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TRANSCRIPT
Notes related to “Scattering theory”
Armin Scrinzi
June 20, 2018
USE WITH CAUTION — DISCLAIMER
These notes are compilation of my “scribbles” (only SCRIBBLES, although typeset in LaTeX). Thereabsolutely no time to unify notation, correct errors, proof-read, and the like. Your primary source mustby your own notes.
For now, it is only an outline to list a few of the topics that will be covered. The notes will be partiallybe filled in as the lecture proceeds.
Contents summary
The lecture gives an introduction to quantum scattering theory, covering contents of practical impor-tance such as the Lippmann-Schwinger equation and Born series, scattering amplitude and cross section,multi-channel scattering and re-arrangement, resonances and inverse scattering. The somewhat distinctproperties of Coulomb scattering will be presented. Occasionally, computational examples will be used forillustrating paper-and-pencil theory. Fundamental concepts that underly scattering theory, such as theMoeller operators and “asymptotic completeness”, the S-matrix, and spectral properties of the Hamil-tonians, will be laid out with some mathematical rigor, but mostly without rigorous proofs. For themore practical content, the main reference is Taylor’s book “Scattering Theory”, while formulation andmathematical background will be in the spirit of Thirring’s Course in mathematical physics: Quantummechanics of atoms and molecules, with occasional references to Reed and Simon.
Purpose of the lecture
Scattering theory is notoriously difficult teach, let alone to implement in serious practical application.Contrary to quantum mechanics of bound states, simple analogies to discrete vector spaces can be mis-leading.
Literature
The course of the lecture is strongly inspired byW. Thirring - Lehrbuch der Mathematischen Physik
The part on unbounded operators follows closelyG. Teschl - Mathematical Methods in Quantum Mechanics
For basic reference we will largely useReed/Simon - Methods of Modern Mathematical Physics
Further literature will we given for selected topics.
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Contents
1 Scattering phenomena and basic concepts 51.1 Rutherford cross-section formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Partially differential, fully differential, and total cross section . . . . . . . . . . . . . . . . 81.4 Center-of-mass vs. lab frame cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Observables in scattering experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Potential scattering, multi-channel scattering, break-up, and rearrangement . . . . . . . . 91.7 Resonances and decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.8 Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.9 Scattering under time-dependent Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Non-relativistic quantum scattering theory 162.1 Mathematical modeling of a (simple) scattering experiment . . . . . . . . . . . . . . . . . 162.2 The Møller operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Hilbert space, operators, norms, spectrum 183.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.2 Adjoint and self-adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.3 Norm of an operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.4 Limits for sequences of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.5 Stone’s theorem: Self-ajointness and unitarity . . . . . . . . . . . . . . . . . . . . . 223.1.6 Spectrum and resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.7 Spectral representation of self-adjoint operators . . . . . . . . . . . . . . . . . . . . 243.1.8 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.9 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.10 Functions of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Spectral properties and dynamical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Spectral subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Fourier transform of smooth functions (Riemann-Lebesgue) . . . . . . . . . . . . . 28
3.3 Classes of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1 (Orthogonal) projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.2 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.3 Relatively bounded operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.4 Relatively compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.5 The RAGE theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.6 Asymptotic completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.7 Short range potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.8 Scattering operators compare two dynamics . . . . . . . . . . . . . . . . . . . . . . 333.3.9 The S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4.1 δ-Function of a self-adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4.2 Abel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Formal derivations 344.1 The Lippmann-Schwinger equation in disguise . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Ω± in the spectral representation of H0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.1 Remark on “ac-spectral eigenfunctions” . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Mapping free to free spaces: the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
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4.3.1 Spectral representation of the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . 394.3.2 The T -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4.1 The LS equation in position space . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 Partial wave scattering solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.6.1 Partial wave expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6.2 Spherical Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.7 Scattering phases and scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 494.7.1 Scattering phases and asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . 50
4.8 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.9 Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.10 Analyticity of S(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.10.1 Rapidly decaying potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.10.2 Poles of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.11 Levinson theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.11.1 Ingredients of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.11.2 Spherically symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.12 The delay operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.12.1 Derivation of the delay operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.12.2 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.13 Properties of the scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.13.1 Optical theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.13.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.13.3 Detailed balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.13.4 Unitarity limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.13.5 Scattering length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.14 Computational techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.14.1 The Born series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.14.2 Kohn variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.15 Standard textbook examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.15.1 Hard sphere scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.15.2 Spherical potential scattering: Ramsauer-Townsend effect . . . . . . . . . . . . . . 65
5 Complex scaling 675.0.1 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.0.2 Analyticity of eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 The origin of resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.1.1 Fermi golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.1.2 Doubly excited states and Feshbach resonances . . . . . . . . . . . . . . . . . . . . 725.1.3 Fano theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Multi-channel scattering 766.1 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.1.1 S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.1.2 LS-equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.1.3 An alternative view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.1.4 Distorted waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2 Two interacting particles in a potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.2.1 Asymptotic completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2.2 Jacobi-coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2.3 The S-matrix (interaction picture) . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
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6.2.4 The S-matrix (Heisenberg picture) . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.3 Graphical representation of the resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7 Coulomb scattering 907.1 Failure of standard scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.2 Compare to a different time-evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.3 Asymptotic constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.4 Complete solution of the Coulomb problem . . . . . . . . . . . . . . . . . . . . . . . . . . 937.5 Exact Coulomb scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5.1 Scattering transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.5.2 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.5.3 Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.6 Coulomb plus short range potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
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1 Scattering phenomena and basic concepts
We do not, in general, approach microscopic systems with microscopic in-struments. Neither do we, in general, observe processes directly on thetime scale on which they evolve. There are notable exceptions to this —such as the scanning tunneling microscopy in space, or attosecond sciencein time — but the bulk of our knowledge about the microscipic world isdrawn from analysis of measurements at lab distances and times muchlarger than the inherent time-scale of a process. Scattering theory is aboutthis connection of macroscopic measurement to microscopic events. Con-sidering large times and distances as “practical” infinity, the simplificationfrom taking the limit to mathematical infinity exposes the simplest possiblerepresentation of actual measurement.
“Simplest possible”, unfortunately, is a relative statement here, and theunwieldiness of scattering theory results in its neglect in standard curricula,notwithstanding its extraordinary conceptual and practical importance.
The abstract procedure of a scattering experiment is simple: direct atmacroscopic beam (of light or other particles, the “projectiles”) onto a“target” (usually an ensemble of microsopic particles, sometimes even sin-gle particles), and detect any changes in the beam or fragments of thetarget that are emitted.
1.1 Rutherford cross-section formula
In that historical experiment of 1909, the beam were the alpha-particlesfrom the decay of Radium (discovered just a few years earlier), the target agold foil. The result was quite unexpected and could be explained as scat-tering by point-like Coulomb charges, upseting the contemporary conceptsof atomic structure.
The celebrated scattering formula can be derived by simple means, if notwith much mathematical rigor. Ironically, it is not amenable to standardmethods of scattering theory as the Coulomb potential “never ends”, a factthat has disquieted cosmolgists on earlier occasions. (For it to really “end”it would need to decay as r−1−ε at the least).
The problem is similar in quantum and classical mechanics, and for afirst exercise you should remind yourself of the following formula’s deriva-
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tion. The differential cross-section for scattering two charge, point-likeparticles is
dσ
dΩ(E, θ) =
(Z1Z2e
2
4E sin2 θ2
)2
. (1)
Problem 1.1: Center-of-mass to lab frame Derive the transformation formulafor the cross section.
dσ
dΩ lab=dσ
dΩcm
(1 + 2λ cos θcm + λ2
)3/2
|1 + λ cos θcm|(2)
with λ = mp/mt the ratio of projectile to target mass and θcm the anglein the center-of-mass system. Specify the arguments in the cross-sectionfunction.Hint: All that is needed is to transform the partial derivatives
∂
∂Ωl=∂Ωl
∂Ω
∂
∂Ω
Problem 1.2: First Born approximation A popular method to investigatescattering is the so-called “first Born approximation”. Its main part is tocompute the “scattering amplitude” in the first Born approximation f (1)
asf (1)(~k,~ki) = −(2π)2〈~k|V |~ki〉, (3)
where |~k〉 and |~k′〉 are δ-normalized plane waves. Compute that integralfor the Yukawa potential
V (r) = V0e−µr
r, µ > 0. (4)
The modulus squared of the scattering amplitude is the cross section
dσ(~k,~ki)
dΩ=∣∣∣f(~k,~ki)
∣∣∣2 . (5)
(a) How does the first Born cross section depend on the sign of V0?
(b) How does the cross section behave with |~k| → 0?
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(c) What is the total cross section∫ 1
−1
d cos θ
∫dφ|f (1)(~k,~k0)|2? (6)
θ and φ are the polar angles of ~k = |~k0|(sin θ cosφ, sin θ sinφ, cos θ).
(d) What is the cross-section in the limit µ→ 0?
Hint: In polar coordinates, the integral can be readily computed.
1.2 Cross-section
This introduces the important concept of cross section, which has an imme-diate interpretation for measurement: suppose we have beam of particlesincident from a fixed direction (say the z-axis) onto a target (one of the goldnuclei in the foil) and the energies are uniformly distributed in as (differ-entiably) small bin [E,E + ∆E]. Our beam being macroscopic, we cannotmake any statement about where the target nucleus is located across thebeam, other than we would like to ensure that alpha particles pass at allreasonable microscopic distances of it. We assume that the beam is homo-geneous across all possible target locations, i.e. the density of flux is equaleverywhere. We characterize the incident beam by (a) the energy E and(b) the number of particles passing through a unit surface perpendicularto the beam direction per unit time: ρi(E). The measurement consistsin detecting particles in all directions relative to the z-direction. As ourmicroscopic system, the gold atom, can be idealized as a point particle andis spherically symmetric (as we know today), the only relevant angle is thepolar angle θ of the spherical coordinate system. Counting the number of“scattered” particles Ns per unit time found at some solid angle
Ω = θ, φ : [θ, θ + ∆θ]× [φ, φ+ ∆φ] =: [Ω,Ω + ∆Ω], (7)
we will find
Ns = ρidσ
dΩ(E, θ)∆Ω. (8)
We see that the name “cross-secton” is fitting, as dσ/dΩ has the dimensionof a surface. You can consider it as the “size” of the surface taken out fromthe beam by that target and re-directed into Ω.
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The pathologies of Coulomb scattering show already here: we mightwant to ask how many particles are “taken out of the beam” and butthen released in beam direction, i.e. θ = 0, the formula gives an answer:infinitely many, it diverges. At first glance a nonsensical result, but it isactually true: our idealization of the beam as homogenous implies that it is“infinitely wide”, so after we have taken all finite pieces for scattering intoany finite angle, there still remains infinitely much beam, which howeverdoes not pass without noticing the presence of the Coulomb potential: therewill be some delay (or advance) of the particle, as it passes the repulsive (orattractive) Coulomb center, not matter at which distances: the forwardscattering amplitude of the Coulomb potential is inifinite: it transversallyis “infinitely large”.
“Reasonable”, i.e. short range potentials, do not have infinite crosssections: this actually is true for any potential falling off stronger than1/r2 at large distances check.
1.3 Partially differential, fully differential, and total cross section
The above example allows to first introduce the important concepts fullydifferential, partial, and total cross section. To the extent that we are deal-ing (in our idealization) with spinless point particles, the Rutherford crosssection differentiates all oservables at large times: direction and energy.The cross section is “fully differential”.
Suppose we cannot measure energy E, but only the angle Ω. Count-ing the particles at that angle results in a partially differential cross-section w.r.t. to angles (the dependence on φ is constant, trivially):
dσ
dΩ(Ω) =
∫ ∞0
dEdσ(E, θ)
dΩ(9)
Likewise, one could integrate over angles and differentiate over energies:
σ(E) =
∫dΩ
dσ(E, θ)
dΩ=
∫ 2π
0
∫ π
0
dθ sin θdσ(E, θ)
dΩ, (10)
which is infinte for the Coulomb scattering, i.e. it is not a good number toask for: Coulomb always scatters.
Finally, if we ask whether there is any scattering at all, i.e. we integrateover all energies and angles, we get the total cross section σt. We have
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noticed already that the total cross section for the Coulomb potential isinfinite.
1.4 Center-of-mass vs. lab frame cross-section
The Rutherfor cross-secton formula holds for a fixed, infinite mass target,or, alternatively, for a in rest frame of the center of mass. In the case of goldatom and alpha-particles, the mass ratio is about 1/60, and although theapproximation is not bad, an accurate description require transformationinto the center of mass frame moving into the lab-frame by the formula
dσ
dΩ lab=dσ
dΩcm
(1 + 2λ cos θcm + λ2
)3/2
|1 + λ cos θcm|(11)
This is left as an entertaining exercise to get you started on the use ofcalculus in this lecture.
1.5 Observables in scattering experiments
Detectors can register:
• direction of emission,
• kinetic energy of arriving particle,
• possibly its internal state (spin, excitation state, mass),
• arrival times.
Note that resolution of arrival times is on a macroscopic time scale (evennano-seconds are “macroscopic” in relation to most microscopic time scales).These times, is measured, are usually used to determine kinetic energyfrom knowing time of emission and distance to the point of arrival (“timeof flight” detector).
1.6 Potential scattering, multi-channel scattering, break-up, and rearrange-ment
The kinematic situation of Rutherford scattering is the simplest possible:a particle gets deflected from a rotationally symmetric scatterer which ap-pears (in center of mass) as a potential. There is no source or sink of energy,
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energy is conserved. This is a case of potential scattering. Rotationalsymmetry reduces the problem to an (infinite) set of one-dimensional prob-lems, each for a given angular momentum, the partial waves.
This a standard situation, also for example in scattering of atoms andmolecules. Rotationally symmetric potential scattering is the case of twoatoms, say He and Ar at energies that are so low that the systems cannotbe transferred into excited states:
He+ Ar −→ He+ Ar. (12)
The lowest excitation energy of Ar is about 12 eV, below that energy theabove process can be considered as potential scattering (a cleaner casewould be electron-proton scattering).
Potential scattering also includes scattering without rotational symme-try, which already poses a significant complication, e.g. scattering frommolcules, say H2:
He+H2 −→ He+H2. (13)
Here potential scattering is an adequate picture only at the lowest ener-gies, as H2 can be brought easily into rotation, with the lowest rotationalexcitation at & 10meV .
If the target can assume different states (say the ground or excited statesof an atom or a nucleus, labelled as A(n)), a new layer of complexity isadded. We have processes where scattering changes the state of the target:
He+ A(m) −→ He+ A(n). (14)
We enter the scattering process in one “channel” (m), but leave through adifferent channel (n). We need to differentiate the multi-channel scat-tering cross-section
dσ(Ein,m, n,Ω)
dΩ, (15)
depending on the ingoing kinetic energy Ein, the initial and final statelabels m,n, and the scattering angle. The outgoing kinetic energy is, ofcourse,
Eout = Ein + Em − En, (16)
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where Em, En are the internal energies of A(m), A(n), respectively. For ex-ample, Ein > 12eV , we can excite Ar(3s23p6) → Ar(3s23p54s), and thecenter-of-mass Eout = Ein = 11.6eV .
Multi-channel scattering is very common in experimental practice, but itentails already a significant mathematical complication. Things get worse,when projectile and/or target can disintegrate, as any of the followingexamples of break up reactions
He+H2 −→He+H +H (17)
He+H2 −→He+H + p+ e (18)... (19)
He with its binding energy of about 24 eV will remain intact at energiesbelow that, but may and will disintgrate at higher scattering energies.
A further variation of the situation is that of both projectile and targetrearrange into new compounds. An important such process is electroncapture in nuclear physics, where, for example an instable C11 nucleus cap-tures and electron and, under emission of a neutrino, converts to Bo11. There-arrangement of constituents, in the end, is essence of reactive chemistryand as such of utmost importance to life. But the underlying processes donot usually occur in typical scattering setups.
A fully differential cross section in such more complex situations canbe defined as follows: Given an ensemble of incident sub-system/particlescharacterized by ~ki, si, i = 1, . . . , I with momenta ~ki, internal states si(e.g. spin, constituents, excitation state etc., if any), and flux Fi (numberof such subsystems per time per surface), one measures the number Ne ofsub-systems/particles emitted per time into momenta and internal states~qe, se, e = 1, . . . , E. The fully differential cross section is the ratio
σ(~k1, s1, . . . ~kI , sI ; ~q1, s1, . . . ~qE, sE
)=Ne
Fi(20)
This is what we ultimately need to know for comparison with observation.We also see that this is an enormous amount of information, both, to mea-sure and to compute. Handling such complex situations is commonplace inparticle physics. In atomic and molecular physics there also exists a pow-erful instrument that delivers such complex data: the COLTRIMS (Cold
11
Target Recoil Ion Momentum Spectroscope), where all outgoing momentacan be collected.
For potential scattering we possess mature and well-founded mathemat-ical theory that also allows practical implementation. As complexity in-creases, mathematical rigor quickly disintegrates and practical implemen-tation becomes cumbersome and rapidly deteriorates in accuracy. The lec-ture will concentrate on the rigorous part, only formulating, rarely solvingthe problems of the more complex situations.
1.7 Resonances and decay
Scattering cross sections can and do very often show striking, well-isolatedpeaks. The phenomenon is rather generic and occurs in the eerie worldof elementary particle physics as well as quantum-toddler’s square-wellpotential. The peak occurs around a certain resonance energy Er (andpossible other parameters as angular momentum or spin), and it has acertain width Γ. Two prominent line-shapes are the Lorentzian (or Breit-Wigner) profile
σ(E,Er,Γ) = σ0(Γ/2)2
(E − Er)2 + (Γ/2)2(21)
and the slightly more complex Fano profile
σ(E − Er) = σ0(qΓ/2 + E − Er)
2
(E − Er)2 + (Γ/2)2(22)
which shows a characteristic dip to (ideally) zero either above or below thepeak, depending on the sign of q. The physics of the Fano resonance is theinterference of alternative paths’ to the final outgoing energy state.
It is inherent in the “fuzzyness” of resonance peaks that they cannot bedirectly associated with standard isolated quantum-mechanical states. Theresonace is a property of a whole surrunding of scattering states (associatedwith the Hamiltonian’s continuous spectrum), but no single state exhibitsa priviledge role and the resonance nature truely only manifests itself bytheir phase relations, a rapid change of “scattering phases” as one passedthe resonance energy.
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Figure 1: The Fano line shape, dependence on the “asymmetry parameter” q
The mathematical essence of the resonance is uncovered by entering thecomplex energy plane. A complex resonance energy
Wr = Er − iΓ/2 (23)
unites the resonance’s two key parameters. Being complex, this cannotbe associated with the normal self-adjoint Hamiltonian. Historically, res-onance energies where identified as position of poles found in the energy-dependent “scattering matrix” upon analytic continuation. If the polesare isolated (no other poles in a surrounding of radius & Γ) and close tothe real axis (small Γ), the Lorentzian peak results. It was also foundthat under suitable conditions the Hamiltonian itself can be analyticallycontinued. The resulting non-Hermitian Hamiltian can have the complexeigen energies Wr with square integrable functions associated with it, whichfairly can be called a resonance state Ψr, although this state has no strictinterpretation in the framework of hermitian quantum mechanics. Reason-ing from the complex plane requires analyticity properties, which we willdiscuss in some detail.
Taking the resonance state a face value, it decays at constant rate:
||Ψr(t)||2 = ||e−itWr/~Ψr(0)||2 = e−Γt/~||Ψr(0)||2 (24)
with a resonance “lifetime”
τr = ~/Γ. (25)
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The idea of decay at constant rate is used widely, although, strictly,there is little in standard quantum mechanics that supports it. It can berigorously proven that at short times no quantum states can decay at con-stant rate (this is a rather simple, but beautiful consequence of assumingthe time-evolution to be differentiable and reversible: as you start time-evolution either forward or backward in time must be the same, so thederivative of probability change must be 0). Less obviously, time-evolutioncannot be exponentially decaying at long times, a fact that holds for anySchrodinger type Hamiltonian that has a ground state (a lower bound).Yet, at intermediate times one does find the near exponential decay asso-ciated with decay width Γ.
The underlying physics of resonances is more often than not the existenceof an unperturbed Hamiltonian H0 with a set of rigorous bound statesthat becomes coupled by an interaction HI to a continuum of some otherHamiltonian HC surrounding some of H0’s bound states. In the simplestpossible case we have the total Hamiltonian
H0 +HC +HI = |Φ0〉E0〈Φ0|+∫dE|E〉〈E|+ |E〉c(E)〈Φ0|+ |Φ0〉c∗(E)〈E|
(26)Due to the coupling c(E) the bound state Φ0 “dissolves” into the contin-uum but it continues a ghost-like life in the complex plane and shapes the“density of states” of the continuum states associated with the real axis.The scattering states for that simple model can be written down in closedform and can be studied for insight into the nature resonances.
You may also recongnize in the above the basic situation described byFermi’s Golden Rule, which is nothing but approximation to Γ for this casein second order perturbation theory.
1.8 Inverse scattering
By sending projectiles onto targets and watching the response, can we re-ally expose the true nature of the target? Not usually. But sometimes thescattering problem can be inverted, i.e. we can reconstruct the scatteringpotential from complete knowledge of the scattering states. This so-called“inverse scattering” method has — to my knowledge — little application
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for the study of actual scattering processes. However, it is turns out to bea powerful method for constructign solutions to non-linear wave equations:the solutions appear as the reconstructed potentials for given (artificial)scattering data. Famous examples are the “non-linear Schrodinger equa-tion” governing the formation of stable solitary waves in optical fibers,
id
dtA(x, t) = −∂2
xA(x, t) + |A(x, t)|2A(x, t) (27)
or the non-linear “Korteweg-de-Vries” equation that describes the forma-tion of unusually stable waves (“tidal bores”) running up rivers at fromsome shore lines:
d
dtU(x, t) + 6U(x, t)∂xU(x, t) + ∂3
xU(x, t) = 0 (28)
Time permitting, we will touch upon this technique.
1.9 Scattering under time-dependent Hamiltonians
Finally I want to mention (and we will surely discuss it) the situationwhere we have a time-dependent Hamiltonian H(t). This usually resultsfrom modelling a physical system which can be, in part, described semi-classically. An important example is the interaction of quantum matterwith intense laser light. Intense here means that absorption of a few pho-tons by the microsopic target from the beam does not change the beamin any relevant amount. Another example is heavy ion scattering, wherea not-so-heavy system just notices the rapidly changing Coulomb field ofheavy ion passing with out the ion’s trajectory being significantly deflected.
For the laser case, at longer wave length, the Hamiltonian is reasonablydescribed in dipole approximation
H(t) = H0 + ~E(t) · ~r, (29)
where ~E(t) the electric field of pulsed or also a contiuous-wave (cw) laser.The time-dependent interaction leads to disintegration of an initially boundsystem. In the case fo cw-lasers, the situation can be converted into a sta-tionary problem and disintegration takes the form of a resonance decayinto a “photon dressed” continuum. In case of finite pulse duration, tech-niques of standard scattering theory are applicable in principle, but often
15
the direct integration of the time-dependent Schrodinger equation is themore feasible approach.
2 Non-relativistic quantum scattering theory
All deeper mathematical discussion in these lectures will be limited to thecase of a single particle that scatters without change of its internal statefrom a fixed, immutable target that is represented by a potential. I.e.1 = I = E and si = se, such that the only relevant two arguments of thecross section are the initial and final momenta ~ki and ~qe. This is what weintroduced as “potential scattering”.
Further we will not treat a statistical ensemble of particles but rathera particle described by a single wave function. This is a minor technicalsimplification useful at the present level.
2.1 Mathematical modeling of a (simple) scattering experiment
Prepare a particle in some “free” state, e.g. free wave packet
Φ(~r,−t) = (2π)−3/2
∫d(3)kei
~k·~rψ(~k) (30)
at some time −t. Prepare the packet such that no part of it is in the rangeof interaction.
Let that wave packet evolve according to the scattering time-evolution.Wait long enough, say, until t, and analyze the wave packet outside therange of the interaction.
The fundamental mathematical issues are:
• What does “preparing the state” mean? Can we actually map any ar-bitrary free wave packet into a scattering wave packet? Is that mappingunique? (Existence and uniqueness of scattering solutions.)
• What is the fate of the scattering state? Can the system just “swallow”an incoming particle? Or can we be certain to find the system againin a (possibly quite different) free wave packet at some large time t?(“Asymptotic completeness”)
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2.2 The Møller operators
Now assume that far back in time a given wave packet can escape theaction of the potential. At that time, free and interacting time evolutionare indistinguishable for the wave packet. Now we can consider the wavepacket as an interacting one and evolve forward in time to the present bythe full time evolution. Under which circumstances can we find times largeenough to do such a mapping? Do the following limits exist:
limt→±∞
U(−t)U0(t) := Ω±? (31)
In which sense may the limits exist? Can it exist in an operator sense?Given ε,
?∃T : ||U(t)U0(−t)− Ω±|| < ε, ∀ ± t > T (32)
(“Uniformly” for all states): probably not, we only need to choose arbi-trarily slow wave packets, i.e. particles that move extremely slowly out ofthe interaction region at a speed v < L/T , where L is some characteristicdimension of the interaction region. However, it may exists “pointwise”for any wave packet
||[U(t)U0(−t)− Ω±]|Φ〉|| → 0 for ± t→∞ (33)
That is, the time-limit can exist in the sense of a strong limit.
2.2.1 Properties
Note that the Møller operators map a wave-packet composed of free elec-trons (plane waves, exp(i~k~r) into a wave packet composed of scattering
solutions Ψ(±)~k
(~r) with asymptotic momenta ~k. We will find that the en-ergies remain unchanged in the process (as it should be). We expect, insome sense,
Ψ(±)~k
(~r) = Ω±ei~k~r. (34)
and
−1
2∆ei
~k~r = H0ei~k~r =
~k2
2ei~k~r and (H0 + V )Ψ
(±)~k
= HΨ~k =~k2
2Ψ~k (35)
We say “in some sense”, as the involved waves are not in Hilbert space.
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For the continuous spectrum we expect an intertwining property ofthe Ω±, i.e.
HΩ± = Ω±H0. (36)
3 Hilbert space, operators, norms, spectrum
3.1 Operators
3.1.1 Domain
We first point out that any operator is only fully defined, if we specify alsoall functions from the Hilbert H space on which it is allowed to be applied,the domain D(H) ⊂ H of H. While everybody understands the neces-sity of excluding some functions from the domain, e.g. non-differentiablefunctions for differential operators, the domain is not, in general, defineduniquely in any natural fasion. It always has an element of a choice init, which means that we really need to choose, which operator exactly wewant to use, even if we already have fully defined it, say, for all functionsof a countable basis. This choice ultimately determines such importantproperties as the spectrum of an operator (and whether it is real). Thedomain is an essential element of theory of unbounded operators and it isdealt with in functional analysis. It is not our ambition to dive into thisother than mentioning that manipulating the domain can easily ruin thephysically relevant properties of an operator.
3.1.2 Adjoint and self-adjoint operator
Given an operator H and its domain D(H), one can introduce the adjointoperator H† and its domain D(H†) through
D(H†) = φ ∈ H| supψ∈D(H)
〈φ|Hψ〉 <∞. (37)
Note that once H and D(H) are given, D(H†) fully defined and there isno further element of choice. The action of H† is then fully determined bydefining all matrix elements
φ ∈ D(H†), ψ ∈ D(H) : 〈H†φ|ψ〉 := 〈φ|Hψ〉. (38)
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In introductory quantum mechanics, the adjoint operator is introducedthrough Eq. (38) only, usually not discussing the domain of the adjoint.For (38) to provide a complete definition of the operator we need the com-pleteness property of the Hilbert space and the continuity property of thescalar product 〈·|·〉. The domain (37) completes the definition of H†. Thebottom line is: given H and D(H), its adjoint H†, D(H†) is fully defined.
A self-adjoint (s.a.) operator is an operator
D(H) = D(H†), and Hψ = H†ψ, ∀ψ ∈ D(H). (39)
The term hermitian operator is limited in mathematics to operators
D(H) ⊂ D(H†), and Hψ = H†ψ, ∀ψ ∈ D(H), (40)
i.e. it does not guarantee the the operator and its adjoint have the samedomain. This is a very important distinction, as only self-adjoint operatorshave strictly real spectrum, a property of great importance in physics.
3.1.3 Norm of an operator
We will need the “norm” of an operator, which can be introduced in severalequivalent ways. We will mostly use the form
||H|| = supψ∈D(H)
||Hψ||||ψ||
(41)
An intuitive picture of the operator norm is the operator’s “largest eigen-value”. Suppose the operator had a complete set of (possbibly infinitelymany) eigenvectors ψn with eigenvalues En, then it is easy to see that thisholds. We immediately see that an operator may (and in QM scatteringtheory often will) be unbounded, i.e. ||H|| =∞.
The intuition about the norm can be carried over to operators with con-tinuous spectrum, where, instead of eigenfunctions, you imagine sequenciesof ever better concentrateted wave-packets in the continuous spectrum.
Example: H = L2(dx, [0, 1]), operator (Xφ)(x) = xφ(x), D(X) = H:||X|| = 1. “Eigenfunctions” are δ-function-like objects, that can be ap-proximated by any sequence of “nice” functions dx0(x) whoses supportincreasingly concentrates around the spectral value x0 (and whose normeare kept below some finite bound).
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Problem 3.3: Operator norm Let A ∈ B(H) be an operator with a completeset of eigenfunctions |n〉 for discrete eigenvalues En. Show that
||A|| = supn|En|
Now let X ∈ B(L2(dx, [0, 1]) be the operator
(Xφ)(x) = xφ(x).
Clearly, this operator has purely continuous spectrum. Show that ||X|| =1.
Properties of the operator norm The operator norm has all properties one ex-pects of a norm
||P|| ≥ 0 (42)
||αP|| = α||P|| (43)
||P|| = 0⇔ P = 0 (44)
||P + Q|| ≤ ||P + Q|| (45)
||PQ|| ≤ ||P|| ||Q|| (46)
(47)
Also, it is compatible with the algebraic structure of (bounded) operatorsand with forming the adjoint, as follows:
||Q†|| = ||Q|| (48)
||QQ†|| = ||Q|| ||Q†|| (49)
||1|| = 1 (50)
The bounded operatators on a Hilbert space, denoted as B(H) form aBanach space, i.e. Cauchy-sequences have a limit that is also a boundedoperator.
3.1.4 Limits for sequences of operators
Already for ordinary funcitions we know of different ways how a squencecan approach a limit (or fail to approach it). Let fn(x) be a sequence of
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functions R→ R. If there is a function f(x) such that
|fn(x)− f(x)| → 0 for any x ∈ R, (51)
we say the funcition converges to f point-wise. You have also met conver-gence w.r.t. to some p-norm on the function space || · ||p, for example L2
with p = 2 or p =∞ : ||f ||∞ = supx |f(x)| meaning
||f − fn||p → 0 (52)
This means that, given an ε there is some N(ε) such that the norm ||f −fn>N ||p never exceeds ε, in case of || · ||∞ the difference between the funcionvalues is nowhere larger than ε. In case of || · ||2, the difference remain finiteat a set of measure 0. For example, fourier series of discontinuous functionsdoes not converge pointwise at the disconituity, but the do converge forp = 2.
Similar distinctions are of special importance in scattering theory. Threedifferent forms of convergence will be important:
Norm-convergence ||H − Hn|| → 0 (similar to p = ∞ uniform convergenceof functions). We also write
s- limn→∞
Hn = H, Hn → H
Strong convergence ||(H −Hn)ψ|| → 0, ∀ψ ∈ D(H) (for simplicity consideronly D(H) = D(Hn)). This is like point-wise convergence of a function.For strong convergence will write
s- limn→∞
Hn = H, Hns→ H
Weak convergence 〈χ|(H − Hn)ψ〉 → 0, ∀χ, ψ ∈ D(H), i.e. convergence ofany legitimate matrix element. For weak convergence will write
w- limn→∞
Hn = H, Hnw→ H.
On finite-dimensional spaces, all three kinds of convergence are equivalent.The situation in scattering theory will be that the limit t→∞ will not
exist in the norm, it will be trivially =0 in weak convergence, and only instrong (ψ-wise) convergence it will be what we really whish.
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3.1.5 Stone’s theorem: Self-ajointness and unitarity
A unitary operator on a Hilbert space is defined as
UU † = U †U = 1. (53)
This property also implies D(U) = H. Selfadjoint operators H and groupsof unitary operators U(t) are intimately related by
Theorem 1. (Stone) Let U(t) be a strongly continuous one-parameterunitary group. Then it has the form U(t) = exp(−itA) for some self-adjoint A.
Here, “strongly continuous” means
limh→0||U(h)Ψ−Ψ|| = 0, (54)
i.e. the U(h) converges to 1 in the strong sense (“Ψ-wise”):
s-limh→0
U(t+ h) = U(t). (55)
This is a theorem of central relevance for quantum mechincs: contrary topopular justifications for the preference of real-value spectra and “hermi-tian” operators, it is this relation between self-adjoint operators and uni-tary groups (symmetry transformations) that singles out the self-adjointoperators as the objects of choice in quantum mechancs.
3.1.6 Spectrum and resolvent
The mathematically solid way to define the spectrum is to consider itscomplement, the resolvent set ρ(H)
ρ(H) = C\σ(H) = z ∈ C|∃(H − z)−1, ||(H − z)−1|| <∞ (56)
i.e. those z for which an inverse can be defined and has finite norm.This reverts the pedestrian definition of spectral value
∃ψE : (H − E)ψE = 0, (57)
which is difficult to make precise in Hilbert space as the continuous eigen-functions are not from Hilbert space.
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The set of operators labelled by z (“operator-valued function”)
RH(z) := (H − z)−1 (58)
is called the resolvent of H. It is a bounded operator for all z from theresolvent set. Its limit as z approaches a spectral value plays a centralrole in scattering theory and goes by the name of “Green’s function” or“propagator”.
This is often a much more pleasant object to deal with than the operator.For example, it is often analytic in z.
Problem 3.4: Convergence of operators Let B(H) denote the space ofbounded operators on a finite-dimensional Hilbert space H and let Bndenote a sequence of operators. Show that the three following definitionsof convergence are equivalent
1. Norm-convergence: Bn → B ⇔ ||Bn − B|| → 0 (Remember: ||B|| =sup||ψ||=1 ||Bψ||)
2. “Strong” convergence: Bns→ B ⇔ ||Bnψ −Bψ|| → 0∀ψ ∈ H
3. “Weak” convergence: Bnw→ B ⇔ 〈φ|(Bn −B)ψ〉 → 0∀φ, ψ ∈ H
On infinite-dimensional Hilbert spaces this is not the case, there is onlythe implication norm-convergence ⇒ strong convergence ⇒ weak conver-gence.
(a) Using H = l2, i.e. the space of infinite length vectors, construct Bns→
B, but Bn 6→ B
(b) Construct Bnw→ B, but Bn 6
s→ B.
Hint: In both cases the trick is to let escape the maps Bn to ever newdirections, e.g. by a sequence of operators that connect ever new pairs ofentries etc.
Problem 3.5: Resolvent Let A ∈ B(H) be a bounded operator on Hilbertspace.
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(a) Show that the formal series expansion
(z − A)−1 = z−1∞∑n
(Az−1)n (59)
converges for |z| > ||A||.Hint: Use the properties of the operator norm to show the the seriesof finite sums is Cauchy.
(b) Conclude from the above that (A− z)−1 ∈ B(H) for ||A|| < |z|.
3.1.7 Spectral representation of self-adjoint operators
One very convenient property of self-adjoint operators is that they have aspectral representation. (Hermitian, but not s.a. operators, btw, do notnecessarily have such a representation, this one reason why the distinctionis important.) This is what you constantly use in quantum mechanics whenyou write things like
H =∑∫
dE|E〉E〈E| =∑n
|En〉En〈En|+∫dE|E〉E〈E|, (60)
where the |E〉 are eigenfunctions
H|E〉 = |E〉E (61)
with the normalization
〈En|Em〉 = δmn for bound states, 〈E|E ′〉 = δ(E − E ′) for continous states.(62)
Self-ajointness is a sufficient, but not necessary condition for the existenceof a spectral representation. In fact, all operators that are bounded and“normal” do have a spectral representation. Normal means here that theoperator commutes with its adjoint:
H†H = HH†. (63)
(For unbounded operator, this definition of “normal” must be modified.)One such case is the unitary time-evolution operator of quantum mechanics
U(t) = exp(−itH), (64)
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which we can write as
U(t) =∑∫
dE|E〉 exp(−itE)〈E|. (65)
The spectral representation is extremely useful as it allows to think ofa self-adjoint operator pretty much like of a multiplication by the spectralvalues. Operators that have a spectral representation and sets of suchoperators that commute are indeed closely related to ordinary functions,where formation of the adjoint maps into taking the complex conjugate.
A somewhat more rigorous statement of the spectral theorem uses thefact that there exists an L2 Hilbert space on the spectrum of H
Hσ = L2(µ(E), σ(H)), (66)
where µ(E) is a measure on the real axis, which is the Lebesgue measure forthe continuous spectrum (usual integration) and the δ-like Dirac measurefor the “point spectrum” of bound state energies. The spectral theoremthan says that there exists a unitary map from the spectral space to theoriginal Hilbert space
UH : Hσ → H : |φσ〉 → |φ〉 = UH |φσ〉, (67)
by which we bring the operatorH to diagonal (multiplication-by-eigenvalue)form
H = UdHU−1 : (dHφσ)(E) = φσ(E)E. (68)
Unitary is here to be understood as ||Uφσ||Hσ = ||φ||H
3.1.8 Fourier transform
The one-dimensional spectral representation of i∂x. Hσ = L2(dk,R)
Ui∂x = F(φ)(k) = (2π)−1/2
∫ ∞−∞
eikxφ(x)dx (69)
Unitarity = Parseval
3.1.9 Degeneracy
Label eigenvectors in a degenerate subspace by α: may be a discrete indexlike, say l,m of angular momentum, or a continuous index, say, if you
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consider the operator −∂2x in L2(dxdy|R2) where α = y,
H =∑∫
dα∑∫
dE|E,α〉E〈E,α|. (70)
A whole (finite or infinite) set of functions is associated with each E. Incase of the point spectrum this is a subspace of H.
Problem 3.6: Spectral representation Make the final steps to get the spectralrepresentation of U(t).Problem 3.7: Spectral representation Write the spectral representation of−∆ as defined in the lecture, using lm for labelling the degeneracy.Hint: Use the following expansion
ei~k~r =
∑lm
Y m∗l (k)Y m
l (r)jl(kr), (71)
where jl are the spherical Bessel functions.
3.1.10 Functions of operators
The spectral representation is also a standard way for defining functionsof operators
f(H) =∑∫
dE|E〉f(E)〈E|. (72)
This works for any function, be it exp(itE) for the exponential, or for thecharacteristic function
χ(H) =∑∫
dE|E〉χ[a,b](E)〈E|. (73)
which is a projector onto a spectral subspace. We will even write thingslike
δ(H − E0) =∑∫
dE|E〉δ(E − E0)〈E| = |E0〉〈E0| (74)
to express a “projector” onto scattering eigenfunctions. (This will not bean operator in Hilbert space!). Computation rules work fine, the meaning,where we will be using this, can be made precise in the mathematical sense,but we will usually rely on the expression’s formal appeal.
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3.2 Spectral properties and dynamical behavior
In physics we like to think of bound states - that never seriously leave somefinite region, and unbound states, that are certain to leave any finite sur-rounding of the system. We also have got accustomed to associating thesetwo types of states with discrete and continuous spectrum, respectively.Not much of a justification is given for this usually. We will first establishthat fact rigorously. The main theorem expressing this fact is called RAGE(for Ruelle, Amrein, Georgescu, Enß).
3.2.1 Spectral subspaces
A general theorem of measure theory (Lebesgues decomposition theorem)implies that the spectrum can be separated into point, absolutly continousand singular spectrum
〈φ|dP (λ)ψ〉 = dµφ,ψ = dµp;φ,ψ + dµac;φ,ψ + dµsing;φ,ψ, (75)
each associated with a subspace of the Hilbert space. These subspaces aremutually orthogonal spaces
H = Hp ⊕Hac ⊕Hsing =: Hp ⊕Hc. (76)
We have lumped singular and ac into “continuous” Hc, as these two partbehave similar in many respects and actually, usual Schrodinger operatorshave empty singular spectrum. There we will assume that our operatorshave no singular spectrum (this is proven for large classes of Schrodingertype operators, especially for the short range potentials considered in scat-tering theory. We may also write
Hp = span(eigenvectors) (77)
andHc = H⊥p (78)
Functions ψ ∈ Hac behave like wave packets, i.e.
limt→±∞
〈e−itHψ|φ〉 = 0, (79)
we can say: after long time, the wave packet becomes orthogonal to anyfixed state. This is at completely generic property of the ac spectral sub-
27
space. We know it as wave-packed spreading in QM. It is related to theac spectrum of ∆.
For functions from the sing spectrum a similar decay only holds in thetime-averaged sense. This kind of spectrum is poorly suitable for scat-tering considerations, as we cannot exclude that the system returns to itsinitial state, even if only for a short time. Note that Hamiltonians withsing spectrum are not necessarily pathological cases: they may possiblybe realized in quasi-crystals (i.e. with quasi-periodic potentials). We willnot consider them for our discussion of scattering theory.
3.2.2 Fourier transform of smooth functions (Riemann-Lebesgue)
We had discussed the absolutely continuous measures as those that areequivalent to the Lebesgue measure in the sense dµac(x) = f(x)dx with anintegrable function f . That means that for the absolutely continuous partof the spectral measure we can take advantage of the following
Theorem 2. (Riemann-Lebesgue) If Lebesgue integral of |f | is finite, thenthe Fourier transform of f satisfies
f(λ) :=
∫Rf(x) exp(−iλx) dx→ 0 as |λ| → ∞.
We leave this important and useful theorem without proof. It makesprecise our intuition that a highly oscillatory integral (|z| → ∞) will even-tually average to zero, if the integrand is otherwise “smooth”, i.e. withoutnon-integrable singularities.
3.3 Classes of operators
We have met hermitian, self-adjoint, and unitary operators, we have dis-tinguished bound and undbound a operators by their norms. Operatorscan be classified by many other criteria. A few of them are particularlyimportant.
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3.3.1 (Orthogonal) projectors
3.3.2 Compact operators
3.3.3 Relatively bounded operators
In scattering theory it is important, how one operator relates to another,e.g. H0 to V , how “big” V is compared to H0. An well known exampleis the fact that reasonable Hamiltonians should have a lower bound, if apotential is added to kinetic energy. V being not too big compared toH0 could be expressed as ||V/H0|| < ∞. The problem, that H0 may besingular is circumvented by using the reoslvent instead.
Definition 1. Let V , H, D(V ) ⊂ D(H). If
||V (H − z)−1|| <∞, for some z ∈ ρ(H)
we call V relatively bounded w.r.t. H.
Under these conditions, perturbation of H + αV theory will be conver-gent. (The condition is NOT fulfilled for popular perturbers such as in theStark effect, for who the perturbations series does actually NOT converge).
3.3.4 Relatively compact operators
Analogously
Definition 2. Let V , H, D(V ) ⊂ D(H). If
V (H − z)−1 for some z ∈ ρ(H)
is compact we call V relatively compact w.r.t. H.
Relatively compact operators play an important role in scattering theory.Relative compactness expresses, in a mathematically precise manner, thatH+V differs from H significantly (more than any given ε) only on a finitesubspace. As a wavepacket spreads, it looses its overlap with any suchfinite subspace and its dynamics differs by arbitrarily little from a freewave packet.
3.3.5 The RAGE theorem
This theorem by named after Ruelle, Amrein, Georgescu, and Enß statesthat there is a strict relation between point spectrum and continuous
29
spectrum of an operator H on the one hand and the long-time evolutionexp(−itH) of functions from the corresponding subspaces Hp and Hc, aswe would expect it: in the long run, functions from Hp will remain withina “finite” region of the Hilbert space (altough that region may need tobe large), while functions from Hc will, in the long run, leave any “finite”region of the Hilbert space. There is a one-to-one relation between thespectral properties and the long-time behavior of the dynamics.
Relative compactness allows us to single out a “finite” subspace of inthe spectral range of an operator. The following theorem tells us, that nowave packet from the continuous spectrum can ever be contained in sucha finite subspace in the long run.
This theorem allows us to formulate the following theorem, that tells usthat time-evolution separates the (at least on average) the Hilbert spaceinto parts that leave any finite subspace and parts that will remain con-tained in a finite subspace.
Theorem 3. (RAGE). Let A be self-adjoint. Suppose there is a sequence ofrelatively compact operators Kn which converges strongly to identity. Then
Hc = ψ ∈ H| limn→∞
limT→∞
∫ T
0
||Kne−itAψ||dt = 0
Hp = ψ ∈ H| limn→∞
supt≥0||(1−Kn)e
−itAψ||dt = 0
This theorem tells us that we can characterize Hc and Hp completely bythe behavior of functions under the long-time limit of quantum mechanicalevolution: the first part says that the fact that wave packets composedof the continuous spectrum will be “diluted” out of any “finite” subspace.“Finite” means the space where Kn differs relevantly from A (relativelycompact). In physics terms: if a wave packet fully spreads (in the time-average), then it completely belongs to the continuous spectrum.
The second part tells us that if we can find a “finite” space (in the senseas above) where the wave-packet remains contained, the wave packet be-longs purely to the point spectrum: spectral properties and time-evolutionbehavior match one-to-one.
check for ac the time-average can be replaced by time-limits
30
3.3.6 Asymptotic completeness
We already discussed that Ω± will not in general be unitary as maps H →H. At finite times t the operator U(t)U0(−t) is unitary. We know thatbound states (any vector from the point-spectrum) cannot appear in therange of Ω± (those states, according to RAGE, would never get out of somefinite range).
Although the Ω± will not be unitary as maps on H, they can still bebijections. We would like them to be bijections between the space of all freestates, i.e. all wave packets formed in Pac(H0)H and all scattering states,which we define as wave packets from Pac(H)H. Note that for H0 = −∆:Pac(H0)H = H. If the Ω± are such bijections, we say that scattering isasymptotically complete.
The physical meaning of asymptotic completeness is that scatteringstates and localized states do not mix and that the dynamics of the un-bound (ac) states of the the two systems are equivalent: localized remainslocalized, scattering remains scattering. It just does not happen, that awave packet goes in and turns into a bound state or gets stuck in someother way. All that goes in comes out again. Neither does it happen thata localized state decides to decay. Of course, in physics such processes doseem to occur, they go by the name of “capture” or “decay” or so: lookingmore closely, in these processes either another constituent goes out (e.g.a photon is emitted) or some long-lived intermediate state is formed (aresonance), that will decay eventually.
As often, an example where the scattering operators exist, but are notasymptotically complete is interesting, (see Reed and Simon III, counterex-ample to Theorem XI.32, p.70).
3.3.7 Short range potentials
The discussion is restricted to H0 = −∆ and H = H0 + V . First re-quirement is that the potential V be “relatively bounded” to −∆, i.e.||V (−∆ + 1)−1|| < ∞. Note that, as σ(−∆) = R+ is postive, we can usethe a real offset instead of the complex z for the general self-adjoint case.
A relatively bounded perturbation V of the Laplacian−∆ is called short
31
range if ∫ ∞0
||V (−∆ + 1)−11R3\Br ||dr <∞ (80)
where Br := x ∈ R3, |x| < r. Relative boundedness guarantees that theintegrand is always finite. This is not as strong a constraint as it may seem:remember that also the Coulomb potential is relatively bounded to −∆.What matters is that the remote parts of the potential drop sufficiently fastcompared to the kinetic energy. That this is not the case for the Coulombpotential.
Problem 3.8: r−1−ε is short range Proof this.Hint: Fourier transform is. . .
Theorem 4. (Volker Enß) Suppose V is short range. Then the Mølleroperators exist and are asymptotically complete.
Examples Potentials behaving asymptotically as r−1+ε are short range forany ε > 0. This leaves the Coulomb potential as the only relevant violatorin situations where we want to talk about “scattering” at al.
Problem 3.9: Relative boundedness Show that the Coulumb potential isbounded relative to −∆. Show that the Coulomb potential is not shortrange.
Technically, the short range property is mostly exploited in the form ofthe compactness of
(H − z)−1 − (H0 − z)−1 and f(H)− f(H0) (81)
Lemma 1. If V is short range, then RH(z)−RH0(z) is compact.
For the proof of the lemma we would need a few results on compactnessthat we have skipped. However, it gives some insight to consider the ideasof the proof
• Denote χr(x) = χx||x|>r(x), the projector onto the remote parts ofspace
• So, 1− χr projects onto a finite range.
32
• We use the (1 − χr)RH0(z) is compact (without proof). In fact, the
product of any two operators g(x)f(p) for integrable functions g, f thatdeay to 0 at large x and p turns out to be compact (Teschl 7.21).
• Also, RH(z)V is bounded: that is plausible, as RH0(z) is bounded and
H = H0 + V , closely related to Kato-Rellich. (Proof in Teschl 6.23).
• Now the short range condition enters
limr→∞
RH(z)V (1− χr)RH0(z)→ RH(z)V RH0
(z)
• Note the identity
RH(z)V RH0(z) = (H0+V−z)−1V (H0−z)−1 = (H0+V−z)−1−(H0−z)−1
Actually, short range implies that for any function that decays at∞, wehave
Lemma 2. Suppose RH(z)−RH0(z) is compact. Then so is f(H)− f(H0)
for any f ∈ C∞(R) and
limr→∞||[f(H)− f(H0)]χr|| = 0. (82)
Proof. Again the proof involves technical results that were skipped (Teschl6.21). But once we know that f(H) − f(H0) is compact, we know thatχr excludes an arbitrarily large space, i.e. any χrφ goes strongly to zero.Multiplying by a compact operator does not change that. As the family ofmaps is bounded and goes to zeor strongly, it goes to zero uniformly, i.e.in the operator norm.
3.3.8 Scattering operators compare two dynamics
We see that the roles of H0 and H enter symmetrically: the scatteringoperators connect two unitarily equivalent dynamical systems. Scatteringtheory compares one known dynamics to another one, maybe less wellknown. From a mathematical point of view, it is irrelevant, which is which.Physically, of course, we know everything about, say, −∆ and often ratherlittle about −∆ + V .
But, for example, we know also a lot about the Coulomb problem (alleigenfunctions and generalized scattering eigenfunctions) and we could just
33
as well use the Coulomb Hamiltonian as our H0 = −∆ − 1/r and com-pare another dynamics, that only asymptotically behaves Coulomb-like.Of course, the ac “eigenfunctions” of −∆ are just plane waves, which aremuch easier to handle than the hypergeometric scattering functions of theCoulomb problem.
Unbound Coulomb dynamics itself is not unitarily equivalent to freemotion. We will need to construct a different dynamics to establish suchan equivalence (Dollard Hamiltonian).
3.3.9 The S-matrix
3.4 Problems
3.4.1 δ-Function of a self-adjoint operator
3.4.2 Abel formula
Problem 3.10: Abel limit
(a) Convince yourself of the Abel-limit:
f(t)→ f∞ ⇒ limε↓0
ε
∫ ∞0
e−tεf(t)dt = f∞
(b) Show the same for the strong limit:
V (t)s→ V∞ ⇒ s- lim
ε↓0ε
∫ ∞0
e−tεV (t)dt = V∞.
4 Formal derivations
In this section we do not even try to give rigorous results. Rather, we showa few manipulations that relate the rigorous results of the previous sectionto what physicists often do.
4.1 The Lippmann-Schwinger equation in disguise
The LS equation is of great practical importance in physics. We introducedit as an equation for the “scattering eigenfunctions” HΨ
(±)~k
= k2
2 Ψ(±)~k
. Un-fortunately, rigorous foundation of “scattering eigenfunctions” and deriva-tion of related formulae of practical importance requires the buildup of a
34
significant mathematical apparatus. Here we will introduce the “dirty”physicist’s formal manipulations that lead from the scattering operators tothe LS equation and further on to cross-sections.
Take an arbitrarily narrow wave free wave packet “ϕ~k ∼ ei~k·~r”, then
|Ψ(±)〉 := Ω±|ϕ~k〉 (83)
is indeed a seemingly much simpler equation than the LS equation to obtainscattering states Ψ(±).
The Ψ(−) is what we want as an “in” state: in the far past it looks like thefree wave packet Ψ~k. Conversely, the “out”-states Ψ(−) are like plane wavesin the remote future. Note that in physics literature, the sign conventionof Ψ(±) is often reverted, the signs relating to the ε-limits to be discussedbelow. We will adhere to the more physical concept of associating + withconditions at large positive times.
The similarity to the LS equation becomes even more evocative, whenwe will show that in the spectral representation of H0 we have
Ω± = 1− lime↓ε
∫ ∞0
(H − E ± ıε)−1V δ(H0 − E)dE (84)
We use the notation for the spectral measure dP (E) = δ(H0 − E)dE (wehave a pure ac spectrum and therefor no trouble with possible point spec-trum. Some signs need adjustment, but most importantly we have theresolvent of H, not H0. Unfortunately, this makes this formula difficult touse in computations, to put it mildly. (Time permitting, we may introducetechniques that allow the direct numerical evaluation of such expressions.)But it can be transformed to the LS equation.
4.2 Ω± in the spectral representation of H0
Ω± are time-independent operators operators obtained as time limits oftime-dependent operators. Here we will replace the limit in time by a limitof resolvents of H0. This is achieved by using a result by Abel:
Let f be a bounded function on (0,∞) and suppose limt→∞ f(t) = f∞.Then
limε↓0
ε
∫ ∞0
e−εtf(t)dt = limt→∞
f(t). (85)
35
In the same way we can evaluate the limits of eitHe−itH0, where we heav-ily rely on the intuition that we can treat operators just like numbers aslong as we respect their non-commuting nature (and take care of domainquestions).
In the spectral representation of H0
H0 =
∫ ∞0
EdP (E) (86)
we write
eitHe−itH0 = eitH∫ ∞
0
e−itEdP (E) =
∫ ∞0
eit(H−E)dP (E) (87)
The integrals exists, as the involved functions and operators are bounded.Now we use the Abel method and formally the time-integral
Ω± = limε↓0
ε
∫ ∞0
e−εt∫ ∞
0
e±it(H−E)dP (E)dt (88)
= limε↓0
ε
∫ ∞0
∫ ∞0
e±it(H−E±iε)dt dP (E) (89)
= limε↓0±iε
∫ ∞0
(H − E ± iε)−1dP (E) (90)
= 1− limε↓0
∫ ∞0
(H − E ± iε)−1V dP (E) (91)
The last step is by some algebra with the resolvent
(H − E ± iε)−1(±iε) = (H − E ± iε)−1[(H − E ± iε)− (H − E)]
= 1− (H − E ± iε)−1(H0 + V − E) (92)
and observing that
(H0 − E)dP (E) = (H0 − E)δ(H0 − E)dE = 0.
For the time integral we have treated H like an ordinary function: as it isselfadjoint, it has a spectral representation
H =
∫GdPH(G) (93)
for the projection valued measure PH belonging to H. Then∫ ∞0
e±it(H−E±iε)dt =
∫ ∫ ∞0
e±it(G−E±iε)dP (G)dt (94)
36
with G ∈ σ(H) ⊂ R. Integrals can be interchanged, the time integralcan be performed and the spectral representation can be replaced by theabstract operator again.
Thus we arrive at
Ψ(±) = Ω±φ = φ− limε↓0
∫ ∞0
(H − E ± iε)−1V dP (E)φ, (95)
which bears great similarity with the LS equation. Actually, it is seeminglybetter, as it is a direct expression for some scattering wave packet Ψ(±)
rather than an integral equation for it. However, it contains (H−E±iε)−1,which usually is rather inaccessible to computation.
Now we remember the symmetric role of H and H0 in scattering theoryand we interchange them
H,H0, H −H0 =: V → H ′ = H0, H′0 = H,H ′−H ′0 = H0−H = −V (96)
to write
φ(±) = Ψ + limε↓0
∫ ∞0
(H0 − E ± iε)−1V dPH(E)ψ, ψ ∈ Pac(H)H. (97)
Next remember that the signs ± just indicate whether the φ(±) and ψ aremapped by going through the the remote past or future. That is, we canjust as well move ± from φ back to ψ, leading to
Ψ(±) = φ− limε↓0
∫ ∞0
(H0 − E ± iε)−1V dPH(E)Ψ(±). (98)
Problem 4.11: Directly derive the Lippmann-Schwinger equation We hadused that on Hac one can interchange H0 ↔ H to obtain the Lippmann-Schwinger equation from the definition of the Møller operators. Alterna-tively one can form the inverses Ω−1
± to arrive at the same result. Do that.
Hint: Show by simple algebraic maninpulations
1− (H − z)−1V = [1 + (H0 − z)−1V ]−1
4.2.1 Remark on “ac-spectral eigenfunctions”
The equation above is almost the shape of the LS equation. However,it is formulated for wave-packets rather than, as physicists like to do it,
37
for “eigenfunctions of the continuous spectrum”. For −∆ these would beplane waves exp(i~k · ~x). Indeed, the path for making this notion precise, isthrough the Fourier transform and using the mapping by Ω±, formally
|~k〉(±)H = Ω±|~k〉H0
:= Ω±ei~k·~x. (99)
The mapping is not unique because of the two alternative signs, but it doesnot need to be, as the energies are degenerate and any linear combinationis a possible choice. The asymptotic condition of matching either in thefuture or in the past fixes the basis in this degenerate subspace. Time-reversal maps between the two signs. Accepting that, we can even givemeaning to the δ-function of the Hamiltonian as the spectral projectoronto the eigenfunctions of given energy∫ ∞
0
δ(H − E)dE :=
∫dαdk2k
2|k, α〉〈k, α| (100)
=
∫ ∞0
dE
∫dαE1/2
2|√E,α〉〈
√E,α| (101)
=
∫ ∞0
dE
∫dα|E,α〉〈E,α| (102)
where
|E,α〉 =E1/4
√2|√E,α〉. (103)
If we do that, formally, the δ(H −E) allows to perform the dE integra-tion and we have the Lippmann-Schwinger equation for the scatteringsolutions
Ψ(±)(E) = φ(E)− limε↓0
(H0 − E ± iε)−1V ψ(±)(E). (104)
where the + corresponds to matching of asymptotics of φ = ei~k~r in the
remote future and − to matching in the remote past.
4.3 Mapping free to free spaces: the S-matrix
With asymptotic completeness the Ω± are unitary bijections Pac(H)H ↔Pac(H0)H and the S-matrix maps Pac(H0)H → Pac(H0)H
S = (Ω+)−1Ω− = Ω∗+Ω− (105)
38
This corresponds to a limit
limt→∞
[U(−t)U0(t)]−1[U(t)U0(−t)] = lim
t→∞U0(−t)U(2t)U0(−t)] (106)
i.e. bring a free packet sufficiently far into the past, let it interact forsufficiently long time (into the future), propagate the free packet it backinto the present: generate the effect of scattering as a mapping betweenfree wave packets.
Some properties of the S-matrix:
• is unitary (asymptotic completeness)
• it “inherits” symmetries from H and H0.
• it conserves energy - intertwining relation
4.3.1 Spectral representation of the S-matrix
In the same spirit as for Ω±, there is a representation of the S-matrix inthe spectral representation of H0:
S = limε↓0
∫ ∞0
dE
1− 2πıδ(H0 − E)[V − V (H − E − ıε)−1V
]δ(H0 − E).
(107)We use the (only slightly) symbolic notation
δ(H0 − E) =∑∫
α
|E,α〉〈E,α|, (108)
where α labels the degeneracy in the energy subspace E.
Derivation of the S-matrix
S = s- limε↓0
ε
∫ ∞0
dt
∫ ∞0
∫ ∞0
dEdE ′δ(H0 − E)e−it(H−E+E′
2 −iε)δ(H0 − E ′)
= s- limε↓0
ε
∫ ∞0
dt
∫ ∞0
∫ ∞0
dEdE ′δ(H0 − E)−iε
H − E+E′
2 − iεδ(H0 − E ′)
Problem 4.12: Resolvent equation An important identity for H = H0 +V :
(H − z)−1 = (H0− z)−1− (H0− z)−1[V − V (H − z)−1V ](H0− z)−1 (109)
39
Verify this!The first term evaluates to
s- limε↓0
ε
∫ ∞0
dt
∫ ∞0
∫ ∞0
dEdE ′δ(H0−E)−iε
H0 − E − iεδ(H0−E ′) =
∫dEδ(H0−E)
(110)FurtherProblem 4.13: δ-function
limε↓0
−iε[(E − E ′)/2− iε][(E ′ − E)/2− iε]
= 2πiδ(E − E ′). (111)
By this, one can integrate over E ′ for the second term and one arrives atEq. (107).
4.3.2 The T -matrix
Let ψ, ψ be two wave packets. The matrix elements of the S-matrix canbe written as
〈ψ, Sψ〉 = 〈ψ, ψ〉 − 2iπ
∫d(3)k ψ(~k)∗t(~k,~k′)δ(~k2 − ~k′2)ψ(~k′) (112)
where ψ(~k) is the representation of ψ in ~k-space with the matrix elementsof the T-matrix
T (z) := V − V (H − z)−1V (113)
taken “on shell”, i.e. z = E + iε:
t(~k,~k′) = 〈~k|V − V (H − E − ıε)−1V |~k′〉, (114)
where |~k〉 = exp(i~k~x)/(2π)3/2 are the δ-normialized plane waves 〈~k|~k′〉 =δ(3)(~k − ~k′) and ψ(k) = 〈~k|ψ〉.t(~k,~k′) contains the relevant scattering information. Also, it is a smooth
function of ~k,~k′, loosely speaking, where solutions of the LS equation exist.It essentially is the same as the scattering amplitude f(k, ~n, ~n′)
f(k, ~n, ~n′) = −(2π)2t(k~n, k~n′) (115)
whose square gives the differential cross section:
σ(~k,~k′) = |f(k, ~n, ~n′)|2. (116)
40
Problem 4.14: δ-function The δ-“function” is fully defined by its property∫dxδ(x)g(x) = g(0) ∀g ∈ C∞0 , (117)
(Here, C∞0 designates the infinitely differentiable functions with compactsupport.) Remind yourself of the following properties of the δ-function andshow them using Eq. (117):
(a) Let fε,∫R fε(x)dx = 1 be a family of functions such that limε↓0
∫ ε−ε fε(x) =
1. Show fε → δ(x).
(b) limε↓0=m 1x−x0−iε = πδ(x− x0)
(c)
δ(ax) =δ(x)
|a|
δ(k′2
2− k′′2
2)δ(k′x − k′′x)δ(k′y − k′′y) = δ(3)(~k′ − ~k′′)/k′z
4.4 Scattering cross section
Typical scattering experiments can be described as aiming a macroscop-ically broad beam onto a microscopic target or a statistical ensemble ofmicroscopic targets. What is measured is at which angle one finds scat-tering products. In the scattering processes that we have discussed sofar, there is only a single particle and usually there is a time-independentHamiltonian. In that case energy is conserved and the only parameter thatneeds to be measured is the angle at which one finds a particle. The num-ber of particles found at a certain scattering angle, if one particle impactsper surface unit is the “scattering cross section”. It tells us, how manyparticles out of a broad beam are deflected into a given direction.
The scattering cross section is the main way how to relate experimentaldata to experiment. This is why we will go through its slightly cumbersomederivation from fundamental scattering theory. (The derivation can alsobe found in Thirring, chap 3.6).
On the microscopic level, we have all scattering information in the S ma-trix. To compute the scattering cross section, we need to model the beam
41
of particles with a macroscopic diameter. We assume that the particlesin the beam are independent of each other. This means in particular thattheir quantum phases are in no fixed relation to each other. The beam isconsidered to be transversally incoherent. In that case, scattering a beamfrom a microscopic target is equivalent to repeating a single scattering pro-cess and averaging over all impact parameters, i.e. the distance at whicha single wave packet passes the target. We do assume that all particles ofthe beam have the same energy, that is, the beam is “mono-energetic”: thebeam is time-coherent.
We will work in the spectral representation of H0. A single momentumwave packet with an approximate momentum ~k0 = (0, 0, k0) is describedby φ(~k) where
∫d(3)k|φ(~k)|2 = 1 and φ(~k) is non-zero only in the vicinity
of ~k0. We can shift this wave packet in ~x-space by a vector ~a = (ax, ay, 0)
perpendicular to ~k0 by the multiplication φ(~k)→ φ(~k)eı~a~k. Remember that
multiplication by eı~a~k in ~p-space is equivalent to applying the shift-operator
exp(ı~a~p) in ~x-space. The scattered shifted wave packet is
ψ~a = Sφ~a. (118)
The probability of finding in the scattered wave packet momenta ~k in acertain solid angle ∆Ω and within a range of magnitudes ∆k is∫
∆Ω
dΩ
∫∆k
dkk2|ψ~a(~k)|2 (119)
Here we use polar coordinates k, θ, φ :
~k = k
sin θ cosφsin θ sinφ
cos θ
(120)
and the notation for the “solid angle” ∆Ω around some Ω1 = (θ1, φ1)∫∆Ω
dΩ
∫∆k
dk =
∫ θ1+∆θ
θ1−∆θ
dθ sin θ
∫ φ1+∆φ
φ1−∆φ
dφ
∫ k1+∆k
k1−∆k
dk (121)
To model the beam, we average the probability by integrating ~a over somelarge beam cross section A and dividing by the surface area
1
A
∫A
d(2)a
∫∆Ω
dΩ
∫∆k
dkk2|ψ~a(~k)|2 (122)
42
If ~k0 is non-zero, the particle is certain to pass somewhere through the sur-face perpendicular to ~k0. The probability for a particle to pass one surfaceunit is therefore 1/A. We divide by 1/A, as we want to find scattering fora beam with one particle per unit surface. Finally, we assume that theangle ∆Ω is far enough from the beam direction (0, 0, 1) such that no un-scattered particles from the beam directly fall into it, i.e.
∫∆Ω |φ(~k)|2 = 0.
With this, 1 in the spectral representation of the S-matrix (107) does notgive a contribution. The cross section is
σ(~k,~k′) :=
∫A
d(2)a
∫∆Ω
dΩ
∫∆k
dkk2|ψ~a(~k)|2
=
∫A
d(2)a
∫∆Ω
dΩ
∫∆k
dkk2
∣∣∣∣∫ d(3)k′∫dE ′δ(
k2
2− E ′)t(~k,~k′)δ(k
′2
2− E ′)φ(~k′)eı~a
~k′∣∣∣∣2
where we have defined
t(~k,~k′) = 2π〈~k|V − V (H − k2/2− ıε)−1V |~k′〉. (123)
This becomes
=
∫A
d(2)a
∫∆Ω
dΩ
∫∆k
dkk2
∫d(3)k′
∫dE ′
∫d(3)k′′
∫dE ′′ (124)
δ(k2
2− E ′)δ(k
′2
2− E ′)δ(k
2
2− E ′′)δ(k
′′2
2− E ′′) (125)
t∗(~k,~k′)t(~k,~k′′)φ∗~a(~k′)φ~a(~k
′′)eı~a(~k′′−~k′) (126)
Integration over E ′ and E ′′ leads to
=
∫A
d(2)a
∫∆Ω
dΩ
∫∆k
dkk2
∫d(3)k′
∫d(3)k′′ (127)
δ(k2
2− k′2
2)δ(
k2
2− k′′2
2)t∗(~k,~k′)t(~k,~k′′)φ∗~a(
~k′)φ~a(~k′′)eı~a(~k′′−~k′) (128)
Integration over dkk2 = d(k2
2 )k gives
=
∫A
d(2)a
∫∆Ω
dΩk
∫d(3)k′
∫d(3)k′′ (129)
δ(k′2
2− k′′2
2)t∗(~k,~k′)t(~k,~k′′)φ∗~a(
~k′)φ~a(~k′′)eı~a(~k′′−~k′) (130)
43
For the integral over d(2)a we observe that∫A
d(2)aeı~a(~k′′−~k′) = (2π)2δ(k′x − k′′x)δ(k′y − k′′y) (131)
and obtain
= (2π)2
∫∆Ω
dΩk
∫d(3)k′
∫d(3)k′′ (132)
δ(k′2
2− k′2
2)δ(k′x − k′′x)δ(k′y − k′′y)t∗(~k,~k′)t(~k,~k′′)φ∗~a(~k′)φ~a(~k′′)(133)
The three 1-dimensional deltas combine to
δ(k′2
2− k′2
2)δ(k′x − k′′x)δ(k′y − k′′y) = δ(3)(~k′ − ~k′′)/k′z (134)
and another integral goes away
= (2π)2
∫∆Ω
dΩ
∫d(3)k′|t(~k,~k′)|2|φ~a(~k′)|2
k′
k′z(135)
Now we use that φ(~k) is very narrowly concentrated around ~k0 such thatk0 ≈ k′ ≈ k′z and t(~k,~k′) ≈ t(~k,~k0) for small range where the integral isnon-zero. Also we use ||φ|| = 1
= (2π)2
∫∆Ω
dΩ|t(~k,~k0)|2 (136)
To connect to conventional notation, we introduce theScattering amplitude
f(~k,~k0) := −(2π)t(~k,~k0) = −(2π)2〈~k|V −V (H − k2/2− ıε)−1V |~k0〉 (137)
and write the scattering cross section
dσ(~k,~k0)
dΩ= |f(~k,~k0)|2 (138)
If our H0 is the free time-evolution with the δ-normalized scattering func-tions |~k〉 = eı
~k~x/(2π)3/2, the scattering amplitude is expressed in terms ofthe plane wave matrix elements
f(~k,~k0) = −(2π)−1〈eı~k~x|V − V (H − k2/2− ıε)−1V |eı~k0~x〉 (139)
44
Post- and prior forms: Using the Møller operator equation in the form
Ψ(±)(~k) = Ω±|~k〉 =[1− (H − ~k2/2± ıε)−1V
]|~k〉 (140)
the scattering amplitude can also be written as
f(~k,~k0) = −(2π)2〈~k|V |Ψ(−)(~k0)〉 “post” form (141)
= −(2π)2〈Ψ(+)(~k)|V |~k0〉 “prior” form (142)
Here the standard bra-ket notation can be deceptive: the Møller operatorsare not self-adjoint, so it is intended that they are applied to the right, i.e.
〈Ψ(+)(~k)|V |~k0〉 = 〈Ω+~k|V |~k0〉 = 〈~k|Ω†+V |~k0〉
= 〈~k|[1− (H − k2/2 + iε)V ]†V |~k0〉= 〈~k|[1− V (H − k2/2− iε)]V |~k0〉
The two forms above are mathematically equivalent and, in case of po-tential scattering, also completely equivalent in practical use. Differencesappear when one approximates the scattering states Ψ(±). In the case ofmulti channel scattering one scattering solution, e.g. Ψ(+), may be easierto approximate that the other. Consequently, in approximate form, postor prior may be a better choice for approximating f(~k,~k′).
The total cross section is the integral of the differential cross sectionover all scattering angles. As |~k| = |~k0| we can write in polar coordinatesf(~k,~k0) = f(k,Ω,Ω0) and integrate over the scattering angle Ω
σtot =
∫dΩ|f(k,Ω,Ω0)|2 (143)
4.4.1 The LS equation in position space
The important point of the LS-equation is that the Green’s function of freemotion (H0 − E ± iε)−1 can be evaluated explicitly (in stark constrast tothe full Green’s function). In position space, the LS equation takes theform
Ψ(±)(~r) = ei~k~r −
∫R3
dr(3) e∓ik|~r−~r′|
4π|r − r′|V (~r′)Ψ(±)(~r′). (144)
where we recognize the form of the solution Ψ(±)(~r) as a plane wave plusa superposition of spherical waves.
45
Problem 4.15: Green’s function in position space Compute the function
G±(~r, ~r′) =1
(2π)3
∫d(3)k
ei~k(~r−~r′)
k20 − k2 ± iε
= − 1
4π
e±ik0|~r−~r′|
|~r − ~r′|(We use the physicists’ convention not to explicitly write the limε).Hint: Use polar coordinates, expansion of a plane wave into sphericalBessel functions, integrate over the angle, use the residue theorem for whatremains.
4.5 Asymptotic behavior
Asymptotically, i.e. for very large |~r| we have
|~r − ~r′| =√~r2 − 2~r~r′ + ~r′2 ∼ |~r| − r~r′ (145)
and thereforeeık|~r−~r
′|
|~r − ~r′|∼ eık|~r|
|~r|e−ıkr~r
′(146)
When applied to |~k〉 this gives
〈~r|[(H0 −
k2
2± ıε)−1
]|~k′〉 ∼ 1
2π
e∓ık|~r|
|~r|
∫d(3)r′e−ıkr~r
′eı~k′~r′/(2π)3/2(147)
∼ 1
2π
e∓ık|~r|
|~r|(2π)3/2δ(3)(~k − ~k′) (148)
where ~k = kr.
Note: Our approximate representation of the integrand only holds for|~r| |~r′|. therefore, we will not get an exact δ-function for any finite ~r.However, making |~r| large enough, we can get as close as we wish to theδ-function. That is what we mean by “asymptotic”: go as far with ~r asneeded to approximate a limiting situation within a preset accuracy.
Inserting∫d(3)k|~k〉〈~k| into the Lippmann-Schwinger equation we obtain
the asymptotic behavior of the scattering solution in the direction ~r ‖ ~k
Ψ(±)~k
(~r) ∼ 1
(2π)3/2
[eı~k~r +
e∓ık|~r|
|~r|f(~k′, ~k)
], ~k′ = kr (149)
46
4.6 Partial wave scattering solutions
4.6.1 Partial wave expansion
Energy conservation follows from the intertwining property of the Mølleroperators
SH0 = Ω−1+ Ω−H0 = Ω−1
+ HΩ− = H0Ω−1+ Ω− = H0S.
It implies that S is a function of H0, i.e. in the spectral representation is amultiplication by a function of E. It may still connect different α, though(after all, we are free to choose the basis in the space of fixed E spannedby the |E,α〉.
In the rotationally symmetric case, S =⊕
lm Sl we know that l,m is notchanged and therefore can be parameterized by the “scattering phases”δl(k), conventionally written with k =
√E:
S|E, l,m〉 = e2iδl(k)|E, l,m〉 (150)
Note that for rotationally symmetric problems there cannot be any depen-dence on m.
The label α can be discrete or continuous. Think of polar angles todenote directions, or, alternatively, the corresponding resolution of therotations in terms of spherical harmonics:
|α〉〈α| = |φ, θ〉〈φ, θ| = P (φ, θ) =∑lm
Y ml (φ, θ)Y m∗
l (φ′, θ′) = δ(θ−θ′)δ(φ−φ′)
(151)
4.6.2 Spherical Bessel functions
In polar coordinates one can better account for the rotational symmetryand expand plane waves in R3 can be expanded into spherical Besselfunctions jl(kr) by the formula
ei~k~r = 4π
∞∑l=0
iljl(kr)l∑
m=−l
Y m∗l (k)Y m
l (r). (152)
The sum over the m’s can be performed leading to
ei~k~r =
∞∑l=0
iljl(kr)(2l + 1)Pl(cosθ). (153)
47
Here we denote r = |~r|, k = |~k|, k = ~k/k, r = ~r/k and cos θ = k · r.The jl obey the radial equation for free motion:[
−∂2r +
l(l + 1)
r2
]jl(kr) = k2jl(kr). (154)
For the first few l we have
j0(x) =sin(x)
x(155)
j1(x) =sin(x)
x2− cos(x)
x(156)
j2(x) =
(3
x2− 1
)sin(x)
x− 3 cos(x)
x2(157)
. . . (158)
They are closely related to the trigonometric functions and, asymptoticallyat large x, behave as sin(x − lπ/2)/x, i.e. they alternate between sin-likeand cos-like asymptotic phase.
The jl are the regular solutions of this equation that behave as ∼ rl asr → 0, which reflects the “angular momentum barrier”. There is a secondtype of solutions of Eq. (154) that diverges as r−l as r → 0. The first fewof these Neumann functions nl(rk) are
n0(x) =cos(x)
x(159)
n1(x) =cos(x)
x2+
sin(x)
x(160)
n2(x) =
(3
x2− 1
)cos(x)
x+
3 sin(x)
x2(161)
. . . (162)
The Neumann functions resemble the jl, but with an additional phase-shiftby π/2, which causes the singularity at r = 0.
The particular linear combinations
h(±)l (x) = nl(x)± ijl(x) (163)
are called Hankel functions and are convenient for the traditional wave-mechanical investigation of the analyticity of the S-matrix and scattering
48
amplitudes. Using these formula, we transform from expressions in k andk′ to expressions in l, where the δl(k) appear as the parameters.
Problem 4.16: Show that the spherical Bessel-functions are δ-normalizedas ∫
drr2jl(kr)jl(k′r) =
π
2k2δ(k − k′)
Hint: Use 〈ei~k~r|ei~k′~r〉 = (2π)3δ(3)(~k−~k′) and compute∫dkY m
l (k)〈ei~k~r|ei~k′~r〉using the plane wave expansion into spherical Bessel functions.
4.7 Scattering phases and scattering amplitude
The scattering amplitude in terms of the scattering phases is
f(~k,~k′) =∑l
2l + 1
kPl(cos γ)eıδl(k) sin δl(k) (164)
where cos γ = k′ · k is the cosine of the angle between ~k and ~k′.
Derivation First we express the scattering amplitude by the S-matrix
S − 1 =− 2ıπ
∫dE
∫d(3)k
∫d(3)k′ (165)
|~k〉δ(k2
2− E)〈~k|V − V (H − E − ıε)−1V |~k′〉δ(k
′2
2− E)〈~k′| (166)
and
〈~k|S − 1|~k′〉 = −2ıπ〈~k|V − V (H − E − ıε)−1V |~k′〉δ(k′2
2− k2
2) (167)
and therefore
δ(k − k′)f(~k,~k′) = −2πık〈~k|S − 1|~k′〉. (168)
Deriving the final expression for the scattering amplitude is left to therereader in the following problem.Problem 4.17: Starting from the basic expression for the scatteringamplitude
δ(k − k′)f(~k,~k′) = −2πık〈~k|S − 1|~k′〉 (169)
49
derive the scattering amplitude expressed through the scattering phases
f(k, k · k′) =∑l
2l + 1
kPl(k · k′)eıδl(k) sin δl(k)δ(k − k′) (170)
Hint:
1. Substitute the plane wave expansion into spherical harmonics into theequation.
2. Use S|k, l,m〉 = e2ıδl(k)|k, l,m〉
3. Use the δ-normalization of the spherical Bessel functions
4. Use 4π∑
m Y∗lm(k)Ylm(k′) = (2l + 1)Pl(cos γ)
4.7.1 Scattering phases and asymptotic behavior
It is plausible, that at large r, any scattering solution of a sperically sym-metric problem for given l behaves as a linear combination of the regularand irregular spherical Bessel and Neumann functions:
ψ(±)l (r) ∼ aljl(kr) + blnl(kr) = cl sin(kr + δl(k))/(kr) (171)
for some constant cl. It is easy to show that the phase-shift in the asymp-totic behavior is exactly the scattering phase introduced earlier:Problem 4.18: Scattering phase and asymptotics Insert the spherical waveexpansion of the scattering amplitude into
Ψ(∓)(~r) ∼ 1
(2π)3/2
[eı~k~r +
e±ık|~r|
|~r|f(~k′, ~k)
], ~k′ = kr (172)
and the expansion of the plane wave in spherical Bessel functions to findfor l’th partial wave asymptotic behavior
〈Pl|Ψ(−)〉 ∼ sin(kr + δl − lπ/2)/kr for large kr (173)
where 〈Pl|Ψ(∓)〉 means integration over cos γ only.Hint: Use the orthogonality of the Legendre polynomials Pl(cos γ) underintegration over γ and the asymptotic form of the spherical Bessel functionsjl(kr) ∼ sin(kr − lπ/2)/kr.
50
4.8 Analyticity
Analyticity the T -matrix is an important property. The starting point israther simple, namely the fact that the resolvent RH(z) is analytic (forsuitable potentials) in the vicinity of any point z0 where the RH(z) exists.Consider the resolvent around some z0 ∈ ρ(H)
(H − z)−1 = (H − z0)−1[1 + (z − z0)(H − z)−1] (174)
= (H − z0)−1
∞∑n=0
[(z − z0)(H − z0)−1]n. (175)
Where it converges, the series is analytic in z. From the spectral repre-sentation of H it is easy to see that ||(H − z0)
−1|| ≤ |=mz0|. Therefore,the series is norm-convergent for all |z − z0| < |=mz0| and, as boundedoperators form a Banach space, defines a family of operators that dependsanalytically on z.
4.9 Analytic functions
The fact that an analytic function has — at least locally — a convergentseries expansion, has far reaching consequences. A brief reminder:
• A function is analytic in a region M , if and only if it is differentiable(as a complex function) at every point.
• Integration along a closed (simply connected) path gives 0 (Cauchytheorem)
• Integration by residue theorem, in case of poles.
• An analytic function is infinitely many times differentiable.
• Any two analytic functions on two distinct, but overlapping regions.If they coincide on a line segment in the overlapping region, they areidentical on the analyticity region that they share. Either is the uniqueanalytic continuation of the other.
• Caution: complex conjugation is not an analytic map, f(z) analyticdoes not imply that f(z)∗ is analytic, but g(z) = [f(z∗)]∗ is analytic.
51
• A simple consquence: if f(z) is anlyitc on a region R that includes asegment of the real axis, then f(z∗)∗ = f(z) on R ∩ R∗ and the twofunctions are the unique analytic continuation of each other
Analytic functions are rather robust under summation and integration:
f(z) =∞∑n=0
f (n)(z) (176)
is analytic, if each f (n) is analytic and convergence is uniform.Let g(z, r) be a function that is analytic w.r.t. z in some region R and
continuous w.r.t. r on r ∈ [a, b]. Then
f(z) =
∫ b
a
drg(z, r) (177)
is analytic. If we can establish uniform convergence on [a, b], the functionmay be singular at the end points and the integral can extend to ∞.
4.10 Analyticity of S(k)
Analyticity is better discussed for the modulus w.r.t. momentum k :=√2E, rather than energy E. By the fact that Ω+ 6= Ω−, we know that
the analytic Ω(E) is discontinuous at E ∈ R+, i.e. the positiv half-axisis a “branch cut” of Ω(E). When considered as a function of k instead,the branch cut is unfolded into the whole real axis: the point where thediscontinuity in E had occured are separated into the positive and negativesides of the real axis. In that way the , the limε↓0, converts to a limit fromthe upper complex k-plane towards the real axis, the ±-signs correspondingto limits to the positive and negative part of the real axis, respectively.Spectral values E < 0 are mapped into k = i
√(2|E|), i.e. points on the
positive imaginary k-axis. Except for these, we can hope for analyticity inthe whole upper half plane.
Rigorously, analyticity in the upper half plane can be proven for a mildlyrestricted class of potentials: one needs that the operator V 1/2(H0−k2/2−iε)−1|V 1/2| is a “Hilbert-Schmidt” operator. That condition is somewhatstronger than “compactness” of V (H0 − k2/2− iε)−1. For negative V , thesquare root are to be understood as V (r)1/2 := sign(V (r))|V (r)|1/2.
52
The fact that the discontinuity at the branch cut disappears opens thechance to analytically continue Ω(k) into the lower half of the complexk-plane. In terms of energy, we slip onto the “second Rieman sheet”.However, the possibility to actually analytically continue has only beenproven for a rather restricted class of short range potentials that fall offas the Yukawa potential. The fact that scattering theory exists (i.e. theMøller operators exist and are asymptotically complete) does by no meansguarantee that analytic continuation is possible.
If we can continue it into the lower half plane, poles in Ω(k) (and withit poles in T ) can (and often will) occur. These are on the second Riemansheet of energy: if we were able to rotate the branch cut, we might uncoverthose energy-poles. Indeed, there is such a procedure, named “complexscaling” of the Hamiltonian.
We can now write the Møller operators as
Ω± =
∫dφdη
∫dkkΩ(iε± k)δ(H − k2/2). (178)
with, for complex k,
Ω(k) = 1− (H − k2/2)−1V = [1 + (H0 − k2/2)−1V ]−1 (179)
Similarly we write representation of the T -operator w.r.t. k as
t(k) = V Ω−(k) = V − V (H − k2/2)−1V. (180)
This is an analytic function of k in the upper complex plane.
Problem 4.19: S how the above form of the S-matrix by formal algebraicmanipulations of t(k)
(a) (assume, for simplicity, that V > 0)
t(k) = [V −1 + (H0 + k2)−1]−1 = t†(−k∗)
(b)t−1(k)− t−1(−k) = 2πiδ(H0 − k2/2)
(c)
53
With that the spectral representation of the S-matrix can be writtenas
S =
∫ ∞0
dkkS(k)δ(H − k2/2) (181)
S(k) = t−1(−k)t(k) (182)
= s- limε↓0
[1 + (H0 −k2
2+ iε)−1V ][1 + (H0 −
k2
2− iε)−1V ]−1 (183)
4.10.1 Rapidly decaying potentials
Analyticity properties have been proven for potentials that fall of rapidly.A typical example is the Yukawa poential
V (r) = V0e−µr
r, µ > 0. (184)
A slightly more technical (and more general) criterion is that the integral
||V 1/2(H0 − k2)−1V 1/2||22 =
∫d(3)rd(3)r′
V (~r)V (~r′)
(4π|~r − ~r′|)2ei(k−k
∗)|r−r′| (185)
remains finite for also for negative imaginary parts of k, i.e. when theexponential in the integrand would be growing.
4.10.2 Poles of S
In traditional representations of scattering theory, the analyitic structureis usually discussed in terms of the “Jost functions”. This is equivalent tostudying S(k) in the form (183). To connect to the traditional discussion(not given here), one defines yet another quantity
D(k) = 1 + V 1/2(H0 − k2/2)−1|V |1/2 = V 1/2t−1(k)|V |1/2 (186)
(we mean V 1/2 := sign(V )|V |1/2) and re-write (183)
S(k) = V −1/2D(−k)−1D−1(k)|V |1/2 (187)
Here we assume that the potential is such that the analytic continuationcan be done into the lower complex half k-plane down to some =mk ≥ κ0 <0. Poles in S(k) will appear, where D(k) has a zero or D(−k) has a pole.
54
The rapidly decaying condition implies that V 1/2(H0 − k2/2)−1|V |1/2 hasat most a finite number of eigenvalues = −1 (for given k?). In the upperhalf plane there are no poles of D(k) (by the analyticity of the resolvent).There are only zeros at strictly imaginary k for the bound state energiesEn = k2/2, En < 0.
As D(k) and D(k∗) are related by time-reversal (i → −i) and D(k) =D†(k∗) (as shown for the t(k)), D(k) and D(−k) assume there poles at thesame k (when and operator is zero, obviously also its adjoint is zero, samefor divergence). This ensures that S(k) remains non-zero and non-singular,where it is analytic.
The zeros of D(k) and D(−k) correspond to
0 = D(k)ψ = ψ + V 1/2(H0 − E)−1V 1/2ψ (188)
Writing φ = (H0−E)−1V 1/2ψ we are lead back to the Schrodinger equation(H0 +V −E)φ = 0. For the kind of short range potentials that we considerhere, the bound states of this appear only at negative E. We have notdeveloped the instruments to show this here, it is related to a theorem thatrelatively compact potentials generate at most a finite number of boundstates at negative energies. In terns of classical physics, such potentialsgenerate an only finite phase spece volume below E = 0.
A priory, D(k) is not defined in the lower half plane, but it may beanalytically continued to there. Upon analytical continuation we may en-counter zeros of D(k) in the lower half complex plane at <(k) 6= 0. Thesewill cause structures in the scattering amplitudes that correspond to reso-nances. Finally, upon analytic continuation, also poles ofD(k) may appear.These are not usually associated with physical meaning.
4.11 Levinson theorem
The fact that the poles in the upper half plane all correspond to boundstates indicates that we should be able to count the bound states by inte-grating arround them in with an infinite half-circle in the upper half plane.For the integral at the infinite half-circle not to contribute we need, asalways, some technical constraints that ensure that the integrand dropsfaster than |z|−1 for large |z|.
55
Theorem 5. (Levinson) Let Nb the number of bound states then
2πNb = i limk→∞
det(S(k)) = i limk→∞
Tr log(S(k)) (189)
under the condition that V is relative compact to H0 and Tr|(H0 − z)−1 −(H−z)−1| < M(z) where M(z) < O(|z|−1−ε) for |z| → ∞ and < O(|=mz|−1+ε)and logS(0) = 0 (requires that there is no bound state of H at 0.)
4.11.1 Ingredients of the proof
Remember the Riesz projectors:
1
2πi
∫Cb
dz(H − z)−1 =∑Eb
∑α
|Eb, α〉〈Eb, α|, (190)
where Eb are the bound states inclosed by the integration path Cb. Thetrace of that is the number of bound states, counting also possible multi-plicities (here labelled by α).
The condition on the trace of the difference is not very strong and astandard part of the proof asymptotic completeness. Here it is used forensuring that the following formal manipulations correct. We write
Q± = 1 + (H0 − E ± iε)−1V, S(E) = Q+(E)Q−1− (E). (191)
We may then calclate inside the Tr just as with ordinary functions (payingattention to not permuting operators) VERIFY
Trd
dElogS(E) = TrQ−Q
−1+
[Q′+Q
−1− −Q+Q
−1− Q
′−Q−1−]
= Tr(Q−1+ Q′+ −Q−1
− Q′−)
= Tr[1 + (H0 − E ± iε)−1V
]−1(H0 − E + iε)−2V − (ε↔ −ε)
= Tr[(H0 − E + iε)−1 − (H − E − iε)−1 − (ε↔ −ε)
]Now we can integrate over E, which we can write as an integral aroundthe real axis:
Tr logS(E) = Tr
∫C
dz[(H − z)−1 − (H0 − z)−1
](192)
(see path). By assumption about M(z), closing the large circle does notgive a contribution, and neither does (H0 − z)−1 as σ(H0) is completelyoutside the path. Refer to the Riesz projectors for completing the proof.
56
4.11.2 Spherically symmetric case
The somewhat scary “trace of a log of an operator” becomes rather trans-parent in a the case of rotational symmetry: for given partial wave, Sl(E)is a function of E and acts as multiplication in a 1-d space:
Sl(E) = exp[2iδl(E)], Tr[logSl(E)] = 2iδl(E) : (193)
the integral of the phase-shift over energies from 0 to ∞ tells us about thenumber of the system’s bound states!.
In the more general case, the meaning of Tr logS is readly seen in thespectral representation.
4.12 The delay operator
One can illustrate a typical resonances as a quasi-bound state, that trapsa scattering particle momentarily and releases it later. Apart form generalhandwaving arguments about associating a line-witdh Γ with a life-timeτ = 2π/Γ, so far no relation between peaks of the scattering cross sectionand any time has been established. To do that, one first defines a “delayoperator” and then shows how it relates to cross-sections, in the simplestcase through phase-shifts δl(E).
Let χR the characerisitic function of a sphere with radius R. Then∫ ∞−∞
dt〈Utψ|χR|Ut(ψ)〉 = 〈ψ|∫ ∞−∞
dtτt(χR)|ψ〉 (194)
is the time that a given wave-packet ψ spends in the sphere. Note that,if ψ has any bound state content, the integral will diverge. Here we havedefined the time-evolution of an operator A (Heisenberg picture) as τt(A) =exp(itH)A exp(−itH), similarly τ 0
t (A) for H0.A delay can now be defined as the difference in time spent in a sphere un-
der free motion τ0 and scattering motion τ . In order to compare equivalentwave packets, we choose the free wave packet as φ and the scattering wavepacket as ψ = Ω−φ, i.e., in the far past they agree. By the intertwiningproperty of the Møller operators we find
〈ψ|τt(χR)|ψ〉 = 〈φ|Ω†−τt(χR)Ω−|φ〉 = 〈φ|τ 0t (Ω†−χRΩ−)|φ〉 (195)
57
and we can compare, for any φ ∈ H the delays at given R:
〈φ|DR|φ〉 = 〈φ|∫ ∞−∞
dtτ 0t (Ω†−χRΩ− − χR)|φ〉. (196)
In order to eliminate the arbitrary choice of R we take R→∞. For seeingthat this makes sense, consider the case that H0 and H agree exactly onR > R0: then the limit is reached already at R0. If this is not the case(an still the limit exists, alas not for Coulomb!) one captures any minisculdelay caused by some increasingly tiny potential and one obtains the
Delay operator
D := w − limR→∞
DR = −iS−1
∫ ∞0
dEδ(H0 − E)dS(E)
dE(197)
The second equation needs to be shown. Before that, we remark that againthere appears something like the derivative of the log of S, which we canidentify with the phase-shift δl(E) in the simple case of spherical symmetry.
4.12.1 Derivation of the delay operator
As usual here, we give a rather formal derivation, which however shouldshow what happens.
• We first reexpress DR using S. For that one uses the fact that, theoperator from infinitely large negative times to 0∫ 0
−∞dtτ 0
t (Ω†−χRΩ− − χR)w→ 0 (198)
Bypassing all important questions about whether we can interchangethe limR→∞ with integration, this becomes a triviality. limR→∞ χR = 1and Ω±† Ω± = 1.
• For the integration [0,∞) we use Ω− = Ω+Ω†+Ω− = Ω+S and the factthat S is invariant unter free time-evolution τ 0
t (S) = S. With that one
58
obtains∫ ∞0
dtτ 0t (Ω†−χRΩ− − χR) =
∫ ∞0
dtτ 0t (S−1χRS − χR)
+S−1
∫ ∞0
dtτ 0t (Ω†+χRΩ+ − χR)︸ ︷︷ ︸
w→0
• For the further steps, one switches back into a time-dependent picturefor the S:
S =
∫dt′S(t′)eit
′H0 (199)
where [S(t), H0] = 0
• One writes DR = S−1∫∞
0
∫∞−∞ S(t′)eit
′H which after some manipulationreduces to
S−1
∫ ∞−∞
∫ ∞−∞
dt′S(t′)eit′H0 = −iS−1
∫ ∞0
δ(H0 − E)dS(E)
dE. (200)
4.12.2 Resonances
If there is a resonance in the sense that D(k) has a zero at k = kr − ib inthe lower half plane, near the real axis, we can write
S(k) = V −1/2D(−k)−1D−1(k)|V |1/2 =(−k − kr + ib)(−k + kr + ib)
(k − kr + ib)(k + kr + ib)×g(k),
(201)where g(k) is some function that varies slowly near kr. Then
−i d
dk2/2log(S(k)) ≈ b
k
[1
(k − k)2r + b2
+1
(k + k)2r + b2
](202)
For small b << kr, this has a δ-like peak which makes a large, the dominantcontribution to the delay, if we choose our wave-packet φ to be concentratednear Er = k2
r/2.
59
4.13 Properties of the scattering amplitude
“On shell”, i.e. for real k, we have
f(k; k, k′)− f ∗(k; k′, k) =ik
2π
∫dk′′f(k, k, k′′)f(k, k′, k′′) (203)
f(k, k′, k) = f(k,−k,−k′) for time-reversible systems (204)
f(k, k′, k) = f(k,−k′,−k) for reflection invariant systems (205)
The first identity results from the Low equation for the t(k):
t(−k)− t(k) = 2πit(k)δ(H0 − k2/2)t(−k). (206)
which can be (formally) obtained from
t(k)[t−1(k)− t−1(−k)]t(−k) = t(k)[2πiδ(H0 − k2/2)]t(−k).
4.13.1 Optical theorem
For k = k′ the rhs. integral in (203) is essentially the total cross section
σt(k) =4π
k=mf(k, k, k) : (207)
the imaginary part of the forward scattering amplitude reflects the totalcross section. (Sounds plausible, if we associate with the imaginary partamplitudes that actually do not continue in that direction). Note thatthis holds for general potential scattering, not just spherically symmetricproblems.
4.13.2 Reciprocity
Time reversal of the original problem guarantees that we can go back theway we came. Of course, we must interchange the roles of incoming andoutgoing angles k ↔ k′ and revert the directions.
4.13.3 Detailed balance
In many simple statistical arguments one uses an asumption about “de-tailed balance”, the fact that the cross section for going from k → k′ equalsthe cross section for the reverse process:
σ(k, k, k′) = σ(k, k′, k). (208)
60
Note that this is more than reciprocity (time-reversal), as it also requiresreflection symmetry. This becomes quite obvious if one considers scatteringfrom targets that are not reflection symmetric.
4.13.4 Unitarity limit
In spherical symmetry we get from (164)
σt =∑
σl, σl(k) =4π
k2(2l + 1) sin2 δl(k). (209)
We see that the partial wave cross sections are bounded by the “unitaritylimit”
σl ≤4π
k2(2l + 1). (210)
The name refers to the fact that the general form of the partial waveamplitude is an immediate consequence of the unitarity of S.
Assume purely resonant scattering, i.e.
Sl(k) = e2iδl(k) =k − kr + ic
k − kr − ic(211)
then (<(eiδl)2 = 1− 2 sin2 δl)
σl =4π
k2(2l + 1)
c2
(k − kr)2 + c2. (212)
It is interesting to note that that limit is just 4 times the surface of anannulus for the classical impact parameters kbl = l~ and bl+1 as its lowerand upper radii
4π
k2= 4π(b2
l+1 − b2l ) (213)
4.13.5 Scattering length
The unitarity limit does not impose any constraint on the behavior fork → 0. Yet, for most standard potentials one can show that δl(k) ∼ k2l+1
for k → 0.This can be thought of as a result of the “centrifugal barrier” and can
be made plausible by looking at the first Born approximation of the lth
61
partial wave of a potential with finite range V (r) = 0 for r > R:
f(1)l ∝
∫ R
0
drr2j2l (kr)V (r)
∫dΩ|Y m
l (Ω)|2 (214)
=
∫ R
0
drr2j2l (kr)V (r) (215)
≈ (4π)2
(2l + 1)!
∫ R
0
drr2(kr)2lV (r) (216)
∝ k2l
∫ R
0
drr2l+2V (r). (217)
Here we have used the behaviour ∼ (kr)l of the jl(kr) for small kr: Thepartial wave contributions go to zero like k2l. The s-wave scattering am-plituede (l = 0) approaches a constant, looking at (164) we see that indeedδ0(k) ∼ k at small k.
We define the scattering length
a = − limk→0
δ0(k)/k = −f(0, k, k). (218)
The sign is chosen such that the positive sign is for repulsive potentials,again referring to the first Born approximation (which, btw. is totally un-justified, although not necessarily totally wrong for low energy scattering):
sign(a) = −sign(f(k = 0)) ≈ −sign(f (1)(k = 0)) = sign(〈~k|V |~k〉) (219)
Low energy scattering is usually only “s-wave scattering”, i.e. scatteringwith l = 0, in traditional terminology the s-wave:
σtot(k → 0) ≈ σ0(k → 0) = 4πa2 (220)
4.14 Computational techniques
4.14.1 The Born series
The Lippmann-Schwinger equation (98) is an integral equation that willalso be difficult to solve directly. However, one can iterate it (signs are
62
easier, when we write −(H0 − E ± iε)−1 = (E −H0 ∓ iε)−1
|Ψ(∓)〉 = |~k〉 (221)
+(k2/2−H0 ± ıε)−1V |~k〉 (222)
+(k2/2−H0 ± ıε)−1V (k2/2−H0 ± ıε)−1V |~k〉 (223)
+(k2/2−H0 ± ıε)−1V (k2/2−H0 ± ıε)−1V (k2/2−H0 ± ıε)−1V |~k〉(224)
etc. . . (225)
If V is “small” in some sense compared to k2/2−H0± ıε one can terminatethe series at finite iterations. Unfortunately, E = k2/2 is in the continuousspectrum of H0, i.e. there are vectors on which E −H0 is arbitrarily smalland (E − H0 ± ıε) would become very large ∼ 1/ε. We are only safe ifthese vectors do not appear in V |~k〉. This is case-dependent. Convergenceor useful approximation properties of the Born series are not guaranteedand, in fact, very often not given in practice.
In spite of these doubts, we can write the scattering amplitude (say, postform)
− 1
2π2f(k, k, k′) = 〈~k|V |Ψ(−)
~k0〉 (226)
= 〈~k|V |~k′〉+ 〈~k|V (k2
2−H0 + ıε)−1V |~k′〉 (227)
+ 〈~k|V (k2
2−H0 + ıε)−1V (
k2
2−H0 + ıε)−1V |~k′〉+ . . . (228)
We have used the first term of this series as the “first Born approximation”f (1) of the scattering amplitude. In real life, one rarely goes beyond theterm second order in V , i.e. f (2).
The popularity of the Born approximation is due to its feasibility andthe lack of alternatives. Even before the feasibility of the computationthere should be the question of convergence. On the other hand, there arewell-known cases, where a series does not converge, but still the first fewterms can give useful information (such is the case in some perturbationseries).
63
4.14.2 Kohn variational principle
We may try to guess an approximate potential Vg ≈ V for which, one wayor another, we can compute the resolvent and wave operators Ωg(k) forHg = H0 + Vg = H + (Vg − V ), and the corresponding Tg(k).Problem 4.20: Kohn variational principle Derive
T (k) = Tg(k)+Ω†g(k)(V −Vg)Ωg(k)−Ω†g(k)(V −Vg)(H−k2
2)−1(V −Vg)Ωg(k)
We get identity for the T -matrices, the Kohn variational principle
T (k) = Tg(k)+Ω†g(k)(V −Vg)Ωg(k)−Ω†g(k)(V −Vg)(H−k2
2)−1(V −Vg)Ωg(k)
(229)The last term is quadratic in V − Vg, so if this is small we may obtain agood approximation by only the computable first two terms.
In some cases this allows analytical estimates. More importantly, itallows for systematic improvement of the result in a numerical approach.
4.15 Standard textbook examples
4.15.1 Hard sphere scattering
Problem 4.21: Hard sphere A hard sphere is described by the potential
V (~x) =
+∞ r < R
0 r ≥ R
Evaluate the s-wave scattering amplitude for this system.Problem 4.22: Give the hard sphere scattering phase δl(k) for l = 1.What is the behavior for small k?
The total cross section at low energies can be calculated by observingthat (1) only s-wave scattering contributes and (2) the s-wave amplitudedoes not depend on the scattering angle:
σtot ≈∫dΩ
dσ0
dΩ= 4π
sin2(−kR)
k2≈ 4πR2 for small k (230)
Note: the total cross section is 4 times the geometrical cross section R2π.
64
Quantum scattering is more than just casting shadows. At the least, italso accounts for the fact that those parts that are missing in the forwarddirection show up in some other direction — which would account for afactor of 2, which is true in the high-energy limit.
4.15.2 Spherical potential scattering: Ramsauer-Townsend effect
We now replace the infinite repulsive potential of the hard sphere by anattractive (negative) finite potential and restrict our considerations to thes wave again.
We have already seen that low-energy scattering is usually dominatedby the s-wave (l=0). In that case the three-dimensional scattering problemis effectively reduced to a 1-dimensional one.
Problem 4.23: Ramsauer-Townsend effect Use the spherical well potential
V (~x) =
V0 < 0 for|~x| < R
0 else(231)
(a) Set up the equations for matching values and derivatives at R.Hint: The problem can be reduced to scattering from a finite potentialwell in one dimesion, which is found in literature.
(b) Starting from V0 = 0, lower the potenial such that√k2 − 2V0R will
pass through (2n+ 1)π/2. Conclude that there is a resonance near by.
(c) Argue that the phase shift goes through 0 at low energies and the crosssection passes through 0 at some point (Ramsauer-Townsend effect).Hint: At low energies, approximate the trigonometric functions bytheir Taylor series.
Resonant scattering: One interesting case arises when we lower V0
such that qR =√k2 − 2V0R passes the a value (2n + 1)π/2: at these
points the inner part of the wave function sin qx has an anti-node at R. IfkR is not too large, i.e. in low energy scattering, δ0 also passes throughπ/2 near these points. The cross section
σ0 =4π
k2sin δ0(k) =
4π
k2(232)
65
reaches the unitarity limit. This is in essence what is a resonance in scat-tering: there is an energy match of the inner range with the scatteringenergy.
Ramsauer-Townsend effect: Another interesting case arises for val-ues of V0 where α− kR = (2n+ 1)π. Using the Taylor expansion of arctanwe see that such points are reached
The scattering phase goes through δ0(k) = 0 and also the s-wave crosssection has a 0 at this point. If energies are low enough
σtot ≈ σ0 =4π
k2sin δ0 = 0! (233)
the object becomes transparent!Historically this is an important effect: such behavior was indeed ob-
served in electron scattering from noble gas atoms. At the time of itsfirst observation, there was no way of explaining the decrease of the cross-section with increasing energy to the “Ramsauer-Townsend minimum”. Wesee why: it is a distinctly wave-mechanical effect as the wave-length qR
must just match the “length” of the potential.
Problem 4.24: W rite down the solutions for spherical well s-wave scatteringfor values of V0 such that qR = nπ and qR = (2n+ 1)π/2, assuming smallk. What is the ratio of the amplitudes of the s-wave function rΨ(r) insideand outside the potential radius R? For which of the two values do we gethigher amplitude in the range r < R? Discuss the meaning of this and
66
make a drawing of the solutions for some (small) k.
Problem 4.25: D raw the functions kR, kR− π and α(k) and show graph-ically the conditions δ0 = π and δ0 = π/2. Use the Taylor expansions ofarctanx = x−x3/3 + . . . and 1/ arctan = x+x3/3− . . . to determine moreprecisely the locations in k of the resonance δ0 = π/2 with lowest k and ofthe Ramsauer-Townsend minimum δ0 = π in low energy scattering.
5 Complex scaling
We have seen that the resolvent Rz(H) = (H − z)−1 plays a central rolefor scattering. While from our discussion we have no direct proof of theconnection between poles of the S-matrix and the analytic structure ofRz(H), we may expect that the analytic structure of H is related to thatof S. Note that we have analytically continued S(k). We can also — insuitable cases — analytically continue the Hamiltonian itself. This startsfrom the observation that we can define a unitary scaling transformationon L2(Rd, d(n)x) by
Uλ : (UλΨ)(x) = enλ/2Ψ(enλx), λ ∈ R. (234)
This amounts to changing the units of length and, as a unitary transforma-tion, does not change any of the “physics” associated with the Hamiltonian.The time-dependent Schrodinger equation transforms as
id
dtΨ(x) = HΨ(x)→ i
d
dtΨλ(x) = HλΨλ(x), Ψλ := UλΨ0, Hλ := UλHUla
†.
(235)Importantly, the σ(Hλ) = σ(H), as the spectrum is invariant under unitarytransformation. The actual shape of Hλ and of Ψλ do depend on λ. Forexample, for the Hydrogen atom
Hλ = −1
2e−2λ∆− e−λ1
r. (236)
This transformation can be exploited to analytically continue the Hamil-tonian to non-selfadjoint values. Naively, we consider Hλ as an operator-valued function for λ on the real axis. In some cases, such as the Hydrogen
67
atom, the formal continuation into the complex plane is simple (the factori in iθ is traditionally kept separate):
Hθ = −1
2e−2(iθ)∆− eiθ1
r, θ ∈ C. (237)
We postpone the discussion of the meaning of “analyticity” for an un-bounded operator and of the domain of the transformed D(Hθ), first con-sider the spectrum. Instead, we naively discuss the spectrum, first thebound states. For the real scaling R 3 λ = −iθ we know that the boundstate energies remain unchanged
En(λ) ≡ En(0). (238)
Assuming analyticity (to be discussed), we see that also for complex θ thebound state spectrum remains unchanged:
En(θ) ≡ En(0) for bound state energies. (239)
What about the continuous spectrum? It does not remain unchanged.The reasoning about an isolated energy that would be constant is notapplicable: although the continuous spectrum as a whole does not change,the σac(H) = [0,∞) can be considered as being stretched (or shrunk)by scaling. E.g., for free motion H = −∆, we may consider the non-Hilbert eigenfunctions eikx, and we immediately see that the eigenvaluegoing with this moves under scaling. We know that σ(−∆) = [0,∞) andwe immedialtely see that
σ(−e2iθ∆) = e−2iθ[0,∞) = e−2i<θ[0,∞) : (240)
the spectrum rotates into the complex plane by the angle −2θ. Inter-estingly, the exact same happens for generaly, suitably analytic Hθ: thecontinous spectrum rotates into the complex plane by −2<(θ).
The explanation for the general case uses a theorem similar to the onethat we had envoked justifiy that a relatively compact potential at mostadds a finite number of bound states to −∆, but leaves the continuousspectrum invariant. This theorem is, in fact, not limited to self-adjointoperators, it requires only that H0 and V are relativ compact: let z 6∈σ(H0) ∪ σ(H0,θ = e−2iθH0). Then, if (H0 − z)−1V is compact, also
(H0,θ − z)−1V = e2iθ(H0 − e2iθz)−1V
68
is compact (this follows from the analyticity in z). So, adding a (rel-atively compact) potential does not change the continous spectrum ofH0 = −e−2iθ[0,∞).
5.0.1 Resonances
The interesting twist is that the extra “bound states” (i.e. point spectrum)can now appear anywhere in the complex plane, as the operator is not self-adjoint. We already know that the bound states will appear where they hadbeen. In addition, it turns out that in the wedge between the positive realaxis and the continuous spectrum, point spectrum can appear at complexvalues. Under sufficiently restrictive conditions on the potential, these canbe exactly identified with the poles of the scattering matrix. Here, thatproof will not be given.
5.0.2 Analyticity of eigenfunctions
All energies in the point spectrum (bound states and resonances) are asso-ciated with eigenfunctions. As a consequence of analyticity, the projectorsonto those eigenfunction are also analytic functions. The proof for this goesthrought the analyticity of the resolvent and the Riesz-projectors: integralsover analytic functions, if they exist, are analytic functions themselves, i.e.∫
Cb
dz(Hθ − z)−1 = 2πiΠb,θ (241)
Actually, can hope that also the eigenfuntions will be analytic funcitions. Inthe case of the hydrogen atom, this is quite obvious: the (radial) solutionshave the form
Qn(eλr) exp(−κneλr), (242)
with some (Laguerre-type) polynomial Qn. This is trivially analytic in λ.On the real axis, the projector onto a scaled bound state is
Πn,λ = |Φn,λ〉〈Φn,λ|. (243)
In the complex plane, one has to consider that complex conjugation is notan analytic map. For s.a. Hamiltonians, eigenfunctions can be chosenas real such that the only source of imaginary parts will be the complex
69
scaling. The corresponding imaginary parts should not change signs, suchthat the complex continued projector is
Πn,θ = |Φn,θ〉〈Φ∗n,θ|, (244)
i.e. the bra-vector is used without complex conjugation. More generally,one can write this by observing that an operator and its adjoint will sharespectral values, but have different left and right eigenvectors:
ΦL 6= ΦR : H|ΦR〉 = |ΦR〉E, H†|ΦL〉 = |ΦL〉E∗ (245)
and writeΠn,θ = |ΦR,n,θ〉〈ΦL,n,θ|, (246)
5.1 The origin of resonances
Resonances can arise from a range of contexts. One example can be seen inscattering at a potential well, as discussed in the context of the Ramsauer-Townsend effect: where the phase shift of the partial wave across the po-tential reaches δl(k) = (2n+ 1)π/2, we will see a peak in the partial wavecross section σl(k) = 4π/k2(2l + 1) sin2 δl(k).
In many cases, the appearance of a resonance can be connected to abound state of some “unperturbed” system. Here we discuss a few of thesecases.
5.1.1 Fermi golden rule
A resonance is behind the famous “Fermi golden rule” for the lifetime of astate in presence of some matrix element connecting it to a continuum atthe same energy. This is a standard textbook example, and it is usuallyderived using time-dependent perturbation theory. In fact, it is betterdescribed using stationary perturbation theory, using complex scaling tojustify the result.
The FGR decay width in its simplest form is given by the square ofthe bound-continum transition matrix element at the initial bound stateenergy E:
Γ = 2π|〈E0|V |Φ0〉|2, (247)
70
where Φ0 is the unperturbed bound state at energy E0 and |E0〉 is theδ-normalized continuum state at that energy. If the continuum at E0 isdegenerated we write
Γ = 2π∑∫
dα|〈E0, α|V |Φ0〉|2. (248)
In literature, you usually find a “density of states” ρ(E) included, whichaccounts for the proper δ-normalization and also for degeneracy in cases,where all α have the same matrix element.
A generic Hamiltonian that can be associated with FGR is
H =H0 + V (249)
H0 = |Φ0〉E0〈Φ0|+∫ ∞
0
dE∑∫
dα|E,α〉E〈E,α| (250)
V =
∫ ∞0
dE∑∫
dα [|E,α〉v(E,α)〈Φ0|+ h.c.] , (251)
and E0 > 0: it is important, that E0 is equal to one of the continuumenergies E.
If V is weak, we may try to apply standard perturbation theory (fornotational simplicity we drop degeneracy of the continuum states). The
first correction is E(1)0 = 〈Φ0|V |Φ0〉. Second order would be
E(2)0 =
∫dE|〈E|V |Φ0〉|2
E − E0= 〈Φ|VΠ⊥(E0 −H0)
−1Π⊥V |Φ〉. (252)
In the second form, we have defined the projector
Π⊥ = 1− |Φ0〉〈Φ0|, (253)
which removes the unperturbed state from the resolvent. The expressionis reminiscent of a second Born amplitude of the T -matrix, and it sharesits problem, viz. that the resolvent is singular at E = E0 > 0. A seeminglyad hoc remedy is to add a −iε to the denominator and approach E0 fromthe complex plane.
We obtain an imaginary part to of E(2)0 as
=m limε↓0
1
E + iε− E0= −πδ(E − E0) : (254)
with this, the imaginary part is just 1/2 of the FGR decay width.A justification for this follows in the next subsection.
71
5.1.2 Doubly excited states and Feshbach resonances
The generic Hamiltonian (249) can be made specific by choosing the Heliumatom as an example:
H0 =2∑i=1
−1
2∆i −
1
ri, V =
1
|~r1 − ~r2|, (255)
i.e. we treat the electron-electron interaction as a “perturbation”. Thecontinuum of H0 starts at −2au, when one electron is unbound and theother is in the ionic ground state. Let us denote such a state as |E〉 :=|n = 1, k〉, omitting all specifications of angular momentum for notationalbrevity. Any state where both electrons are in an excited ionic state,denoted as |Φx〉 = |n1, n2〉, n1, n2 > 1 will have an energy Ex = E1 +E2 >
− 2n21− 2
n22, i.e. Ex is immersed into the continuum of |E〉-states. In absence
of V , the electrons are unaware of their mutual existence and such a doublyexcited state is strictly bound. As soon as there is any mechnamism forthe two electrons to exchange energe (i.e. V ), one electron drops to theground state and the other is promoted into the continuum: we have anon-vanishing matrix element
〈n = 1, k|V |Φx〉 6= 0, (256)
which enters the FGR decay rate. Of course, one needs to question, whethera perturbative treatment is legitimate here, considering that the V raisesthe ground state binding energy from Eth − E0 = −2 − (−4) = −2 forH0 to the Helium ground state binding energy for H0 + V with a valueof 0.9037. Indeed FGR gives good values only when the two principalquantum numbers differ significantly. (This is easy to understand if weconsider that the two electrons reside in different spatial regions and barelyoverlap.)
We can readily complex scale H0 and also V . The spectrum of the com-plex scaled H0,θ is the sum of the spectra of the two independent ionicHamiltonians: at each bound state of one ionic Hamiltonian, the spectrumof the other ionic Hamiltonian is attached. The doubly excited boundstates |Φx〉 are located between thresholds. We can now compute pertur-bative corrections to these bound states using αV , keeping α sufficiently
72
small for now. Perturbation theory does not require operators to be self-adjoint. It is applicable whenever an energy is isolated from the rest of thespectrum.
The perturbative correction is
E(2)x,θ = 〈Φ∗x,θ|VθΠ⊥,θ(H0,θ − Ex)
−1Π⊥,θVθ|Φx,θ〉. (257)
Here, we do not need to add an iε, is the resolvent is not singular at E0
and perturbation theory remains directly applicable.Again, we have E
(2)x,θ as an integral of a product of analytic functions,
i.e. it is analytic. We can add an iε and compare the result for real scalingwith the complex scaled result: it is constant for real scaling, therefore E
(2)x,θ
coincides with the FGR result.Of course, complex scaling provides also the exact, non-perturbative
position Ex,θ of the resonance: you can view as the original E(0)x,θ pulled
into the complex plain by the interaction with the complex continua in itsvicinity. Note that Ex,θ is independent of θ as long as no continuum sweepsover it: it is invariant if we add a real part to the scaling parameter (i.e.unitary transformation), therefor invariant in a whole surrounding of theoriginal θ.
In addition we can now associate a unique funtion from Hilbert spacewith it, let us call it Ψx,θ and we can solve the eigenvalue problem for thatin Hilbert space.
The generalization of this concept is the so-called Feshbach resonance:we have a few-body problem that exhibits an exact bound state immersedinto a continuum, with which it does not interact in absence of pertur-bation. When a perturbation is added, the boundstate converts into acontinuum and acquires an imaginary part. This is amply exploited inultracold systems. There, an external handle (often a magnetic field) is in-troduced to manipulate the position of the resonace, in first approximationthe bound state energy.
5.1.3 Fano theory
An interesting case is the theory of Fano resonances. This, basically, is asimple interference phenomenon. From our point of view, the interesting
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thing is that the resolvent of model Hamiltonian and therefore the scatter-ing amplitude and cross-section can be calculated exactly.
The Fano process is a transition from some bound state into a continuumwhich has a FGR-like resonance. Typically, a system in its initial state |i〉would be excited to a continuum state |E±〉 by absorbing one photonwith energy ~ω = E − Ei. By |E±〉 we indicate that the continuumconsists of non-trivial scattering states that are modified by the presenceof an immersed resonance. The cross section for photo-absorption has acharacteristic line shape
σ(∆) = σ0|qΓ + 2∆|2
Γ2 + (2∆)2, (258)
where ∆ parameterizes is the “detuning” of the energy E from the reso-nance position Er and q is the “Fano q-parameter”. At ∆ = 0 the cross-section peaks, and, depending on the sign of q it dips to 0 at some positiveor negative ∆, i.e. above or below the resonance energy.
In lowest order of the photon-interaction, the cross section is given as
σ(E) ∝ |〈E±|T |i〉|2, (259)
where T is the transition operator, typically the dipole operator ~r. Thecharacteristic Fano dip arises from the energy dependence of the continuumstates |E±〉. These are the scattering solutions of a FGR-type Hamiltonian
H =: H0 + V = |0〉E0〈0|+∫dE|E〉E〈E|+ |v〉〈0|+ |0〉〈v| (260)
The dip in the line-shape can be understood as the destructive interferenceof a transition from the initial state |i〉 into the embedded bound state |0〉that decays and a “direct” transition into the continuum states |E〉, whichare considered as structureless, i.e. with weak E-dependence.
(further refer to fano-theory.pdf).
Problem 5.26: The Woodbury formula is a useful identity for the inverse ofan operator of the form A = A0 + UMW †:
A−1 = A−10 − A−1
0 U[W †A−1
0 U +M−1]−1
W †A−10 (261)
Show this!
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Problem 5.27: Fano states and profile
(a) Compute the Møller operator for
H =: H0 + V = |0〉E0〈0|+∫dE|E〉E〈E|+ |v〉〈0|+ |0〉〈v|,
considering the last two terms as the V . Bound and continuum statesof H0 are orthogonal: 〈0|E〉 = 0. Explicitly take the limit ε ↓ 0. Forthe scattering solution, you should obtain
|E±〉 =1
E0 + F (E)− E ± iπ|V (E)|2×[
(E0 + F (E)− E)|E〉 − V (E)∗(|0〉+ P
∫dE ′|E ′〉V (E ′)
E − E ′
)]with
P
∫dE ′|V (E ′)|2
E − E ′=: F (E), V (E) := 〈E|v〉.
Hint: For computing the resolvent (E − H ± iε)−1 write H = H0 +
Wσ1W† with W := (|0〉, |v〉) and σ1 =
(0 11 0
), then use the Wood-
bury formula. It can be useful to introduce short hand notation forthe diagonal elements of the 2× 2-matrix appearing in the Woodburyformula. Also note V |E〉 = |0〉〈v|E〉 = |0〉V ∗(E).
(b) Show
〈E−|T |I〉 ∝1
π〈0|V |E〉〈Φ(E)|T |I〉 sin ∆− 〈E|T |I〉 cos ∆
with
∆(E) = − arctanπ|〈E|V |0〉|2
E − E0 − F (E),
and the resonant contribution Φ(E).
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6 Multi-channel scattering
6.1 Inelastic scattering
So far we have assumed that there is only a single particle and no internalstructure of the scatterers. In reality, scattering may proceed in-elastically.Imagine a positron scattering from an atom: it will interact with the atom’selectron and, in principle, it can put the atom into an excited state. (Wechoose a positron rather than an electron to avoid questions arising withindistinguishable particles.) How can we describe this situation in theframework of S-matrix theory?
As the simplest model, assume our target is a two-level system, whichcan be put from the ground into the lowest excited state, but we excludeexcitation to higher excited states or ionization. Let
Hc =
∫d(3)k|~k, c〉(k2/2 + Ec)〈~k, c|, c = g, x (262)
be the Hamiltonians for free positron and the atom in the ground andexcited states, respectively. The states |~k, c〉 = |~k〉 ⊗ |c〉 describe a freepositron with momentum ~k and the atom in state c = x or gE is the total energy, Eg and Ex are the ground and excited state energies
of the atom, and E−Ec is the energy of the positron scattering in “channel”c.
We write a scattering potential in the form
HI =
∫d(3)k
∫d(3)k′δ(k′ − k)
[|~k, g〉Vgg〈~k′, g|+ |~k, x〉Vxx〈~k′, x|
](263)
+δ(k′ − q(k))[|~k′, x〉Vxg〈~k, g|+ |~k, g〉Vgx〈~k′, x|
]. (264)
with q(k) =√k2 + 2Eg − 2Ex and integration ranges are adjusted such
that the argument of the root remains ≥ 0.The first line describes elastic scattering processes where the atom re-
mains in |g〉 or |x〉 and the second line generates transitions from the groundto excited state and back. The total Hamiltonian is
H = Hg +Hx +HI (265)
Note that the S-matrix formalism does not make any assumptions aboutthe exact nature of the Hamiltonians involved. The only assumption is
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that at long times the time-evolutions by H0 and H become equal. Thuswe choose for the comparison Hamiltonian H0
H0 = Hg +Hx (266)
and the scattering “potential” V is
V := HI . (267)
The Vcc′ = 〈~k, c|V |~k′, c′〉 are functions of ~k and ~k′.The concept of scattering cross section must be extended to include the
fact that the atom can be either in its ground state |g〉 or in the excitedstate |x〉. For each pair of channels c = g or x and c′ = g or x, we define ascattering cross section as follows. Assume you have an incident beam ofelectrons that hits the atom in the state state |c〉. What is the probabilityof finding in the scattered wave packet the atom in the state |c′〉 in theenergy range ∆E?
The procedures are essentially the same as for a single particle.
• Set up a normalized incident wave packet, preferably with the atom ineither state |g〉 or |x〉.
• Write down the probability for finding a given final range of finalpositron momenta and atomic state |g〉 or |x〉.
• Average over all impact parameters keeping the number of particlesper surface unit constant, i.e. do the integral
∫R2 d
(2)a.
• Handle the δ-functions. Here arises a new feature as total energy E =k2/2 + Ec is conserved, but not the energy of the scattering positronalone k2/2. This adds a factor k/k0 to our equation.
In this way one finds for the channel scattering cross section
dσcc0dΩ
=k
k0|fcc0(~k,~k0)|2 (268)
with
fcc0(~k,~k0) = −(2π)2〈~k, c|Vc + Vc(
k2
2+ Ec −H + ıε)−1Vc0|~k0, c0〉 (269)
77
Where Vc := H − Hc. Note that because of conservation of total energyE = k2 + Ec = k2
0 + Ec0
〈~k, c|Vc0|~k0, c0〉 = 〈~k, c|H −Hc0|~k0, c0〉 = 〈~k, c|H − E|~k0, c0〉 (270)
= 〈~k, c|H − E|~k0, c0〉 = 〈~k,Hcc|Vc − E|~k0, c0〉 (271)
= 〈~k, c|Vc|~k0, c0〉 (272)
Post and prior forms: In this case the distinction between the “post”and “prior” forms of the scattering amplitude becomes relevant. We canwrite the
f(~k,~k0) = −(2π)2〈~k, c|Vc|Ψ(−)c0
(~k0)〉 “post” form (273)
= −(2π)2〈Ψ(+)c (~k)|Vc0|~k0, c0〉 “prior” form (274)
and each of the two forms may allow different approximations, dependingon the exact shape of Vc and Vc0 or the scattering waves Ψ
(±)c (~k). E.g. one
of the channels may have very high or very low scattering energy, whichmakes replacing the scattering wave function by plane waves seem morelegitimate.
6.1.1 S-matrix
The S-matrix for this inelastic scattering is
S = Ω†+Ω−. (275)
In potential scattering the S matrix conserved asymptotic energy of thefree particle k2/2 and we could represent it as a function of k. We need,in addition the asymptotic kinetic energy also the bound state energy.Matrix elements of the S matrix will therefore depend on k as well as onthe integral state before and after the scattering:
S =
(Sgg SgxSxg Sxx
). (276)
6.1.2 LS-equation
Generalization for inelastic scattering is straight forward:Problem 6.28: Multi-channel LS-equation
78
• Derive|ψ(±)
c,~k〉 = |~k, c〉 − (Hc − E ± iε)−1Vc|ψ(±)
c,~k〉 (277)
with Vc = H −Hc for the inelastic case.Hint: Start from the LS equation for the full problem and identifythe c-component.
• Show that the same form applies for general multi-channel scattering.
Hint: Start from the definition of Ω(α)± and observe that the stationary
form (ε-limit) is the exact same as in potential scattering.
Above, we were able to trivially solve scattering problem for Hc andtherefore we were able to use (Hc− k2/2± iε)−1 in the spectral amplitude.In more realistic problems, already solution of the channel problem maypose a problem.Problem 6.29: Multi-channel Born approximation Write the multi-channelLS-equation in first Born approximation w.r.t. Hα. Rewrite (Hα−E±iε)−1
using the (exactly known) (−∆/2−E±iε)−1 and the Møller operators Ω(a)± .
Approximate the Møller operators by the first two terms of the resolventseries and consider the result up to first and second order of potentials.
6.1.3 An alternative view
We will now formalize the expectation
Scc′ = (Ω(c)+ )†Ω
(c′)− , c, c′ ∈ g, x (278)
Above we have used the formalism exactly as developed so far, as a com-parison between two time evolutions H0 and H, where
H0 = −1
2∆⊗
(1 00 1
)(279)
describes free motion with the scatterer in either g or x.and the Møller operators
Ω(c)± = lim
t→±∞exp(−itH) exp(itH0)Pc. (280)
79
Here we have added a projector
Pg = 1⊗(
1 00 0
), Px = 1⊗
(0 00 1
), (281)
that admits only asymptotic states in the respective channel.The channel Hamiltonians commute [Hc, Hc′] = 0 and therefore the two
asymptotic subspaces are (rather trivially) orthogonal
PcPc′ = δcc′Pc. (282)
H0 is a sum of the channel Hamiltonians Hc:
H0 = HgPg +HxPx (283)
SimilarlyΩ± = Ω
(g)± + Ω
(x)± (284)
and we can read off the scattering matrix elements (278).
6.1.4 Distorted waves
We are free to choose the alternative comparison Hamiltonian
H0 =∑c
Hc =∑c
−1
2∆ +
∫d(3)k|~k, c〉Vcc〈~k, c|, (285)
assuming that we either can exactly solve or independently approximatethe scattering problem for Hc.
6.2 Two interacting particles in a potential
In inelastic scattering we could essentially straight forwardly extend thetheory of potential scattering. However, the general multi-particle caseshows fundamental complications due to the fact the spaces Hac(Ha) maynot be orthogonal.
The simplest system that shows all complications are two particles in apotential that interact by a force VI . Assume all forces are short range.The Hamiltonian on L2(R3) ⊗ L2(R3) (with proper domain, not essentialfor this discussion) is
H(1, 2) = H1 ⊗ 12 + 11 ⊗H2 + VI(1, 2) (286)
80
where H1 and H2 are Hamiltonians on L2(R3) and VI(1, 2) is a multiplica-tion operator on L2(R3)⊗ L2(R3), say with kernel
V (~x1 − ~x2). (287)
There are several different asymptotic situations
• (012) . . . all particles bound, state in the point spectrum of H
• (01)+2 . . . 1 bound, but 2 free, state in the continuous spectrum σc(H),but 1 in the bound spectrum σp(H1)
• (02)+1, 1↔ 2 of the above
• (12)+0 . . . 1 and 2 bound by VI
• ()+0+1+2 . . . both particles free
To specify the asymptotic situation fully, we must for each case define theasymptotic bound state (if any), say |a〉 := ψ
(ij)n where n labels the bound
states of (ij) = (01), (02), (12). a designates the asymptotic channel. Thecomparison Hamiltonian for channel a is then
Hα = −∆α ⊗∑a
|a〉Ea〈a| (288)
where α denotes the coordinates from the center-of-mass of particles i andj to the remaining particle and
− 1
2mij∆ij + V (~ri − ~rj)|a〉 = |a〉Ea, (289)
with the reduced mass of particles i and j. For simplicity, we have assumedparticle 0 as infinitely heavy. If this is not the case, appropriated reducedmasses need to be used.
In addition there is the breakup channel α = () with the channel Hamil-tonian.
H() = − 1
2m1∆1 −
1
2m2∆2. (290)
We identify a channel as subset of coordinates w.r.t. to which all par-ticles remain bound and the (appropriately constructed) remaining coor-dinates, where the motion is free.
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The construction of a single wave operator “eitHe−itH0” to describe allthese situations is not possible, since there is no single H0 that can describeall channels simultaneously. We can readily give the projectors Pα ontoHac(Hα)
Pα = 1α ⊗∑a
|a〉〈a|, P() = 1, P(012) = 0. (291)
The ac spectral spaces Pac(Hα)H are not mutually orthogonal. Simplest tosee by looking at the total breakup 0 + 1 + 2 with H(), whose ac-spectrumspectral subspace is the whole L2(R3)⊗L2(R3), while obviously any otherchannel is part of that: P(ij)P() = P(01). So, if we cannot decomposethe initial wave packet uniquely into components Pac(Hα)H, for which Hα
should we compute the Møller operators and the scattering wave packetΨ(±)?
The solution is that, the if the channel Møller operators
Ω(α)± := lim
t→±∞eitHe−itHαPα (292)
exist, one can show that they map onto mutually orthogonal subspaces ofHac(H).
That means that while a given wave-packet cannot be unambiguouslydecomposed into contributions belonging to a given channel from its behav-ior in the remote past (alternatively: remote future) alone, the scatteringwave packets
Ψ(±)α = Ω
(α)± φ, φ ∈ H (293)
are orthogonal:〈Ψ(±)
α |Φ(±)β 〉 = 0 for α 6= β. (294)
We can uniquely decompose the corresponding scattering wave packet intoscattering (not channel) contributions with the respective asymptotic be-havior. We need the information of the actual scattering process for re-solving the ambiguity contained in the asymptotic behavior alone. This isdifferent here from the case of inelastic scattering, where the asymptoticscould be unambiguously related to a given Hamiltonian and therfore bedecomposed into “incoming ground” or “incoming excited” channels etc.without any reference comparison to the actual scattering.
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The physical reason for this orthogonality is that, by construction, α 6= βimplies that the two channels differ in a least one coordinate, where, say, αwould be bound, but β is free. Wave packet spreading of free motion (i.e.Φβ(t)
w→ 0) while Ψα(t) remains localized w.r.t. to the same coordinate.That implies that the overlap of β with the α w.r.t. to that coordinate→ 0.
Let us consider the channels α = (01) and β = (02) in our presentexample. We have assumed that the channel Møller operators exist astrong limits, i.e.
|| limt→±∞
e−itHeitHαPαφ− Ω(α)± φ|| = 0∀φ ∈ H. (295)
From this follows
||Q(α)± φ||2 = lim
t→±∞〈Ωα(t)φ|Ωα(t)φ〉 =
= limt→±∞
〈e−itHeitHαPαφ|e−itHeitHαPαφ〉 = ||Pαφ||2
and similarly[Q
(α)± ]2 = Q
(α)± (296)
Note that although Pa 6= Qa, the norms agree for all φ: the differencebetween them is an isometric map from Hac(Hα) to Hac(H). In morephysical terms it can be understood such that for Q−α at some pointin the remote past the norms agree. As the Møller operators are time-invariant also the Q
(α)− are, therefore the norms always agree. For Q
(α)+ , use
the agreement in the remote future.For our two channels we can write the Hamiltonians as
H(01) = −1
2∆⊗
∑a
|a〉〈a|, H(01) =∑b
|b〉〈b| ⊗ −1
2∆, (297)
where a and b run over all bound states of the (01) and (02) coordinate,respectively.
〈Ω(α)± φ|Ω(β)
± φ〉 = limt→±∞
Pαφ|e−itHαeitHβPβφ〉 := limt→±∞
〈φα|e−itHαeitHβφβ〉
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For showing the orthogonality it is sufficient to show that e−itHαeitHβw→ 0:[∑
a
|a〉e−itEa〈a| ⊗ e−it∆/2][
e−it∆/2 ⊗∑b
|b〉e−itEb〈b|
]=[∑
a
|a〉e−itEae−it∆/2e−it∆/2〈a|
]⊗
[e−it∆/2
∑b
|b〉e−itEb〈b|
]Now, to the separate tensor factors, the wave-packet spreading argumentapplies. (If you like: RAGE theorem). Similar reasoning can be carriedthrough for all partitionings.
Mutual orthogonality of the images of Ω(α)± means
(Ω(α)± )†Ω
(β)± = δαβPα. (298)
We remember thatΩ
(α)± : Hac(Hα)→ Hac(H) (299)
The map is no longer surjective, i.e. on to all of Hac(H), but onto asubspace. The projector onto that subspace can be constructed: As theimage of, say, Ω
(a)± ⊂ Hac(H) and the Møller operators do not change the
norm when considered as maps from Hac(Hα), the operators
Q(α)± := Ω
(α)± (Ω
(α)± )† (300)
are projectors onto the respective subspaces of Hac(H). Orthogonality ofthese subspaces means that
Q(α)± Q
(β)± = δαβQ
(β)± . (301)
Note thatQ
(α)− 6= Q
(α)+ (302)
in general, the two projectors differ. For a given channel α, it matterswhether we construct the subspace by going back or forward in time. It iseasy to understand on physical grounds why this cannot be: suppose wehave an incoming state in the channel α = (01) channel, i.e. particle 1bound. In the reaction, all three channels can be generated: (01), (02), ().If we want to create that scattering state by going into the remote future,we need components in all three channels, not just in the original α = (01).
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Of course, the two scattering state subspaces are related by time-reversal:
Q(α)+ = KQ
(α)− K, (303)
if K denotes the time-reversal operator.
6.2.1 Asymptotic completeness
For the few-body situation completeness means that, going through allpossible channels α, we can build all scattering states in Hac(H). That
means all Q(α)+ (or Q
(α)− ) together project onto the complete ac spectral
subspace of H ∑α
Q(α)± H = Pac(H)H. (304)
6.2.2 Jacobi-coordinates
With different clustering of particles (01), (02), (12), the choice of coordi-nates is not necessarily obvious: with an infinite mass for particle 0, thecoordinates
~r01 = ~r1 − ~r0, ~r02 = ~r2 − ~r0 (305)
are natural choices. Already in the (12) channel, it is advisable to choosethe relative coordinates
~r12 = ~r2 − ~r1 (306)
and the coordinate from ~r0 to the center-of-mass of (12):
~rJ =m1~r1 +m2~r2
m1 +m2− ~r0. (307)
This is the simplest form of “Jacobi coordinates” which connect the centers-of-masses of a hierarchy of sub-clusters. Only with such coordinates, thechannel Hamiltonians factor in the simple shape that we used. We couldconveniently use the fact that for the channels (01) and (02), the Hamilto-nians factorize into tensor products with respect to the same tensor factors
H = L2(R3, ~r1)⊗ L2(R3, ~r2) (308)
with kinetic energy in the unbound channels in the form −∆.
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Already at finite mass for particle 0, we loose that convenience. In thatcase we can either keep the factorization (308), but than obtain contribu-tions to kinetic energy terms that of the form
~∇1 ⊗ ~∇2, (309)
sometimes called “mass polarization terms”.Alternatively, we can choose Jacobi coordinates in all channels and the
corresponding reduces massed for the energy. This is the standard choicefor abstract scattering theory, as it simplifies the discussion of free motion.
Already with three particles, the choice of Jacobi coordinates is notunique for the α = () channel, as the sequence of grouping the particles isnot unique, one for each pair-wise clustering (ij).
6.2.3 The S-matrix (interaction picture)
This is what we have discussed so far, a map from one channel β whereparticles are bound w.r.t. to the subset of coordinates β and free for therest into channel β:
Sαβ = Ω(α)+ Ω
(β)− . (310)
6.2.4 The S-matrix (Heisenberg picture)
Problem 6.30: S-matrix in the Heisenberg picture One can define an S-matrix as
S =∑α
Ω(α)− (Ω
(α)+ )†. (311)
(a) On which subspace(s) does S operate?
(b) Show that Ω(α)+ (Ω
(α)− )†Ψ
(−)
β,~k= δαβΨ
(+)
α,~k.
(c) Show that S is related to Sαβ by
〈~k, α|Sαβ|~k′, β〉 = 〈Ψ(−)
α~k|S|Ψ(−)
β~k′〉
(d) Assuming asymptotic completeness, show that S is unitary on Hac(H).
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The S of the Heisenberg picture has the virtue of being unitary which Sαβis not, as it maps between different subspaces. For using it in practice, oneneeds access to the scattering functions Ψ
(±)
α,~k.
6.3 Graphical representation of the resolvent
With the resolvent playing the lead role in scattering, the resolvent equa-tion, its extension to multi-channel, and expansion into a series (as in theBorn-series) play a central role. This is, in fact, what is behind the popularFeynman diagrams with their intuitive physical interpretation.
Resolvents are linear operators (and bounded), therefore one can addand multiply them: they form an algebra. As we only want to make youaware of this formally, we do not discuss questions of convergence of aseries.
Also, there are certain elements of choice in using the resolvent for-mula: we may subsume parts of the interaction into the resolvent. TheKohn variational principle was an example where one isolates a solvablepart and remains with a correction, that can be neglected or be treatedapproximately.
Let us remain with our 3-particle problem, but adjust notation for thepurpose
H = T + v1(~r1 − ~r0) + v2(~r2 − ~r0) + v3(~r2 − ~r1) (312)
then we can use the resolvent series to first, e.g., expand
1
z −H=
1
z − (H − v1)− v1=
1
z − (H − v1)
∞∑n=0
[v1
1
z − (H − v1)
]n(313)
and repeat the process for [z − (H − v1)]−1 = [z − (H − v1 − v2)− v2)]
−1,etc. We end up with a multiple sum of the form (R0 := (z − T )−1)
(z −H)−1 =∞∑n=0
3∑γ1=1
. . .3∑
γn=1
R0vγ1R0vγ2R0 . . . vγnR0, (314)
which we have sorted by powers of R0. This is clearly a rather large sumand proper book keeping of all terms is needed. We do this by introducingthe graphical notation:
87
Absorbing part of the interaction into the resolvents amounts to notcarrying the expansion to the point where all R0 are free motion, butrather include the effect of, say, an external potential by using some Rx
instead of R0. In the corresponding graphics, one can distinguish this bya different kind of line, e.g. fat.
A typical and extensively used example of such a corrected resolvent isthe “self-energy” correction. The terms missing to the full resolvent arethen, more often than not, neglected altogether.
6.3.1 Example
The importance of this is, however, not only a matter of book-keeping.The symbols can be distinguished into those that do involve an interactionbetween them (or to the outside, the infinitely heavy nucleus), and thosethat are connected. It turns out that operators corresponding to connectedgraphs are compact (like, e.g. (p2 − z)−1V for a single particle). We hadused several times, that such “relatively compact” operators leave the con-tinuous spectrum unchanged, which is obviously an important propertyfor scattering problems. The graphical representation allows to easily dis-tinguish the compact parts from the non-compact. In our example, wedraw:
where we have collected disconnected parts as
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And finally cast this into a formula (Weinberg-Van Winter):
R = D + JR = (non-compact, spectrum known) + (compact) R. (315)
As it holds that (compact)×(bounded)=(compact), we know that the con-tinuous spectrum for R is the same as the continuous spectrum for J . Thelatter is known and thus we know the spectrum for R.
89
7 Coulomb scattering
The Møller operators for the pair H = −∆/2 − 1/r and H0 = −∆/2do not exist, which renders useless, at first glance, the instruments devel-oped so far. There are several ways out of this predicament, each withits own merits and drawbacks. First, any truncation of the potential at aremote distance will alleviate the problem. Second, we actually know theexact (although a bit unwieldy) scattering solutions for the Coulomb prob-lem. Thirdly, for the same reason that we know the exact solutions, wealso can obtain the relevant scattering information without ever computingthe scattering wave functions. Finally, we may choose for the comparisonHamiltonian H0 into something for that the Møller operators do exist andwhose spectral representation we still know.
7.1 Failure of standard scattering theory
An abstract way how the failure manifests itself is the fact that
w- limt→±∞
eit∆/2+1/re−it∆/2 = 0, (316)
which means, as strong convergence would imply weak convergence, thatwe would obtain Ω± = 0, certainly not a suitable operator to obtain asymp-totic completeness.
The problem is fundamental and easy to understand. Consider the clas-sical motion on a trajectory that directs radially away from an attractiveCoulomb potential −1/r. At radius r and given total energy E, the classi-cal momentum is
√2E + 2/r. We also know that the velocity is bounded,
therefore the position cannot grow more than linearly in time,
r(t) = ct+O(t). (317)
Using this we see
1√2
d
dtr =√E
√1 +
1
E
1
ct+O(t)≈√E
[1 +
1
2E
1
ct+O(t−1)
](318)
where the expansion becomes valid where potential energy is only a smallpart of of total energy.
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We find that the particle never moves like a free particle:
r(t) ∼ ct+ d log t. (319)
For attractive potentials, it will always “outrun” the free particle by anydistance. Embarrassingly, our concept of scattering as comparing to freetime-evolution fails fundamentally for this well-studied problem.
The slightly paradoxical impression that an attractive potential seemsto cause the particle to move faster is readily dissolved when rememberingthat, at given asymptotic energy E, the kinetic energy in an attractive, i.e.negative potential is higher than E.
7.2 Compare to a different time-evolution
We will assert below that a meaningful asymptotic momentum exists for theCoulomb potential. This motivates the construction of comparison time-evolution that is based only on momentum, but differs from free motion.Such a Hamiltonian was proposed by Dollard
HD(t) = −∆− 1
2|t|√−∆
θ(−4|t|∆− 1) (320)
and it turns out
Ω(D)± = s- lim
t→±∞eit(−∆−1/r)e−i
∫ t0HD(s)ds (321)
exists. Again Volker Enss (1979) proved asymptotic completeness for Dol-lard comparison operators (not in the 1972 Reed and Simon book).
The functions of ∆ are well defined, consider a momentum represen-tation, and we know the eigenfunctions. An important drawback is thatwe cannot use the formulae of stationary scattering theory because of thetime-dependence of HD(t). This may be the reason why such techniqueshave found little practical application.
7.3 Asymptotic constants
From the above we learn, that the requirements of the beautiful appara-tus of scattering theory are too strict: the exact comparison of the time-evolution is not what we usually seek. Instead, we want to know in which
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direction (from some microscopic center) and at which momentum/energya fragment arrives. Scattering theory where it exists, also requires, thatmotion towards the detector is essentially free: but this is not the casefor Coulomb. When we do not require that, we can ask for “asymptoticobservables”, i.e. observables that become constant at “infinite time” andat “infinite distance”, i.e. when we actually measure at our detector.
In strict terms, an asymptotic constant A± is the (strong) limit
s- limt→±∞
e−itHAeitH = A± (322)
I.e., at sufficiently large times and for a given wave packet, the observabledoes not change any more. As always, the limit can work only in thestrong sense, i.e. on a wave-packet wise level. Imagine momentum, which,we expect, will become constant eventually in a scattering process. But for(arbitrarily) slow fragments will take (arbitrarily) long time to leave theregion of non-zero potential, and there will be no time when all, includingthe slowest, momenta have already assumed there final values within somepreset ε.
Of course, any conserved observable [A,H] = 0 is also an asymptoticconstant, e.g. angular momentum A = L2. The non-obvious constantsof motion for the Coulomb potential are, as we would hope, momentum,direction, and the Coulomb potential itself, if we restrict them to the acspectrum by the projector PH = Hac(H):
s- limt→±∞
(P~pP, P~r
|~r|P, P
1
rP ) = (~p±,±
~p±|~p±|
, 0) (323)
The sign of ±~p±/|~p±| reflects time-reversal: for the same value of momen-tum, we find a particle at negative times in the opposite direction frompositive times. This expected result can be proven rigorously, but theproof involves surprisingly much effort.
For us, the message is: in the Coulomb problem, the momentum doesassume a constant value asymptotically, although there is never strictlyfree motion. In that sense, the Coulomb force does become negligible atlarge distances.
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7.4 Complete solution of the Coulomb problem
The high symmetry of the Coulomb problem allows for its exact solutionby purely algebraic means. We know 7 constants of motion, energy H,angular momentum ~L = ~r × ~p, and the Runge-Lenz vector
~F =1
2(~p× ~L− ~L× ~p) +mα
~r
|~r|. (324)
As [~p, ~L] 6= 0, the explicitly hermitian form of the ×-product appears here.We have emphasized here how mass m and charge α = qZ (positive αfor repulsive potentials) enter the constants. The constants are not alge-braically independent but there are relations between them. This is to beexpected from the corresponding classical system: there, the constants arefunctions in phase space and there cannot be more independent constantsthen the 6 dimensions of the phase space.
In fact, the 7 constants of motion generate an algebra that is isomorphicto the algebra of two sets of angular momenta in the case of the boundstates.
Problem 7.31: Symmetries of the H-atom
(a) The constants of motion fulfill the following commutation relations
[H,Ln] = 0, [Ln, Fl] = iεnlsFs
[H,Fn] = 0, [Fn, Fl] = −2mH iεnlsLs~L · ~F = ~F · ~L = 0
~F 2 = 2mH(~L2 + 1) +mα2
Verify this by explicit calculation.
(b) Let P =∑
nlm |nlm〉〈nlm| be the projector onto the bound state spec-trum. Then
An := Ln + Fn√−2mHP/2, Bn := Ln − Fn
√−2mHP/2 (325)
are well-defined and fulfill the algebra of two independent angular mo-mentum operators:
[Ai, Aj] = iεijkAk, [Bi, Bj] = iεijkBk, [Ai, Bj] = 0. (326)
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Verify this.
(c) Convince yourself of
~A2 = ~B2 = −(
1
4+mα2
8H
)P (327)
(d) Remember that for any vector of operators ~X obeying an angular mo-mentum algebra one has the spectra
σ(Xm) =1
2. . .− 2,−1, 0, 1, 2 . . .
σ( ~X2) =1
40, 1(1 + 2), 2(2 + 2), 3(3 + 2), . . .
and use (327) to derive the Balmer formula.
Note that the ~A, ~B form the Lie-algebra of the SO(4) symmetry group,i.e. rotations in R4.
7.5 Exact Coulomb scattering
In the limit of large times t → ±∞, we can in all of these constants ofmotion replace the operators ~p and ~r/|~r| by their asymptotic values, ifwe restrict ourselves to the continuous spectrum by using the projectorQ = (1 − P ). But as they are constants, the limit value is equal to thevalue at any time, i.e.
QHQ =1
2~p2±
Q~FQ =1
2(~p± × ~L− ~L× ~p±)±mα ~p±
|~p±|.
=1
2[~p±, L
2]±mα ~p±|~p±|
Note: we have exactly the same algebra in classical mechanics, replacingcommutators with Poisson brackets.
From these equations, one can derive a relations between ~p+ and ~p−
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(show by evaluationg ~F × ~L):
~p+ =~p−L2 + iη − η2
L2 − iη + η2+ (~p− × ~L− ~L× ~p−)
η
L2 − iη + η2(328)
= ~p−L2(1 + iη) + iη − η2
L2 − iη + η2− iηL2~p−
1
L2 − iη + η2(329)
η(H) := Qmα√2mH
(330)
.
7.5.1 Scattering transformation
S =Γ(1
2 −√L2 + 1
4 + iη(H))
Γ(12 −
√L2 + 1
4 − iη(H))(331)
with the property~p+ = S−1~p−S. (332)
This has the form A†A−1 with A being “normal” ([A,A†] = 0, it is thereforeunitary.
This can be shown as follows: In a angular expansion and spectral rep-resentation w.r.t. E = 2k2 it is
S|l,m〉 ⊗ |φ(k)〉 = e2iδl(k)|l,m〉 ⊗ |φ(k)〉 (333)
The phase can be determined by computing matrix elements for the z-components of ~p±: this conserves m, but changes angular momentum byone:
〈l + 1,m|(~p+)z|l,m〉
= 〈l + 1,m|(~p−)zl(l + 1)(1 + iη) + iη − η2
l(l + 1)− iη + η2− iη(l + 1)(l + 2)(~p−)
1
l(l + 1)− iη + η2|l,m〉
= 〈l + 1,m|(~p−)z|l,m〉l(l + 1)− iη(2l + 1)− η2
l(l + 1)− iη + η2= 〈l + 1,m|(~p−)z|l,m〉
(l + 1− iη)(l − iη)
(l + 1 + iη)(l − iη)
from which one obtains
〈l + 1,m|(~p+)z|l,m〉〈l + 1,m|(~p−)z|l,m〉
=l + 1− iηl + 1 + iη
= e2i(δl−δl+1) (334)
The scattering phase is fixed only up to a given overall phases and we fixit by
e2iδ0(k) =Γ(1 + iη(k))
Γ(1− iη(k)). (335)
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By the property of the zΓ(z) = Γ(z + 1) we obtain the result for l > 0.
7.5.2 Scattering cross section
Although we have not run the full program of scattering theory, we canstill compute the exact cross section using the S-operator. Note that S isin the Heisenberg picture, i.e. it maps within Hac(H), not within Hac(H0).Therefore, we cannot consider an arbitrary φ ∈ H = Hac(H0) 6= Hac(H)and apply S.
However, we can simply identify S with some S00 (in the interactionpicture) of a scattering problem that has the same mapping of momentawhich is all we care about usually. The difference between S and S00 couldbe that S maps a given wave-packet φ ∈ Hac(H) to a larger distance atthe outgoing side than S00, but into the same direction, which is the onlything that matters, as energy is guaranteed to be conserved.
Considering this fictitious camparison system, we can use the usual ap-paratus of scattering theory to determine the cross sections. In particularwe can use teh relation between scattering phases and scattering amplitude:
f(k, ~n, ~n′) =∞∑l=0
2l + 1
2ikPl(cos θ)
[Γ(1 + l + iη)
Γ(1 + l − iη)− 1
](336)
This actually evaluates by several non-elementary manipulations to
f(k, ~n, ~n′) =iη
2k
(4
|~n− ~n′|2
)1+iηΓ(1 + iη)
Γ(1− iη)− 1
2ikδ(2)(~n− ~n′) (337)
Note that all complex contributions here are just phases and, avoiding~n = ~n′, we recover the classical cross section
σ(k, ~n, ~n′) = |f(k, ~n, ~n′)|2 =α2
16E
1
sin4(θ/2). (338)
7.5.3 Coulomb functions
For completness we mention the Coulomb functions that solve the eigen-value equation [Abramowitz 1954]
d2
dρ2W +
[1− 2η
ρ− l(l + 1)
ρ2
]W = 0. (339)
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Here, all parameter- and energy dependence has been absorbed into η,which is exactly the η we had defined previously:
η(E) =meZ√2mE
=meZ
k. (340)
Inserting into (339) and setting ρ = kr and multiplying by −k2/2, werecover the familiar equation for hydrogen-like problems:[
−1
2
d2
dr2+meZ
r− l(l + 1)
2r2− k2
2
]W = 0. (341)
There are regular Fl(η, ρ) and irregularGl(η, ρ) solutions with the asymp-totics
Gl(η, ρ) + iFl(η, ρ) ∼ exp[i(ρ− η log(2ρ)− lπ/2 + σl)] (342)
where scattering phases are σl = arg Γ(l+ 1 + iη). As to be expected, thisis not asymptotically free motion, rather there is a logarithmic phase slip.
As is known, these can be expressed in terms of confluent hypergeo-metric functions, in various forms and a range of asymptotic expressionsare known. (See, e.g., the “Digitial Library of Mathematical Functions”,hosted at NIST. Also, computer codes are provided, e.g. in the Gnu Sci-entific Library to evaluate them. Unforatunately, handling these functionsis far more difficult than handling plane waves with the related Fouriertransforms.
7.6 Coulomb plus short range potentials
A large class of scattering problems can be described by potentials
V (r) = Vs +q
r. (343)
For these problems we expect scattering theory to exist for the comparisonHamiltonian
Hq = −1
2∆ +
q
r. (344)
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