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3. Scattering Theory 1 A Scattering Theory 1 (last revised: September 5, 2014) 3{1 3. Scattering Theory 1 a. Reminder of basic quantum mechanical scattering The dominant source of information about the two-body force between nucleons (protons and neutrons) is nucleon-nucleon scattering. You are probably familiar from quantum mechanics classes of some aspects of non-relativistic scattering. We will review and build on that knowledge in this section. Nucleon-nucleon scattering means n{n, n{p, and p{p. While all involve electromagnetic inter- actions as well as the strong interaction, p{p has to be treated specially because of the Coulomb interaction (what other electromagnetic interactions contribute to the other scattering processes?). We'll start here with neglecting the electromagnetic potential Vem and the difference between the proton and neutron masses (this is a good approximation, because (m m )=m 10−3 with n − p ≈ m (m + m )=2). So we consider a generic case of scattering two particles of mass m in a ≡ n p short-ranged potential V . While we often have in mind that V = VNN, we won't consider those details until the next sections. p2 p2′ P/2 + k P/2 + k0 = p ) 1 p′ P/2 k P/2 k0 1 − − Figure 1: Kinematics for scattering in lab and relative coordinates. If these are external (and 2 02 therefore \on-shell") lines for elastic scattering, then when P = 0 we have k = k = mEk = 2µEk (assuming equal masses m). Let's set our notation with a quick review of scattering (without derivation; see the linked references to fill in the details). The Hamiltonian (in operator form) is p2 p2 H = 1 + 2 + V; (1) 2m 2m where we label the momentum operators for particles 1 and 2. We switch to relative (\rel") and center-of-mass (\cm") coordinates: r + r r = r r ; R = 1 2 (2) 1 − 2 2 p p k = 1 − 2 ; P = p + p ; (3) 2 1 2 so that the Hamiltonian becomes (still operators, but the P, k, and k0 in Fig. 1 are numbers) P2 k2 H = T + H = + + V; (4) cm rel 2M 2µ Scattering Theory 1 (last revised: September 5, 2014) 3{2 with total mass M = m1 + m2 = 2m and reduced mass µ = m1m2=M = m=2. (Note: Hrel is sometimes called the intrinsic Hamiltonian.) Now we note that V depends only on the cm variables (more on this in the next section!), which means that we can separate the total wave function into a plane wave for the center-of-mass motion (only kinetic energy) and the wave function for relative motion: Ψ = P : (5) j i j i j reli As a result, the two-body problem has become an effective one-body problem. From now on, we are only concerned with rel , so we will drop the \rel" most of the time. 42 j i Scattering theory Incoming plane wave exp(ikz) R = 0 Beam direction +z Outgoing spherical waves exp(ikR)/R Fig. 3.1. A plane wave in the +z direction incident on a spherical target, giving Figure 2: Scatteringrise problem to spherically-outgoing for an incident scattering plane waves wave in the +z direction on a spherical target (in the text r is used instead of R). [From F. Nunes notes.] 3.1.1 Partial wave scattering from a finite spherical potential Thus we can considerWe start relative our development motion with of scattering total P theory= 0 (see by finding Fig. 1) the and elastic describe scat- elastic scattering 0 0 from a potential fromtering incoming from a potentialk to outgoingV (R) thatk iswith sphericallyk = symmetrick and E and= sok can2=(2 beµ) (with ~ = 1). We written as V (R). Finite potentials will bej dealtj withj j first: those for which describe this quantumV (R mechanically)=0forR R in, wheretermsR ofis an the incoming finite range plane of the wave potential. and This an outgoing scattered ≥ n n (\sc") wave: excludes Coulomb potentials, which will be dealt with later. ik·r We will(+) examine the solutions at positive energye of the time-independent (r) = in(r) + sc(r) = + sc(r) ; (6) Schr¨odingerE equation with this potential, and(2 showπ)3 how=2 to find the scattering amplitude f(θ, φ) and hence the differential cross section σ(θ, φ)= f(θ, φ) 2 | | which (assuming V fallsfor elastic off with scattering.r faster We will than use 1 a=r decomposition), means that in parti faral way waves fromL=0, the 1, potential the wave , and the spherical nature of the potential will mean that each partial function has the asymptotic··· form wave function can be found separately. The time-independent Schr¨odinger equation for the relativikre motion with (+) e c.m. energy E(r,) from Eq.(2 (2.3.18),π)−3=2 is eik·r + f(k; θ; φ) ; (7) E r−!!1 r [Tˆ + V E]ψ(R, θ, φ)=0, (3.1.1) − where the scatteringusing angle polarθ is coordinates given by (θ cos, φ)θ such= k^ thatk^0z.= AR schematiccos θ, x = R picturesin θ cos φ isand shown in Fig. 2 with · ˆ k = kz^. The scatteringy = amplitudeR sin θ sin φ.f Inmodulates Eq. (3.1.1), the the kinetic outgoing energy spherical operator T waveuses according the to direction reduced mass µ, and is and carries all the physics information. !2 Tˆ = 2 The differential cross section is calculated−2µ∇R from 1 !2 ∂ ∂ Lˆ2 = R2 + , (3.1.2) 2 dσ number of particles2µ − scatteringR2 ∂R ∂ intoR dRΩ2 per unit time Ssc;rr (k; θ; φ) = ! " # $ = ; (8) dΩ number of incident particles per unit area and time Sin where Ssc;r and Sin are the scattered and incoming probability current densities, respectively. Scattering Theory 1 (last revised: September 5, 2014) 3{3 Taking k z without loss of generality (and with ~ = 1 and suppressing common (2π)3 factors), / ∗ 1 d −ikz 1 ikz k Sin = Reb in = Re e ike ; (9) in iµ dz iµ / µ 1 d e−ikr 1 eikr 1 k 1 S = Re ∗ = Re f ∗ ikf + f 2 + ; (10) sc;r sc iµ dr sc r iµ r O r3 / µr2 j j O r3 so the differential cross section is dσ (k; θ; φ) = f(k; θ; φ) 2 : (11) dΩ j j If we do not consider spin observables (where the spin orientation of at least one of the particles is known|polarized), then f(k; θ; φ) is independent of φ; we consider only this case for now and drop the φ dependence. We expand the wave function in spherical coordinates as 1 1 u (r) u (r) (r; θ) = c l Y (θ; φ) = c~ l P (cos θ) ; (12) l r l0 l r l Xl=0 Xl=0 where because there is no φ dependence, only the m = 0 spherical harmonic Y (θ; φ) = θ; φ l; m = l l0 h j l 0 appears and we can use Y = 2l+1 P (cos θ). The radial function u (r) satisfies the radial i l0 4π l l Schr¨odingerequation (if there is noq mixing of different l values), d2u l(l + 1) l + 2µV k2 u (r) = 0 : (13) dr2 − r2 − l We have freedom in choosing the normalization of ul, which we will exploit below. For a central potential V (we'll come back to non-central potentials in the section on nuclear forces!), we resolve the scattering amplitude f into decoupled partial waves (the definition of fl sometimes has a extra k factor): 1 f(k; θ) = (2l + 1)fl(k)Pl(cos θ) : (14) Xl=0 [We can do a partial wave expansion even if the potential is not central, which means the potential does not commute with the orbital angular momentum (as in the nuclear case!); it merely means that different l's will mix to some degree (in the nuclear case, this includes triplet S and D waves). The more important question is how many total l's do we need to include to ensure convergence of observables of interest.] Using the plane wave expansion of the incoming wave, 1 1 l+1 −ikr ikr ik·r ikr cos θ l ( 1) e + e e = e = (2l + 1)i jl(kr)Pl(cos θ) (2l + 1)Pl(cos θ) − ; (15) r−!!1 2ikr Xl=0 Xl=0 where we have used the asymptotic form of the spherical Bessel function jl: π l ikr l −ikr sin(kr l 2 ) 1 ( i) e i e jl(kr) − = − − ; (16) r−!!1 kr kr 2i Scattering Theory 1 (last revised: September 5, 2014) 3{4 we can resolve the asymptotic wave function into incoming (e−ikr) and outgoing (eikr) spherical waves (so now effectively a one-dimensional interference problem): 1 l+1 −ikr ikr (+) −3=2 ( 1) e + Sl(k)e (r) (2π) (2l + 1)Pl(cos θ) − ; (17) E r−!!1 2ikr Xl=0 which identifies the partial wave S-matrix Sl(k) = 1 + 2ik fl(k) : (18) [Note: in scattering theory you will encounter many closely related functions with different nor- malizations, often defined just for convenience and/or historical reasons. Be careful that you use consistent conventions!] So part of eikr is the initial wave and part is the scattered wave; the latter defines the scattering amplitude, which is proportional to the on-shell T-matrix (see below). The conservation of probability for elastic scattering implies that S (k) 2 = 1 (the S-matrix is unitary). j l j The real phase shift δl(k) is introduced to parametrize the S-matrix: iδl(k) 2iδl(k) e Sl(k) = e = ; (19) e−iδl(k) (the second equality is a trivial consequence but nevertheless is useful in manipulating scattering equations) which leads us to write the partial-wave scattering amplitude fl as S (k) 1 eiδl(k) sin δ (k) 1 f (k) = l − = l = : (20) l 2ik k k cot δ (k) ik l − We will see the combination k cot δl many times.
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