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Y. D. Chong PH4401: Quantum Mechanics III

Chapter 0: Scattering Theory

I. SCATTERING EXPERIMENTS ON QUANTUM PARTICLES

Quantum particles exhibit a feature known as wave-particle duality, which can besummarized in the quantum double-slit thought experiment. As shown in the figurebelow, a source emits electrons with energy E, which travel towards a screen with a pairof slits. A detector is positioned on the other side of the screen. By moving the detectoraround, we can measure the rate at which electrons are detected at different positions.

According to quantum theory, the experiment reveals the following: (i) the electrons arrivein discrete units—one at a time, like classical particles; (ii) when we move the detector aroundto measure how the detection events are statistically distributed in space, the resultingdistribution matches an interference pattern formed by a classical wave diffracted by theslits. The wavelength λ is related to the electron energy E by

λ =2π

k, E =

~2k2

2m, (0.1)

where ~ = h/2π is Dirac’s constant, and m is the electron mass. (These relations imply,by the way, that we can deduce the spacing of the slits from the diffraction pattern, if theincident energy E is known.)

Wave-particle duality arises from quantum theory’s distinction between a particle’s stateand the outcomes of measurements performed on it. The state is described by a wavefunctionψ(r), which can undergo diffraction like a classical wave. Measurement outcomes, however,depend probabilistically on the wavefunction. In a position measurement, the probability oflocating a particle in a volume dV around position r is |ψ(r)|2 dV .

In this chapter, we will study a generalization of the double-slit experiment called a scat-tering experiment. The idea is to take an object called a scatterer, shoot quantumparticles at it, and measure the resulting particle distribution. Just as the double-slit in-terference pattern can be used to deduce the slit spacing, a scattering experiment can beused to deduce various facts about the scatterer. Scattering experiments, as a class, consti-tute a large proportion of the methods used to probe the quantum world—from electron-and photon-based laboratory experiments for measuring the properties of materials, to hugeaccelerator experiments that study high-energy phenomena like the Higgs boson.

We will focus on a relatively simple scenario, consisting of a single non-relativistic quan-tum particle and a classical scatterer. Consider a continuous and unbounded d-dimensional

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space, describable by coordinates r. Somewhere around the origin, r = 0, is a finite-sizedscatterer. An incoming quantum particle, with energy E, is governed by the Hamiltonian

H = H0 + V (r), H0 =p2

2m. (0.2)

Here, H0 describes the particle’s kinetic energy, m is the particle’s mass, r and p are positionand momentum operators, and V is a scattering potential describing how the scattereraffects the quantum particle. We assume that V (r) → 0 as |r| → ∞, i.e., the scatteringpotential becomes negligible far from the origin.

We prepare an incoming particle state with energy E, and want to see how the particle isscattered by the potential. However, converting these words into a well-defined mathematicalproblem is a bit tricky! We will give the formulation first, before discussing its meaning:

1. The particle state |ψ〉 obeys the time-independent Schrodinger equation

H|ψ〉 = E|ψ〉, (0.3)

where E is the incoming particle energy.

2. This state can be decomposed into two terms,

|ψ〉 = |ψi〉 + |ψs〉, (0.4)

where |ψi〉 is called the incident state and |ψs〉 is called the scattered state.

3. The incident state is an eigenstate of H0 with energy E:

H0|ψi〉 = E|ψi〉. (0.5)

4. Lastly, we require the scattered state to be an “outgoing” state. This is the mostsubtle requirement, and we will describe how to deal with it later.

The first condition says that the scattering process is elastic; since the scatterer takes theform of a potential V (r), its interaction with the particle is conservative (i.e., the total energyE is fixed). The second condition, viewed in the position basis, says that the particle’swavefunction ψ(r) = 〈r|ψ〉 can be taken as a superposition of an incoming wave and ascattered wave. The third condition says that far from the scatterer (|r| → ∞), the incidentwave has a fixed wavelength determined by the energy E. The final condition says that thescattered part of the wavefunction must correspond to a wave moving out to infinity.

Given |ψi〉, E, and V (r), we are to solve for |ψs〉. Note that this is not an eigenproblem!Usually, when dealing with the time-independent Schrodinger equation, we treat it as aneigenproblem and solve for the energy eigenvalues and eigenstates. Here, however, E is notan output of the calculation, but part of the input—it is a parameter describing the energyof the incoming particles in the experiment.

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II. RECAP: POSITION AND MOMENTUM STATES

Before proceeding, let us review the properties of quantum particles in free space. In ad-dimensional space, a coordinate vector r is a real vector of d components. A quantumparticle can be described by the position basis—a set of quantum states |r〉, one for eachpossible r. The r’s form a continuum, and so, like the real numbers, this set of states isuncountably infinite. If we are studying a particle trapped in a finite region (e.g., a particlein a box), r is restricted to that region; otherwise, r is any real d-dimensional vector.

In either case, |r〉 spans the state space, so the identity operator can be resolved as

I =

∫ddr |r〉〈r|, (0.6)

where the integral is taken over all allowed r. It follows that

〈r|r′〉 = δd(r− r′). (0.7)

The position eigenstates are said to be “delta-function normalized”, rather than being nor-malized to unity, because the r vectors form a continuum rather than a discrete set. In theabove equation, δd(· · · ) denotes the d-dimensional delta function; for example, in 2D,

〈x, y |x′, y′〉 = δ(x− x′) δ(y − y′). (0.8)

The position operator r is defined by taking |r〉 and r as the eigenstates and eigenvalues:

r|r〉 = r |r〉. (0.9)

Momentum eigenstates are constructed from position eigenstates via Fourier transforms.First, suppose the allowed region of space is a box of length L on each side, with periodicboundary conditions in every direction. Define the set of wave-vectors k corresponding toplane waves that satisfy the periodic boundary conditions at the boundaries of the box:

k∣∣∣ kj = 2πm/L, m ∈ Z, for each j = 1, . . . , d

.

Note that so long as L is finite, the k vectors form a discrete set. Next, define

|k〉 =1

Ld/2

∫ddr eik·r|r〉, (0.10)

where the integral is taken over the box. These states can be shown to satisfy the following:

〈k|k′〉 = δk,k′ , 〈r|k′〉 =1

Ld/2eik·r, I =

∑k

|k〉〈k|. (0.11)

The momentum operator is defined as an operator that has |k〉 as its eigenstates:

p|k〉 = ~k |k〉. (0.12)

Thus, for finite L, the momentum eigenstates are discrete and normalizable to unity; themomentum component in each direction is quantized to a multiple of ∆p = 2π~/L.

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Now we take the limit of an infinite box, L → ∞. In this limit, ∆p → 0, meaning thatthe momentum eigenvalues coalesce into a continuum. To properly handle the momentumeigenstates in this limit, we re-normalize them by taking

|k〉 →(L

2π

)d/2|k〉. (0.13)

In the L→∞ limit, the re-normalized momentum eigenstates satisfy

|k〉 =1

(2π)d/2

∫ddr eik·r|r〉, (0.14)

|r〉 =1

(2π)d/2

∫ddk e−ik·r|k〉, (0.15)

〈k|k′〉 = δd(k− k′), 〈r|k〉 =1

(2π)d/2eik·r, I =

∫ddk |k〉〈k|. (0.16)

The integrals are now taken over infinite spaces. The position and momentum eigenstatesare now on a similar footing; both are delta-function normalized. Note that in deriving theabove equations, it is helpful to use the formula∫ ∞

−∞dx exp(ikx) = 2π δ(k). (0.17)

For an arbitrary quantum state |ψ〉, a wavefunction is defined as the projection onto theposition basis: ψ(r) = 〈r|ψ〉. Using the momentum eigenstates, we can show that

〈r|p|ψ〉 =

∫ddk 〈r|k〉 ~k 〈k|ψ〉

=

∫ddk

(2π)d/2~k eik·r〈k|ψ〉

= −i~∇∫

ddk

(2π)d/2eik·r〈k|ψ〉

= −i~∇ψ(r).

(0.18)

This result can also be used to prove Heisenberg’s commutation relation [ri, pj] = i~δij.

III. SCATTERING FROM A 1D DELTA-FUNCTION POTENTIAL

We are now ready to solve a simple scattering problem. Consider a 1D space with spatialcoordinate denoted by x, and a scattering potential that consists of a “spike” at x = 0:

V (x) =~2γ2m

δ(x). (0.19)

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The form of the prefactor ~2γ/2m is chosen for later convenience; the parameter γ, whichhas units of [1/x], controls the strength of the scattering potential.

If you are disturbed by the idea of a delta function potential, just regard the delta functionas a limiting case of a family of non-singular functions, consisting of increasingly tall andnarrow gaussians centered at x = 0. For any non-singular potential, each solution ψ(x) iscontinuous and has well-defined first and second derivatives. In the delta function limit, thesecond derivative of ψ(x) blows up at x = 0, while the first derivative becomes discontinuousbut finite; however, ψ(x) itself remains continuous.

To formally describe this, let us integrate the Schrodinger wave equation over an infinites-imal range around x = 0:

limε→0+

∫ +ε

−εdx

[− ~2

2m

d2

dx2+

~2γ2m

δ(x)

]ψ(x) = lim

ε→0+

∫ +ε

−εdx Eψ(x)

= limε→0+

− ~2

2m

[dψ

dx

]+ε−ε

+

~2γ2m

ψ(0) = 0

(0.20)

Hence,

limε→0+

dψ

dx

∣∣∣∣x=+ε

− dψ

dx

∣∣∣∣x=−ε

= γ ψ(0). (0.21)

To proceed, consider a particle incident from the left, with energy E. This is described byan incident state proportional to a momentum eigenstate |k〉, where k =

√2mE/~2 > 0.

We said “proportional”, not “equal”, for it is conventional to adopt the normalization

|ψi〉 =√

2πΨi|k〉 ⇔ ψi(x) = 〈x|ψ〉 = Ψi eikx. (0.22)

The complex constant Ψi is called the “incident amplitude.” Plugging this into theSchrodinger wave equation gives[

− ~2

2m

d2

dx2+

~2γ2m

δ(x)

] (Ψi e

ikx + ψs(x))

= E(Ψi e

ikx + ψs(x)). (0.23)

Taking E = ~2k2/2m, and doing a bit of algebra, simplifies this to[d2

dx2+ k2

]ψs(x) = γδ(x)

(Ψi e

ikx + ψs(x)), (0.24)

which is an inhomogenous ordinary differential equation (ODE) for ψs(x), with the potentialterm on the right acting as a “driving term”.

To find the solution, consider the two regions x < 0 and x > 0. Since δ(x)→ 0 for x 6= 0,the equation in each half-space reduces to[

d2

dx2+ k2

]ψs(x) = 0. (0.25)

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This is the Helmholtz equation, whose general solution may be written as

ψs(x) = Ψi

(f1 e

ikx + f2 e−ikx) . (0.26)

Here, f1 and f2 are complex numbers that can take on different values in the two differentregions x < 0 and x > 0.

We want ψs(x) to describe an outgoing wave, moving away from the scatterer towardsinfinity. So it should be purely left-moving for x < 0, and purely right-moving for x > 0.To achieve this, let f1 = 0 for x < 0, and f2 = 0 for x > 0, so that ψs(x) has the form

ψs(x) = Ψi ×

f− e

−ikx, x < 0

f+ eikx, x > 0.

(0.27)

The complex numbers f− and f+ are called scattering amplitudes. They describe themagnitude and phase of the wavefunction scattered backwards into the x < 0 region, andscattered forward into the x > 0 region, respectively.

Recall from the discussion at the beginning of this section that ψ(x) must be continuouseverywhere, including at x = 0. Since ψi(x) is continuous, ψs(x) must be as well, so f− = f+.Moreover, we showed in Eq. (0.21) that the first derivative of ψ(x) is discontinuous at thescatterer. Plugging (0.21) into our expression for ψ(x), at x = 0, gives

Ψi [ik(1 + f±)− ik(1− f±)] = Ψ(1 + f±)γ. (0.28)

Hence, we arrive at the result

f+ = f− = − γ

γ − 2ik. (0.29)

For now, let us focus on the magnitude of the scattering amplitude (in the next chapter,we will see that the phase also contains useful information). The quantity |f±|2 describesthe overall strength of the scattering process:

|f±|2 =

[1 +

8mE

(~γ)2

]−1. (0.30)

The result is plotted above. There are several notable features. First, for fixed potentialstrength γ, the scattering strength decreases monotonically with E—i.e., higher-energy par-ticles are scattered less easily. Second, for given E, the scattering strength increases with|γ|, with the limit |f |2 → 1 as |γ| → ∞. Third, an attractive potential (γ < 0) and arepulsive potential (γ > 0) are equally effective at scattering the particle.

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IV. SCATTERING IN 2D AND 3D

We now wish to consider scattering experiments in spatial dimension d ≥ 2, which havea new and important feature. For d = 1, the particle can only scatter forward or backward,but for d ≥ 2 the scattering can occur “to the side”.

Far from the scatterer, where V (r) → 0, the scattered wavefunction ψs(r) satisfies thefree-space Schrodinger wave equation:

− ~2

2m∇2ψs(r) = Eψs(r). (0.31)

Here, ∇2 denotes the d-dimensional Laplacian. Let E = ~2k2/2m, where k ∈ R+ is thewave-number in free space. Then the above equation can be written as[

∇2 + k2]ψs(r) = 0, (0.32)

which is the Helmholtz equation in d-dimensional space.

One set of elementary solutions to the Helmholtz equation are plane waves exp(ik · r),whose wave-vectors satisfy |k| = k. But we’re looking for an outgoing solution, and a planewave can’t be said to be “outgoing”. Therefore, we turn to curvilinear coordinates.

In 2D, the relevant curvilinear coordinates are polar coordinates (r, φ). We will skip themathematical details of how to solve the 2D Helmholtz equation in these coordinates. Theresult is that the general solution can be written as a linear combination

ψ(r) =∑±

∞∑m=−∞

c±m Ψ±m(r, φ), where Ψ±m(r, φ) = H±m(kr) eimφ. (0.33)

This describes a superposition of elementary circular waves Ψ±m(r, φ), with arbitrary coeffi-cients c±m ∈ C. Each circular wave Ψ±m(r, φ) is a solution to the 2D Helmholtz equation fora given angular momentum, indexed by m ∈ Z. Its r-dependence is given by the specialfunction H±m, called a Hankel function of the “first kind” (+) or “second kind” (−).

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The H+m functions are plotted in the figure above; the H−m functions are their complex

conjugates. For large values of the input, the Hankel functions have the following behavior:

H±m(kr)r→∞−→

√2

πkrexp

[±i(kr −

(m+ 12)π

2

)]∼ r−1/2e±ikr. (0.34)

Therefore, the ± index specifies whether the circular wave is an outgoing wave directedoutward from the origin (+), or an incoming wave directed toward the origin (−).

The 3D case follows a similar logic. We use spherical coordinates (r, θ, φ), and the generalsolution to the Helmholtz equation has the form of a superposition of outgoing and incomingspherical waves:

ψ(r) =∑±

∞∑`=0

∑m=−`

c±`m Ψ±`m(r, θ, φ) where Ψ±`m(r, θ, φ) = h±` (kr)Y`m(θ, φ). (0.35)

The c±`m factors are complex coefficients. Each h±` is a spherical Hankel function, andeach Y`m is a spherical harmonic. The ` and m indices specify the angular momentum ofthe spherical wave. For large inputs, the spherical Hankel functions have the limiting form

h±` (kr)r→∞−→ ±

exp[±i(kr − `π

2

)]ikr

. (0.36)

Hence, the ± index specifies whether the spherical wave is outgoing (+) or incoming (−).More discussion about these spherical waves can be found in Appendix A.

It is now clear what we need to do to get a scattered wavefunction ψs(r) that is outgoingat infinity. We take the general solution and discard all incoming (−) wave components,keeping only the outgoing (+) terms:

ψs(r) =

∑m

c+mH+m(kr) eimφ, d = 2∑

`m

c+`m h+` (kr)Y`m(θ, φ), d = 3.

(0.37)

For large r, the outgoing wavefunction has the r-dependence

ψs(r)r→∞∼ r

1−d2 exp (ikr) . (0.38)

For d > 1, the magnitude of the wavefunction decreases with distance from the origin.This is to be expected, because with increasing r, an outgoing wave spreads out over a widerarea. Consider the probability current density J = (~/m)Im [ψ∗s∇ψs]; its r-component is

Jrr→∞∼ Im

[r

1−d2 e−ikr

∂

∂r

(r

1−d2 eikr

)]= Im

[1− d

2r−d + ikr1−d

]= k r1−d.

(0.39)

In d dimensions, the area of a wave-front scales as rd−1, so the probability flux goes asJr r

d−1 ∼ k, which is positive and independent of r. This describes a constant probabilityflux flowing outward from the origin. Note that if we plug d = 1 into the above formula, wefind that Jr scales as r0 (i.e., a constant), consistent with the results of the previous section:waves in 1D do not spread out with distance as there is no transverse dimension.

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https://en.wikipedia.org/wiki/Spherical_harmonics


V. THE SCATTERING AMPLITUDE AND SCATTERING CROSS SECTION

We can use the results of the previous section to systematically characterize the outcomesof a scattering experiment. Let the incident wavefunction be a plane wave,

ψi(r) = Ψi eiki·r, (0.40)

in d-dimensional space. Here, Ψi ∈ C is the incident wave amplitude, and ki is theincident momentum. We let k = |ki| denote its magnitude, so that the particle energy isE = ~2k2/2m. We adopt coordinates (r,Ω), where r is the distance from the origin. For1D, Ω ∈ ± which specifies the choice of “forward” or “backward” scattering; for 2D polarcoordinates, Ω = φ; and for 3D spherical coordinates, Ω = (θ, φ).

Far from the origin, the scattered wavefunction reduces to the form

ψs(r)r→∞−→ Ψi r

1−d2 eikr f(Ω). (0.41)

The complex function f(Ω), called the scattering amplitude, is the fundamental objectof interest in scattering experiments. It describes how the particle is scattered in variousdirections, and it depends on the inputs to the scattering problem, including ki and thescattering potential.

Sometimes, we write the scattering amplitude using the alternative notation

f(ki → kf ), where kf = kr. (0.42)

This emphasizes firstly that the incident wave-vector is ki; and secondly that the parti-cle is scattered in some direction which can be specified by the unit position vector r, orequivalently by the momentum vector kf = kr, or by the angular coordinates Ω.

From the scattering amplitude, we can define two other important quantities of interest:

dσ

dΩ=∣∣f(Ω)

∣∣2 (the differential scattering cross section) (0.43)

σ =

∫dΩ∣∣f(Ω)

∣∣2 (the total scattering cross section). (0.44)

In the second equation,∫dΩ denotes the integral(s) over all the angle coordinates; for 1D,

this is instead a discrete sum over the two possible directions, forward and backward.

The term “cross section” comes from an analogy with the scattering of classical particles.To understand this, consider the probablity current density associated with the scatteredwavefunction:

Js =~m

Im[ψ∗s∇ψs

]. (0.45)

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Let us focus only on the r-component of the current density, in the r →∞ limit:

Js,r =~m

Im

[ψ∗s

∂

∂rψs

]r→∞−→ ~

m|Ψi|2 |f(Ω)|2Im

[(r

1−d2 eikr

)∗ ∂∂r

(r

1−d2 eikr

)]=

~ km|Ψi|2 |f(Ω)|2 r1−d.

(0.46)

The total flux of outgoing probability is obtained by integrating Js,r over a constant-r surface:

Is =

∫dΩ rd−1 Js,r =

~km|Ψi|2

∫dΩ∣∣f(Ω)

∣∣2. (0.47)

We can assign a physical interpretation to each term in this result. The first factor, ~k/m,is the particle’s speed (i.e., the group velocity of the de Broglie wave). The second factor,|Ψi|2, is the probability density of the incident wave, which has units of [x−d] (i.e., inversed-dimensional “volume”). The product of these two factors represents the incident flux,

Ji =~km|Ψi|2. (0.48)

This has units of [x1−dt−1] (i.e., rate per unit “area”).

Let us re-imagine this incident flux Ji as a stream of classical particles, and the scattereras a “hard-body” scatterer that only interacts with those particles striking it directly:

In this classical picture, the rate at which the incident particles strike the scatterer is

Is = Ji σ, (0.49)

where σ is the exposed cross-sectional area of the scatterer. Comparing this expression toEq. (0.47), we see that

∫dΩ |f |2 plays a role analogous to the classical hard-body cross-

sectional area. We hence call

σ ≡∫dΩ |f |2 (0.50)

the total scattering cross section. Moreover, the integrand |f |2 is called the differentialscattering cross section, for it represents the rate, per unit of solid angle, at whichparticles are scattered in a given direction.

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VI. THE GREEN’S FUNCTION

The scattering amplitude f(Ω) can be calculated using a variety of analytical and nu-merical methods. We will discuss one particularly important approach, based on a quantumvariant of the Green’s function technique for solving inhomogenous differential equations.

Let us return to the previously-discussed formulation of the scattering problem:

H = H0 + V

H|ψ〉 = E|ψ〉|ψ〉 = |ψi〉 + |ψs〉

H0|ψi〉 = E|ψi〉.

(0.51)

These equations can be combined as follows:(H0 + V

)|ψi〉+ H|ψs〉 = E (|ψi〉+ |ψs〉)

⇒ V |ψi〉+ H|ψs〉 = E|ψs〉

⇒(E − H

)|ψs〉 = V |ψi〉

(0.52)

To proceed, we define the inverse of the operator on the left-hand side:

G =(E − H

)−1. (0.53)

This operator is called the Green’s function. Using it, we get

|ψs〉 = GV |ψi〉. (0.54)

Note that G depends on both the energy E and the scattering potential. To isolate thedependence on the scattering potential, let us define the Green’s function for a free particle,

G0 =(E − H0

)−1. (0.55)

This will be very useful for us, for G0 can be calculated exactly, whereas G typically has nofinite exact expression. We can relate G and G0 as follows:

G(E − H0 − V ) = I and (E − H0 − V )G = I

⇒ GG−10 − GV = I and G−10 G− V G = I.(0.56)

Upon respectively right-multiplying and left-multiplying these equations by G0, we arriveat the following pair of equations, called Dyson’s equations:

G = G0 + GV G0 (0.57)

G = G0 + G0V G (0.58)

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These equations are “implicit”, as the unknown G appears in both the left and right sides.

Applying the second Dyson equation, Eq. (0.58), to the scattering problem (0.54) gives

|ψs〉 =(G0 + G0V G

)V |ψi〉

= G0V |ψi〉+ G0V GV |ψi〉= G0V |ψi〉+ G0V |ψs〉= G0V |ψ〉.

(0.59)

This is a useful simplification, since it involves G0 rather than G. The downside is that theequation is still implicit, for the right-hand side involves the unknown total state |ψ〉 ratherthan the known incident state |ψi〉.

We can try to solve this implicit equation by using Eq. (0.59) to get an expression for|ψ〉, then repeatedly plugging the result back into the right-hand side of Eq. (0.59). Thisyields an infinite series formula:

|ψs〉 = G0V(|ψi〉+ G0V |ψ〉

)=

...

=[G0V + (G0V )2 + (G0V )3 + · · ·

]|ψi〉.

(0.60)

Or, equivalently,

|ψ〉 =[I + G0V + (G0V )2 + (G0V )3 + · · ·

]|ψi〉. (0.61)

This is called the Born series.

To understand the meaning of the Born series, let us go to the position basis:

ψ(r) = ψi(r) +

∫ddr′〈r|G0|r′〉V (r′)ψi(r

′)

+

∫ddr′ddr′′〈r|G0|r′〉V (r′) 〈r′|G0|r′′〉V (r′′)ψi(r

′′)

+ · · ·

(0.62)

This formula can be regarded as a description of multiple scattering. Due to the presenceof the scatterer, the particle wavefunction is a quantum superposition of terms describingzero, one, two, or more scattering events, as illustrated in the figure below:

Each successive term in the Born series involves more scattering events, i.e., higher multiplesof V . For example, the second-order term is∫

ddr′ddr′′〈r|G0|r′〉V (r′) 〈r′|G0|r′′〉V (r′′)ψi(r′′).

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This describes the particle undergoing the following process: (i) scattering of the incidentparticle at point r′′, (ii) propagation from r′′ to r′, (iii) scattering again at point r′, and(iv) propagation from r′ to r. The scattering points r′ and r′′ are integrated over, with allpossible positions contributing to the result; since the integrals are weighted by V , thosepositions where the scattering potential are strongest will contribute the most.

For a sufficiently weak scatterer, it can be a good approximation to retain just the firstfew terms in the Born series. For the rest of this discussion, let us assume that such anapproximation is valid. (The question of what it means for V to be “sufficiently weak”—i.e., the exact requirements for the Born series to converge—is a complex and interestingtopic, but one that is beyond the scope of our present discussion.)

VII. THE GREEN’S FUNCTION FOR A FREE PARTICLE

We have defined the free-particle Green’s function as the operator G0 =(E − H0

)−1. Its

representation in the position basis, 〈r|G0|r′〉, is called the propagator. As we have justseen, when the Born series is written in the position basis, the propagator appears in theintegrand and describes how the particle “propagates” between discrete scattering events.

The propagator is a solution to a partial differential equation:

〈r|(E − H0

)G0|r′〉 = 〈r|I|r′〉

=

(E +

~2

2m∇2

)〈r|G0|r′〉 = δd(r− r′)

⇒(∇2 + k2

)〈r|G0|r′〉 =

2m

~2δd(r− r′).

(0.63)

As before, k =√

2mE/~2 where E is the energy of the incident particle. Therefore, up toa factor of 2m/~2, the propagator is the Green’s function for the d-dimensional Helmholtzequation (see Section IV). Note that the ∇2 acts upon the r coordinates, not r′.

To solve for 〈r|G0|r′〉, we can use the momentum eigenstates:

〈r|G0|r′〉 = 〈r|G0

(∫ddk′|k′〉〈k′|

)|r′〉

=

∫ddk′ 〈r|k′〉 1

E − ~2|k′|22m

〈k′|r′〉

=2m

~21

(2π)d

∫ddk′

exp [ik′ · (r− r′)]

k2 − |k′|2.

(0.64)

To proceed, we must specify the spatial dimension d. Here, we will take d = 3; the calcula-tions for other d are fairly similar. To calculate the integral over the 3D wave-vector space,we adopt spherical coordinates (k′, θ, φ), with the coordinate axes aligned so that r − r′

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points along the θ = 0 direction. We can now do the integral:

〈r|G0|r′〉 =2m

~21

(2π)3

∫d3k′

exp [ik′ · (r− r′)]

k2 − |k′|2

=2m

~21

(2π)3

∫ ∞0

dk′∫ π

0

dθ

∫ 2π

0

dφ k′2

sin θexp (ik′|r− r′| cos θ)

k2 − k′2

=2m

~21

(2π)2

∫ ∞0

dk′∫ 1

−1dµ k′

2 exp (ik′|r− r′|µ)

k2 − k′2(letting µ = cos θ)

=2m

~21

(2π)2

∫ ∞0

dk′k′2

k2 − k′2exp (ik′|r− r′|)− exp (−ik′|r− r′|)

ik′|r− r′|

=2m

~21

(2π)2i

|r− r′|

∫ ∞−∞

dk′k′ exp (ik′|r− r′|)(k′ − k)(k′ + k)

This looks like something we can handle with contour integration techniques. But there’sa snag: the integration contour runs over the real-k′ line, and since k ∈ R+, there are twopoles on the contour (at ±k). Hence, the value of the integral, as written, is singular.

To make the integral non-singular, we must “regularize” it by tweaking its definition.One way is to displace the poles infinitesimally in the complex-k′ plane, shifting them offthe contour. We have a choice of whether to move each pole upwards or downwards, andthis choice turns out to be linked to whether the Green’s function describes waves that areincoming or out-going (or behave some other way) at infinity. It turns out that the rightchoice for us is to move the pole at −k infinitesimally downwards, and move the pole at +kinfinitesimally upwards:

This means replacing the denominator of the integrand as follows:

(k′ − k)(k′ + k) → (k′ − k − iε)(k′ + k + iε) = k′2 − (k + iε)2, (0.65)

where ε is a positive infinitesimal. This is equivalent to replacing E → E+iε in the definitionof the Green’s function. The integral can now be computed as follows:∫ ∞−∞

dk′k′ exp (ik′|r− r′|)(k′ − k)(k′ + k)

→ limε→0+

∫ ∞−∞

dk′k′ exp (ik′|r− r′|)

(k′ − k − iε)(k′ + k + iε)(regularize)

= limε→0+

∫C

dk′k′ exp (ik′|r− r′|)

(k′ − k − iε)(k′ + k + iε)(close contour above)

= 2πi limε→0+

Res

[k′ exp (ik′|r− r′|)

(k′ − k − iε)(k′ + k + iε)

]k′=k+iε+

= πi exp (ik|r− r′|) .

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Plugging this into Eq. (0.64) yields the propagator 〈r|G0|r′〉. The final result is given below,along with the results for d = 1 and d = 2 (which are obtained in a similar fashion):

〈r|G0|r′〉 =2m

~2×

1

2ikexp (ik|x− x′|) , d = 1

1

4iH+

0 (k|r− r′|), d = 2

− exp (ik|r− r′|)4π|r− r′|

, d = 3.

(0.66)

The propagator can be viewed as a function of the position r, describing a wave propa-gating isotropically outwards from a source point r′. This behavior is a consequence of ourabove choice of regularization, which tweaked the definition of the Green’s function to be

G0 = limε→0+

(E − H0 + iε

)−1. (0.67)

This is called an outgoing or causal Green’s function. The word “causal” refers to theconcept of “cause-and-effect”: i.e., a source at one point of space (the “cause”) leads to theemission of waves that head out toward infinity (the “effect”).

Different regularizations produce Green’s functions with alternative features. For in-stance, we could flip the sign of iε in the Green’s function redefinition, which displacesthe k-space poles in the opposite direction. The resulting propagator 〈r|G0|r′〉 is complex-conjugated, and describes a wave moving inwards from infinity, “sinking” into the point r′.Such a choice of regularization thus corresponds to an incoming Green’s function. Inthe scattering problem, we will always deal with the outgoing/causal Green’s function.

VIII. SCATTERING AMPLITUDES IN 3D

The propagator can now be plugged into the scattering problem posed in Sections V–VI:

ψi(r) = Ψi eik·r,

ψs(r) = 〈r|G0V |ψ〉r→∞−→ Ψi r

1−d2 eikr f(ki → kr).

(0.68)

Our goal is to determine the scattering amplitude f . We will focus on the 3D case; the 1Dand 2D cases are handled in a similar way.

In the r →∞ limit, the propagator can be simplified using the Taylor expansion

|r− r′| = r − r · r′ + · · · , (0.69)

where r denotes the unit vector pointing parallel to r. (This is the same “large-r” expansionused in deriving the electric dipole moment in classical electromagnetism.) Applying this to

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the 3D outgoing propagator gives, to lowest order,

〈r|G0|r′〉r→∞≈ −2m

~2eikr

4πrexp (−ik r · r′) (0.70)

Hence, the scattered wavefunction is

ψs(r) =

∫d3r′ 〈r|G0|r′〉V (r′)ψ(r′)

r→∞≈ −2m

~2eikr

4πr

∫d3r′ exp (−ik r · r′) V (r′)ψ(r′)

= −2m

~2eikr

4πr(2π)3/2

⟨kf∣∣V ∣∣ψ⟩, where kf ≡ kr.

(0.71)

We now combine this with the Green’s function relation from Section VI,

|ψ〉 =(I + GV

)|ψi〉. (0.72)

This yields

ψs(r)r→∞−→ −2m

~2eikr

r

√π

2

⟨kf∣∣V + V GV

∣∣ψi⟩= −2m

~2Ψi

eikr

r2π2

⟨kf∣∣V + V GV

∣∣ki⟩. (0.73)

This can be compared to the earlier definition of the scattering amplitude,

ψs(r)r→∞−→ Ψi

eikr

rf(ki → kf ). (0.74)

Hence, we find that

f(ki → kf ) = −2m

~2· 2π2

⟨kf∣∣V + V GV

∣∣ki⟩= −2m

~2· 2π2

⟨kf∣∣V + V G0V + V G0V G0V + · · ·

∣∣ki⟩, (0.75)

subject to the elasticity constraint |ki| = |kf |. In deriving the last line, we used the Bornseries formula (0.61).

This result is the culmination of the numerous definitions and derivations from the pre-ceding sections. On the left side is the scattering amplitude, the fundamental quantity ofinterest in scattering experiments. The right side contains quantities that are known to us,or that can be calculated: the initial and final momenta, the scattering potential, and theGreen’s function. Although this result was derived for the 3D case, very similar formulashold for other dimensions, but with the 2π2 factor replaced with other numerical factors.

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IX. EXAMPLE: UNIFORM SPHERICAL WELL IN 3D

Let us test the Born series against a simple example, consisting of the scattering potential

V (r) =

−U, |r| ≤ R

0, |r| > 0.(0.76)

We will assume that U > 0, so that the potential is attactive and describes a uniformspherically symmetric well of depth U and radius R, surrounded by empty space. For thispotential, the scattering problem can be solved exactly, using the method of partial waveanalysis described in Appendix A. The resulting scattering amplitudes are

f(ki → kf ) =1

2ik

∞∑`=0

(e2iδ` − 1

)(2`+ 1

)P`(ki · kf ),

where δ` =π

2+ arg

[kh+`

′(kR) j`(qR)− qh+` (kR) j′`(qR)

],

q =√

2m(E + U)/~2

k ≡ |ki| = |kf |.

(0.77)

This solution is expressed in terms of various special functions; j` and h` are the sphericalBessel function of the first kind and spherical Hankel function, while P` is the Legendrepolynomial (which appears in the definition of the spherical harmonic functions).

We will pit this exact solution against the results from the Born series:

f(ki → kf ) ≈ −2m

~2· 2π2

[⟨kf∣∣V |ki⟩+

⟨kf∣∣V G0V

∣∣ki⟩+ · · ·

]. (0.78)

The bra-kets can be evaluated in the position representation. Let us do this for just the firsttwo terms in the series:

f(ki → kf ) ≈ −2m

~22π2

[∫d3r1

exp(−ikf · r1)(2π)3/2

V (r1)exp(iki · r1)

(2π)3/2

+

∫d3r1

∫d3r2

exp(−ikf · r2)(2π)3/2

V (r2) 〈r2|G0|r1〉V (r1)exp(iki · r1)

(2π)3/2

]

=1

4π

[2mU

~2

∫|r1|≤R

d3r1 exp [i(ki − kf ) · r1]

+

(2mU

~2

)2 ∫|r1|<R

d3r1

∫|r2|<R

d3r2exp [i(k|r1 − r2| − kf · r2 + ki · r1)]

4π|r1 − r2|

].

If we use only the first term in the Born series, the result is called the “first Born ap-proximation”; if we use two terms, the result is called the “second Born approximation”.Higher-order Born approximations can be derived in a similar fashion.

The most expedient way to calculate these integrals is to use the numerical method knownas Monte Carlo integration. To find an integral of the form

I =

∫|r|<R

d3r F (r), (0.79)

17

https://en.wikipedia.org/wiki/Legendre_polynomials

https://en.wikipedia.org/wiki/Legendre_polynomials

https://en.wikipedia.org/wiki/Monte_Carlo_integration


we randomly sample N points within a cube of volume (2R)3 centered around the origin,enclosing the desired sphere of radius R. For the n-th sampled point, rn, we compute

Fn =

F (rn), |r| < R

0, otherwise.(0.80)

The Fn’s give the values of the integrand at the sampling points, omitting the contributionfrom points outside the sphere. Then we estimate the integral as

I ≈ (2R)3 〈Fn〉 =(2R)3

N

N∑n=1

Fn. (0.81)

The estimate converges to the true value asN →∞; in practice, N ∼ 104 yields a good resultfor typical 3D integrals, and can be computed in around a second on a modern computer.Similarly, to calculate the double integral appearing in the second term of the Born series,we sample pairs of points; the volume factor (2R)3 is then replaced by (2R)6.

This method for calculating the Born series can be readily generalized to more complicatedscattering potentials, including potentials for which there is no exact solution.

The figure below shows the results of the Born approximation for the uniform potentialwell, compared to the “exact” solution computed from partial wave analysis. It plots |f |2versus the scattering energy E, for the case of 90 scattering (i.e., kf perpendicular to ki),with wells of different depth U and the same radius R = 1. We adopt computational units~ = m = 1, and each Monte Carlo integral is computed using 3× 104 samples.

The first thing to notice in these results in that |f |2 diminishes to zero for large E. Thismakes sense. The scattering potential has some energy scale (U), and if the incident particleis highly energetic (E U), it will just zoom through, with little chance of being deflected.

Looking more closely at the plots, we see that for the shallower well (U = 0.1), thefirst Born approximation is in good agreement with the exact results; the second Bornapproximation is even better, particularly for small E. But for the deeper well (U = 1), theBorn approximations do not match the exact results at all. One way to interpret this is thatas a scattering potential gets stronger, an incident particle has a higher chance to undergomultiple-scattering (i.e., bouncing around the potential multiple times before escaping),which means that higher terms in the Born series become more important. In fact, if the

18


potential is too strong, even taking the Born approximation to higher orders might not work,because the Born series can become non-convergent. In those cases, different methods mustbe brought to bear. We will see an example in the next chapter, in the form of phenomenaknown as “scattering resonances”.

Exercises

1. Using the results for the 1D delta-function scattering problem described in Section III,calculate the probability current

J(x) =~

2mi

(ψ∗dψ

dx− ψdψ

∗

dx

), (0.82)

where ψ(x) is the total (incident + scattered) wavefunction. Explain the relationshipbetween the values of J on the left and right side of the scatterer.

2. Derive the Green’s function for a free particle in 1D space:

〈x|G0|x′〉 =2m

~2· 1

2ikiexp (iki|x− x′|) . (0.83)

3. In Section VIII, the scattering amplitude f(k→ k′) for the 3D scattering problem wasderived using the Born series. Derive the corresponding expressions for 1D and 2D.

Further Reading

[1] Bransden & Joachain, §13.1—13.3 and §13.5—13.6.

[2] Sakurai, §7.1–7.3, 7.5–7.6

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