹⁹ In the latter expression,

{(x)}_{k}

is the Pochhammer symbol, which can be defined as

{(x)}_{k} \equiv Γ (x + k) / Γ (x)

⁠. It follows that at large values of

ρ s = k (r - z) = ρ (1 - cos θ)

⁠, using Eq. with

q = i s ρ

in Eq. , one finds at leading order

\begin{matrix} \tilde{ψ} \sim e^{i k z + i γ ln ρ s} (1 - i \frac{γ^{2}}{ρ s}) - \frac{γ}{ρ s} \frac{Γ (1 + i γ)}{Γ (1 - i γ)} e^{i ρ - i γ ln ρ s}, \end{matrix}

(21)

which is valid well outside the region

ρ s = 1

and, therefore, not valid for θ = 0. In this equation, one usually identifies an incident component (first term) and a scattered spherical wave (second term). The tilde symbol has been used to stress that this is an asymptotic approximate solution, and to distinguish it from the exact one. We observe that in Eq. the correction to the incoming wave proportional to

γ^{2}

can be relevant for large values of γ, and can be seen as a backreaction of the scattered wave on the incoming wave (when going beyond the Born approximation lowest order). Hence, we define the distorted incoming plane wave as

ψ_{in, γ^{2}} \equiv e^{i k z + i γ ln ρ s} (1 - i \frac{γ^{2}}{ρ s}) .

(22)

In any realistic configuration of Rutherford scattering (and also in the case of scattering of classical waves, as in Sec. V), we, however, typically have

γ ≪ k r

⁠, and this is the regime we will focus on from now on. Hence, in the analytical computations that follow, we will neglect the backreaction correction

\propto γ^{2} / (ρ s)

to the incoming term, assuming that we are working in a regime where this correction is very small.

Notice that the incoming wave is distorted from a pure plane wave by the presence of the interaction, which gives the appearance of logarithmic phase shifts. A similar statement can be made for the phase corrections to the scattered wave. Logarithmic corrections to the phase are a specificity of $1 / r$ potentials.²⁰ For the latter, the radial decay of the potential is too slow to have standard plane waves as solutions of the wave equation. Even at asymptotic radial distance, where we expect the scattered waves to become vanishingly small in amplitude, these phase corrections remain present as part of the incoming wave. A logarithmic feature in the phase is also present in the context of scattering of classical waves off a $1 / r$ potential, as is briefly discussed in Sec. V.

We further notice that the approximate solution diverges in the scattered part for $θ \to 0$ ⁠, unlike the exact solution (19), as Fig. 1 illustrates. It clearly follows that the divergence in the cross section is a direct consequence of using the approximate solution (21) outside its regime of validity.

Fig. 1.

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Comparison between the absolute value of the exact solution (Eq. (17)) (solid line) and the approximate solution (Eq. (21)) (dashed) for γ = 1 and $ρ = 100, 10, 1$ ⁠. Numerical parameter values are for illustration purposes, and the units are arbitrary. The approximate solution departs from the exact one at an angle θ whose value decreases as we increase ρ.

Let us see this more explicitly. Using the approximate Rutherford solution of Eq. found by expanding the exact solution for

ρ s ≫ 1

⁠, one identifies the scattering amplitude

f (θ)

as the r-independent constant multiplying the (distorted) spherical term

f (θ) = \frac{- γ}{2 k {sin}^{2} (θ / 2)} \frac{Γ (1 + i γ)}{Γ (1 - i γ)},

(23)

which combined with Eq. gives the usual Rutherford cross section. However, this derivation shows explicitly that this result rigorously holds only when ρs is asymptotically large, which clearly excludes the region θ = 0. Experimental conditions, however, typically satisfy

ρ s ≫ 1

⁠,²¹ ensuring the validity of the usual cross section result of Eq. , even in an ideal case without screening.

We conclude this section by observing that at leading order in ρs, the extra logarithmic phase in Eq. does not affect the currents defining the cross section. To be more precise, if

ψ_{in} \equiv e^{i k z + i γ ln k (r - z)}

⁠, the associated current is given as

J_{in} = \frac{ℏ k}{m} [(cos θ + \frac{γ}{ρ}) e_{r} - (sin θ - \frac{γ}{ρ} \frac{sin θ}{1 - cos θ}) e_{θ}] .

(24)

This gives the standard

ℏ k e_{z} / m

at large distance but it has a

1 / ρ

tail so that

J_{in} = \frac{ℏ k}{m} [1 + O (1 / ρ)],

(25)

compared to the expression given after Eq. . Similar considerations can be made for the asymptotic outgoing spherical part, allowing us to have an asymptotic cross section of the form

\propto | f (θ) |^{2}

⁠, as expected.

C. Small angle behavior

We compare the standard approach in which one takes the large-r limit and then faces a singularity at θ = 0 to the one in which we look at the exact solution for

ρ s ≪ 1

⁠, which includes

θ \to 0

and arbitrary r. From the expansion

_{1} F_{1} [- i γ, 1; i ρ s] = 1 + γ ρ s + \frac{γ ρ^{2}}{4} (i + γ) s^{2} + O (s^{3}),

(26)

the exact solution is approximated when

ρ s ≪ 1

by

\tilde{ψ} = e^{i ρ (1 - s)} e^{- π γ / 2} Γ (1 + i γ) (1 + γ ρ s) .

(27)

In the θ = 0 direction, we have exactly s = 0 and the solution reduces to

\tilde{ψ} |_{θ = 0} = e^{i k z} e^{- π γ / 2} Γ (1 + i γ) .

(28)

For small angles, the functional form of the total solution does not show a natural splitting between an incoming-like and a scattered-like term, but rather a full superposition of the two, corresponding to the forward direction interference. As a result, there is no natural notion of scattering differential cross section in the $ρ s ≪ 1$ region, because $d σ / d Ω$ relies on the existence of well-defined incoming and scattered currents. However, the total current associated with the full solution (19) remains well defined and divergence-free everywhere in space, and smoothly transitions between the small and large angle regimes. Currents are further discussed in Sec. III E.

D. Interpretation of the forward direction

The large ρs limit can always be fulfilled for every $θ \in (0, π]$ ⁠. Indeed, in this case we have s > 0, such that there always exists a distance r large enough to satisfy $ρ s ≫ 1$ ⁠.

Given that $ρ s = k (r - z)$ by definition, writing r in Cartesian coordinates explicitly allows to rewrite the condition $ρ s = 1$ as $k (z + 1 / 2 k) = \frac{1}{2} k^{2} (x^{2} + y^{2})$ ⁠, which defines a paraboloid of revolution. The transverse section of the paraboloid increases as we go toward larger positive z, but its angular size, seen from the origin, decreases. The limit $ρ s ≫ 1$ corresponds to the region sufficiently far outside this paraboloid.

Considering an incoming plane wave along z, we follow a point on the wavefront identified by a set of coordinates (x, y). Such a wavefront point will enter the paraboloid once

k (z + \frac{1}{2 k}) \geq \frac{1}{2} k^{2} (x^{2} + y^{2}) .

(29)

The entrance point thus depends on the point (x, y) that one is following, and is located at a further distance from the scattering center for larger

| x |, | y |

⁠. We illustrate this in Fig. 2.

Fig. 2.

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Top: exact solution on slices parallel to the z axis $(x = const ., y = 0)$ ⁠. Fixing a higher kx, the entrance inside the paraboloid (and thus the asymptotic damped plane wave regime) is reached at a higher z. The black line is the expected asymptotic amplitude $| Γ (1 + i γ) | e^{- π γ / 2}$ ⁠. We plot $Im [ψ]$ ⁠, but analogous considerations hold for $Re [ψ]$ ⁠. Bottom: two-dimensional slice of the exact solution $| ψ |$ at y = 0, for γ = 1. The paraboloid region corresponding to $ρ s < 1$ is clearly visible.

We consider the exact solution of the Rutherford problem, Eq. (17), and we look at the evolution of a point in the (x, z) plane. As Eq. (21) shows, the exact solution is normalized such that ψ is a traveling wave with unit amplitude when $z \to - \infty$ ⁠. However, sufficiently far inside the paraboloid, Eq. (28) shows that ψ is oscillating with reduced amplitude $\sim e^{- π γ / 2} | Γ (1 + i γ) |$ which tends to 0 for $γ ≫ 1$ ⁠. This corresponds to a damped-wave regime. This can be understood to mean that the initial plane wave has transferred a part of its initial magnitude to the scattered wave outside the paraboloid (or inside the paraboloid if $γ < 0$ ⁠). The paraboloid marks the transition between the two asymptotic regimes.

E. Outgoing current and interference

When computing the cross section, we need to compute the current of the scattered wave. Obviously this is not equivalent to the difference between the current of the total wave and the current of the incoming wave, since there are interference terms between the incoming and outgoing wave. Explicitly, the current associated with the total wave (approximate solution)

\tilde{ψ}

can be written as

\tilde{J} = {\tilde{J}}_{in} + {\tilde{J}}_{scat} + {\tilde{J}}_{\times},

(30)

where

{\tilde{J}}_{in} \equiv J [ψ_{in}], {\tilde{J}}_{scat} \equiv J [{\tilde{ψ}}_{scat}]

and the current operator

J

is defined in Eq. . The last term

{\tilde{J}}_{\times}

is the current resulting from the interference between the incoming and scattered waves.

Given an initial waveform

ψ_{in}

⁠, a scattered current can also be computed from the exact solution of Eq. by subtracting the incoming wave contribution, expressed as

J_{out} \equiv J [ψ - ψ_{in}] .

(31)

If we chose instead to subtract the distorted plane wave of Eq. , we define

J_{γ^{2} out} \equiv J [ψ - ψ_{in, γ^{2}}] .

(32)

In the top panel of Fig. 3, we plot the radial component of the total current that can be computed from the exact and asymptotic solutions. The latter diverges as expected at small angles, while the exact solution tends to a constant value. The larger angular range for which the asymptotic solution is accurate is larger for larger values of ρ. In the middle panel of Fig. 3, we plot the various contributions to the radial component of the current given in Eq. , while in the bottom panel we compare the outgoing current obtained from the exact solution to the approximated one. In the bottom panel of Fig. 3, we compare the radial part of

J_{out}

and

J_{γ^{2} out}

⁠.

Fig. 3.

Top: comparison between the total currents J r ≡ J [ ψ ] · e r (solid) and J ̃ r ≡ J [ ψ ̃ ] · e r (dashed) for ρ = 10 (black) and 100 (orange). The current computed from ψ ̃ (Eq. (21)) diverges for small angles, unlike the exact current. Middle: total current computed from ψ ̃ (black, solid), along with its different components as in Eq. (30). The divergence at small angles comes from the divergence of the approximate solution at small angles. Bottom: scattered current from the asymptotic (red) and exact solutions. The latter are computed from ψ in without (gray-blue, solid line) and with (green, dash-dotted) backreaction corrections. Vertical bars mark the angle at which ρ s = 1 (below which ψ ̃ and quantities derived from it are outside their region of validity). For all the plots, we chose γ = 0.4.

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Top: comparison between the total currents $J_{r} \equiv J [ψ] \cdot e_{r}$ (solid) and ${\tilde{J}}_{r} \equiv J [\tilde{ψ}] \cdot e_{r}$ (dashed) for $ρ =$ 10 (black) and 100 (orange). The current computed from $\tilde{ψ}$ (Eq. (21)) diverges for small angles, unlike the exact current. Middle: total current computed from $\tilde{ψ}$ (black, solid), along with its different components as in Eq. (30). The divergence at small angles comes from the divergence of the approximate solution at small angles. Bottom: scattered current from the asymptotic (red) and exact solutions. The latter are computed from $ψ_{in}$ without (gray-blue, solid line) and with (green, dash-dotted) backreaction corrections. Vertical bars mark the angle at which $ρ s = 1$ (below which $\tilde{ψ}$ and quantities derived from it are outside their region of validity). For all the plots, we chose $γ = 0.4$ ⁠.

We note that using the distorted plane wave of Eq. (22), which includes a $γ^{2}$ correction, to define $J_{γ^{2} out}$ makes the latter closer to ${\tilde{J}}_{scat}$ in the sense that it removes a small oscillatory pattern. This pattern is due to the interference between the $γ^{2}$ correction to the incoming wave and the outgoing part in $J_{out}$ ⁠.

As we have seen, the interference terms between the incoming wave and the scattered one are neglected in the standard picture, where any observed outgoing flux is identified with the flux of the scattered part of the wave (total minus incident one). The leading order contribution to the interference current in the radial direction is

{\tilde{J}}_{\times, r} ≃ - \frac{ℏ γ}{m r} {cot}^{2} (θ / 2) cos [ρ s - 2 γ ln (ρ s) + 2 δ_{0}],

(33)

where δ₀ is a phase shift defined by

e^{2 i δ_{0}} = Γ (1 + i γ) / Γ (1 - i γ)

⁠. Though

{\tilde{J}}_{\times, r}

decays as

1 / r

⁠, and thus slower than

{\tilde{J}}_{scat, r}

⁠, its angular dependence is oscillatory due to the argument of the last factor. This oscillatory behavior is manifested in the middle panel of Fig. 3. The oscillation length (in the orthoradial direction) is typically

λ_{osc} ≃ \frac{2 π}{k sin θ (1 - \frac{2 γ}{ρ s})},

(34)

hence, as long as the detector is larger than this length, which is proportional to the wavelength of the scattered wave, this contribution of the current is averaged out.

The interference contribution may be discarded for a further reason that is not specific to Rutherford scattering but rather lies in the scattering formalism itself. While it is standard to solve the Schrödinger equation with an incoming plane wave condition and find the associated scattered wave as in Eq. (2), the experimentally relevant quantities are wave packets. The steps passing from plane wave to wave packet scattering are carried out in Ref. 22, where it is shown that the scattering cross is identical in both contexts (the argument is explicitly outlined for short range potentials and subsequently generalized to the Coulomb case). An important point is that, while in the context of plane waves, interference may be expected at every point in space due to the overlap of the incoming and scattered parts, such an interference does not occur for wave packets of finite support. A region of interference may still exist around the forward direction, but only in a very restricted angular range of $θ \in [0, 10^{- 3}]$ for a collimated beam with a 1 mm transverse size and detector placed at 1 m from the target.²²

This is to be contrasted with the case of classical wave scattering discussed in Sec. V, where a long wavelength signal may be of macroscopic size and, thus, produce a large region of interference, as do classical waves on a surface of water.

IV. RUTHERFORD IN $ℓ$ SPACE

The Rutherford cross section can also be derived in the multipole space using an expansion of the angular dependence in Legendre polynomials

P_{ℓ} (cos θ)

⁠. A plane wave is expanded as

ψ_{plane} = e^{i k r cos θ} = \sum_{ℓ} i^{ℓ} (2 ℓ + 1) j_{ℓ} (k r) P_{ℓ} (cos θ),

(35)

which we shorten to

\sum_{ℓ} ψ_{plane, ℓ} P_{ℓ} (cos θ)

⁠. In scattering problems, one usually wants to identify a spherical incoming and a spherical outgoing wave. For a given

ℓ ≪ k r

⁠, one can expand the spherical Bessel into the sum of incoming and outgoing spherical waves and obtain

ψ_{plane, ℓ} ≃ \frac{2 ℓ + 1}{2 i k r} [{(- 1)}^{ℓ + 1} e^{- i k r} + e^{i k r}] .

(36)

A similar decomposition can be introduced for a generic wave. For example, the total wave in real space can be written asymptotically as

ψ_{tot} ≃ e^{i k r cos θ} + f (θ) \frac{e^{i k r}}{r} .

(37)

The decomposition in a basis of Legendre polynomials must be of the following form:

ψ_{tot, ℓ} ≃ \frac{2 ℓ + 1}{2 i k r} [{(- 1)}^{ℓ + 1} e^{- i k r} + e^{2 i δ_{ℓ}} e^{i k r}],

(38)

where the functions

δ_{ℓ}

are called phase shifts. They encode the effects of scattering, which by construction affects only outgoing terms. Phase shifts describe the departure of the wave function from the plane wave, i.e., for

δ_{ℓ} = 0, ψ_{tot}

coincides with the plane wave. The scattering amplitude is traditionally defined as the amplitude of the scattered wave (subtracting the transmitted component),

f (θ) \equiv \sum_{ℓ} \frac{2 ℓ + 1}{2 i k} (e^{2 i δ_{ℓ}} - 1) P_{ℓ} (cos θ) .

(39)

We now discuss how the above procedure can be applied to Rutherford scattering. Let us expand ψ as

ψ (ρ, θ) = \sum_{ℓ} ψ_{ℓ} (ρ) P_{ℓ} = \sum_{ℓ} \frac{Ψ_{ℓ} (ρ)}{ρ} P_{ℓ},

(40)

where

P_{ℓ}

are evaluated at

cos (θ)

⁠. The differential equation of Eq. becomes

[\partial_{ρ}^{2} + \frac{2}{ρ} \partial_{ρ} - \frac{ℓ (ℓ + 1)}{ρ^{2}} + 1 - 2 \frac{γ}{ρ}] ψ_{ℓ} = 0,

(41)

or equivalently

Ψ_{ℓ}^{″} + [1 - \frac{2 γ}{ρ} - \frac{ℓ (ℓ + 1)}{ρ^{2}}] Ψ_{ℓ} = 0 .

(42)

This is known as the Coulomb wave equation, which is a Whittaker equation whose solutions are known. Rewriting it with a proper rescaling of

e^{- i ρ} ρ^{ℓ + 1}

⁠, we see that the solutions are expressed directly in terms of confluent hypergeometric functions as

Ψ_{ℓ} = C (ℓ, γ) ρ^{ℓ + 1} e^{- i ρ}_{1} F_{1} (ℓ + 1 - i γ, 2 ℓ + 2; 2 i ρ),

(43)

where

C (ℓ, γ)

is at this stage an arbitrary constant. There exists another independent solution of Eq. that we discard since it is not regular as

ρ \to 0

⁠. Apart from normalization conventions, the function

Ψ_{ℓ}

given by Eq. is usually called the regular Coulomb wave function in the literature (see, e.g., Ref. 9).

We now consider the asymptotic behavior of Eq. for large ρ, where large means typically

ρ ≫ ℓ (ℓ + 1) + γ^{2}

since the expansion of Eq. is typically valid when

| (b - a) (1 - a) / q | ≪ 1

and

| a (a - b + 1) / q | ≪ 1

⁠. After some algebra, we get

\frac{Ψ_{ℓ}}{ρ} \sim \frac{\tilde{C} (ℓ, γ)}{2 i ρ} [{(- 1)}^{ℓ + 1} e^{- i ρ_{c}} + e^{2 i δ_{ℓ}} e^{i ρ_{c}}],

(44)

with

ρ_{c} = ρ - γ ln (2 ρ),

(45)

and where the factors

e^{i δ_{ℓ}}

are now directly coming from the asymptotic expansion of the

_{1} F_{1}

function, and defined such that

Γ (ℓ + 1 + i γ) \equiv | Γ (ℓ + 1 + i γ) | e^{i δ_{ℓ}},

(46)

or equivalently

e^{2 i δ_{ℓ}} = \frac{Γ (ℓ + 1 + i γ)}{Γ (ℓ + 1 - i γ)} .

(47)

The overall constant

\tilde{C} (ℓ, γ)

consists of the original

C (ℓ, γ)

with additional factors coming from the asymptotic expansion, namely,

\tilde{C} (ℓ, γ) \equiv C (ℓ, γ) \frac{1}{{(2 i)}^{ℓ}} \frac{Γ (2 ℓ + 2)}{Γ (ℓ + 1 + i γ)} e^{γ π / 2} .

(48)

The freedom in

C (ℓ, γ)

translates to a freedom in the choice of

\tilde{C} (ℓ, γ)

⁠. For consistency with our initial conditions, we must require that our solution asymptotically matches Eq. in the limit of the absence of scattering, i.e., for γ = 0 (implying

δ_{ℓ} = 0

⁠, as well as

ρ = ρ_{c}

⁠). The result is

\tilde{C} (ℓ, γ = 0) = 2 ℓ + 1

⁠, which can be generalized for

γ \neq 0

to

\tilde{C} (ℓ, γ) = 2 ℓ + 1 .

(49)

This self-consistent choice implies

ψ_{ℓ} \sim \frac{2 ℓ + 1}{2 i k r} [{(- 1)}^{ℓ + 1} e^{- i ρ_{c}} + e^{2 i δ_{ℓ}} e^{i ρ_{c}}],

(50)

which has the same structure as Eq. . (The two are identical in the absence of scattering.) For

γ \neq 0

⁠, the solution of our scattering problem differs from that in Eq. by the presence of ρ_c instead of ρ in the exponent. This reflects the fact that the scattering problem does not have solutions in the exact form of a superposition of a plane wave and a spherical one, but instead has logarithmic modifications in the phases with respect to the latter, as we have seen in Eq. .

In the supplementary material, we compute the scattering amplitude using Eqs. (47) and (39). The multipole summation is nontrivial, but we rigorously prove it to lead to the unique result of Eq. (23).

V. CLASSICAL WAVE SCATTERING

Problems of scattering classical waves off matter or spacetime structures have the same mathematical form as quantum Rutherford scattering.^5,7,8 It follows that several mathematical techniques developed in the framework of quantum scattering can be imported in this context, with a few caveats.

Let us consider the simple case of classical scattering of a massless scalar wave off a Schwarzschild black hole. The process is described by

\bar{□} Φ = 0,

(51)

where the d'Alembertian is defined on the (curved) black hole background, and we are using Schwarzschild coordinates,

d s^{2} = {\bar{g}}_{μ ν} d x^{μ} d x^{ν} = - A (r) d t^{2} + A^{- 1} (r) d r^{2} + r^{2} d Ω^{2},

(52)

with

A (r) = 1 - r_{s} / r

⁠, where

r_{s} \equiv 2 M

⁠, and M is the black hole mass. The speed of light and the gravitational constant are conventionally set to

c = 1 = G

here. The problem is static and spherically symmetric, and we assume to have a monochromatic wave with frequency ω. As in Sec. II, we can choose our coordinate system with the polar axis aligned with the propagation direction of the incoming wave. With this choice, the problem depends only on the polar angle θ, hence we can introduce the decomposition

Φ = \sum_{ℓ} P_{ℓ} (cos θ) \frac{e^{- i ω t}}{r} u_{ℓ} (r, ω) .

(53)

We will omit indices on u for simplicity. It can be verified that⁴

\frac{d^{2} u}{d r_{*}^{2}} + (ω^{2} - V_{eff}) u = 0,

(54)

where

r_{*} = r + r_{s} ln (r / r_{s} - 1)

and

V_{eff} = \frac{1}{r^{2}} (1 - \frac{r_{s}}{r}) (\frac{r_{s}}{r} + ℓ (ℓ + 1)) .

(55)

If we use the replacement

u = \frac{\bar{u}}{{(1 - r_{s} / r)}^{1 / 2}},

(56)

then in the limit

r \to \infty

⁠, we get

\frac{d^{2} \bar{u}}{d r^{2}} + [ω^{2} + \frac{4 M ω^{2}}{r} + \frac{12 M^{2} ω^{2}}{r^{2}} - \frac{ℓ (ℓ + 1)}{r^{2}} + O (r^{- 3})] \bar{u} = 0 .

(57)

In the long wavelength limit, i.e.,

ℓ (ℓ + 1) > 12 {(M ω)}^{2}

⁠, we can neglect the third term in the square bracket and we are left with (for

ℓ \neq 0

⁠)

\frac{d^{2} \bar{u}}{d r^{2}} + [ω^{2} + \frac{4 M ω^{2}}{r} - \frac{ℓ (ℓ + 1)}{r^{2}}] \bar{u} = 0,

(58)

which has the same structure as the equation describing Rutherford scattering, see Eq. .

Let us use the notation

γ = - 2 M ω

⁠, and rewrite our radial equation in terms of the rescaled variable

ρ \equiv ω r

⁠. We obtain

\frac{d^{2} \bar{u}}{d ρ^{2}} + [1 - \frac{2 γ}{ρ} - \frac{ℓ (ℓ + 1)}{ρ^{2}}] \bar{u} = 0 .

(59)

Then all the results derived for Rutherford scattering hold in this case. In particular, if the incoming wave is a (distorted) plane wave, the resulting asymptotic solution will be of the form

\frac{u}{r} = \frac{2 ℓ + 1}{2 i ω r} [{(- 1)}^{ℓ + 1} e^{- i ω r_{c}} + e^{2 i δ_{ℓ}} e^{i ω r_{c}}],

(60)

where

r_{c} \equiv (ρ - γ ln (2 ρ)) / ω

and phase shift .

In the literature, this analogy with a quantum scattering problem is often pushed forward, and a differential cross section is defined as the ratio of the outgoing and incoming flux. While this is a perfectly well-defined object from a mathematical point of view, it does not directly correspond to a physical observable for classical waves. Indeed, when considering scattering of classical waves, the quantity that we measure is often related to the wave itself rather than a cross section. Moreover, as we have seen, when introducing a cross section we implicitly neglect the effect of interference between the incoming and outgoing wave. While this is well-justified in the context of a quantum scattering experiment, in the context of the scattering of a classical wave, there is no reason to assume that interference is small because the detector is typically smaller than the wavelength. On the contrary, the wave observed coming out of the scattering will be the superposition of a transmitted wave and a diffused one, and the two will interfere in an extended region after the scattering center.

VI. DISCUSSION AND CONCLUSIONS

In this paper, we have reviewed, with a pedagogical approach, the problem of quantum scattering off a Coulomb-like potential. We have shown that in real space it is possible to find an exact solution, and we have identified the regime of validity of the so-called Rutherford solution, used in the computation of the scattering cross section. We have stressed that the divergence of the scattering cross section in the forward direction is a consequence of extrapolating the Rutherford solution outside its regime of validity. As a consequence, there is no need to invoke physical mechanisms to explain such a divergence, which is a pure mathematical artifact. We have also presented Rutherford scattering in multipole space and made apparent the similarities that this problem shares with the scattering of a classical scalar wave by a black hole.

Finally, we have commented on the role of interference and explained that, while in a quantum context interference effects can be legitimately neglected, this is not the case in the context of classical scattering of waves off mass singularities. This leads to some important differences between the problems of quantum and classical scattering. Indeed, while from a formal point of view, the problem of classical scattering is very similar to a quantum scattering in $ℓ$ -space, the physically observable quantities in the two contexts are not the same. For a Rutherford-like quantum scattering, it is a flux (incoming and outgoing) that is observed. Interference can be neglected since in any observational setting the incoming beam is typically sufficiently well collimated. As a consequence, the notion of a cross section can be introduced, and it corresponds to an observable quantity. However, in a classical context, what may be observed is a waveform, which consists of a superposition of transmitted and scattered waves in a broad region after the target. In this case, the notion of a cross section does not correspond to any physical observable; while, of course, one can formally introduce and compute it, some care has to be taken when assigning it a physical meaning and when using it to describe measurable effects.

SUPPLEMENTARY MATERIAL

Please click on this link to access the supplementary material, which includes details on the derivation of the exact Rutherford scattering solution, and a rigorous treatment of the multipole expansion, at https://doi.org/10.60893/figshare.ajp.c.7241929.

ACKNOWLEDGMENTS

The authors thank Ruth Durrer, Djibril Ben Achour, and Guillaume Faye for discussions. M.P. and G.C. acknowledge support from the Swiss National Science Foundation (Ambizione grant, Gravitational wave propagation in the clustered universe). The work of G.C., C.P., and J-P.U. is founded by CNRS.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

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Rutherford scattering of quantum and classical fields

I. INTRODUCTION

II. REVIEW OF RUTHERFORD SCATTERING

III. FROM EXACT SOLUTION TO THE RUTHERFORD SOLUTION

A. Exact solution

B. Approximate solution

C. Small angle behavior

D. Interpretation of the forward direction

E. Outgoing current and interference

IV. RUTHERFORD IN $ℓ$ SPACE

V. CLASSICAL WAVE SCATTERING

VI. DISCUSSION AND CONCLUSIONS

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

REFERENCES

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Rutherford scattering of quantum and classical fields

I. INTRODUCTION

II. REVIEW OF RUTHERFORD SCATTERING

III. FROM EXACT SOLUTION TO THE RUTHERFORD SOLUTION

A. Exact solution

B. Approximate solution

C. Small angle behavior

D. Interpretation of the forward direction

E. Outgoing current and interference

IV. RUTHERFORD IN ℓ SPACE

V. CLASSICAL WAVE SCATTERING

VI. DISCUSSION AND CONCLUSIONS

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

REFERENCES

Citing articles via

Submit your article

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IV. RUTHERFORD IN $ℓ$ SPACE