ψ( ) k (r) which take the asymtotic form far away from the scattering center: k (r) = E kψ (±) φ k (r) = e ikr

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1 Scattering Theory Consider scattering of two particles in the center of mass frame, or equivalently scattering of a single particle from a potential V (r), which becomes zero suciently fast as r. The initial state is, and the nal state after scattering is. The scattering matrix (S-matrix) describes probabilities that scattering events between dierent pairs of channels happen: S = ψ ( ) ψ(+) Here, the states are plane waves, while the states ψ ( ) and ψ(+) are incomming and outgoing spherical waves (see below). S-matrix is a unitary matrix due to orthonormality conditions: = ψ ( ) ψ( ) = ψ (+) ψ(+) = ()3 δ( ). The incomming and outgoing states are solutions to the full Schrodinger equation Hψ (±) (r) = (H 0 + V (r))ψ (±) (r) = E ψ (±) (r) which tae the asymtotic form far away from the scattering center: where ψ (±) (r) = φ (r) + f(ω) e ±ir r φ (r) = e ir The function f(ω) of the solid angle is called scattering amplitude. It can be related to the scattering matrix in the following way. Start from the Lippmann-Schwinger equation: ψ (±) (r) = φ (r) + E H 0 ± i0 + V ψ(±) (r) which can be easily derived from the full Schrodinger equation and Hφ = E φ. By recursive substitution of the left-hand side into the right-hand side, one obtains: ) ( ) ψ (±) ( = + E H ± i0 + V P = + E H V iπδ(e H)V Thus: ψ (+) ψ( ) = iδ(e H)V Now, using this expression we nd that the S-matrix taes form: S = ψ ( ) ψ(+) = iδ(e E ) ψ (+) = iδ(e E )T where the T-matrix is dened by In the operator form: T = V ψ (+) S = iδ(e E )T Next, we explicitly rewrite the Lippmann-Schwinger equation in the coordinate representation. To this end, we need to express the operator (E H 0 ± i0 + ) in the coordinate representation (E = 2 /2m): E H 0 ± i0 + = d d 3 rd 3 r 3 e i (r r ) r () 3 E 2 /2m ± i0 + r = d 3 rd 3 r r 2mei r r 4π r r r Substituting this into the Lippmann-Schwinger equation, and approximating r r r r cos θ for distances r much larger than the range of the scattering potential (θ is the angle between r and r ), we nd:

2 2 ψ (+) (r) = φ (r) 2meir 4πr d 3 r e i r V (r )ψ (+) (r ) Here, the vector has the same magnitude as, but a dierent direction - the same as r, which is that of the incident wave. Therefore, we can identify the scattering amplitude as: f(ω) = m d 3 r φ (r )V (r )ψ (+) (r ) = m V ψ (+) = m T Finally, the relationship between the scattering matrix and the scattering amplitude is: S = () 3 δ( ) iδ(e E ) ( m ) f(ω) Using δ( ) = ( 2 sin θ) δ( )δ(θ θ)δ(φ φ) and δ(e E ) = m δ( ) we can write: ( S = () 3 δ( ) δ(θ θ)δ(φ φ) 2 + i ) sin θ f(ω) which facilitates derivation of the fowrard-scattering matrix elements - these will be diagonal elements of the S-matrix. The diagonal S-matrix elements are readily uncovered by integrating out, θ and φ at Ω = 0 (this removes the Dirac-function normalization and isolates the magnitude of the matrix elements): S = ( + 2if(0)) δ(0) At the end, it is worth noting that unitarity of the S-matrix implies the following equation for the T-matrix: from which follows the optical theorem (see below) T T = iδ(e f E i )T T Scattering Cross-Section In scattering, a dierential cross section is dened by the probability to observe a scattered particle in a given quantum state per solid angle unit, such as within a given cone of observation, if the target is irradiated by a ux of one particle per surface unit. Again, for a single-particle scattering on a potential, the asymptotic wavefunction is: ψ(r) e iz + f(ω) e ir r where the rst term is an incomming plane-wave, and the second term is an outgoing radial wave, the scattered s-wave. The scattering amplitude f(ω) is related to the dierential scattering cross-section (by denition): dσ dω = f(ω) 2 The scattering amplitude for a spherically symmetric potential can be expanded in powers of the angular momentum l: f(θ) = (2i) l=0 (2l + )(e 2iδ l )P l (cos θ) where δ l are phase shifts (phase dierences between the scattered waves at angular momenta l and the incident wave at innity), and P l (x) are Legendre polynomials.

3 3 The total cross-section is the solid-angle integral of the dierential cross-section: and satises the optical theorem: σ = dω dσ dω = 4π 2 (2l + )(e 2iδ l ) l=0 σ = 4π Im{f(0)} These expressions establish the following relationships: dσ ( m ) 2 dω = T 2 S = δ iδ(e E )T = δ + i ()2 m δ(e E )f(ω) Re{S } = 2Im{f(0)} = 2 σ Scattering on a hard-sphere; scattering length If the scattering potential is innite for r < a and zero for r > a, then two limits can be identied. For a (ν /a), (a) 2l+ tan δ l (2l + )((2l )!!) 2 f(θ) σ ν i 2 ν 2 cos2 ( θ 2 ) 4πa 2 (a) 2 ( + 2 cos θ) This limit is obtained for a general short-ranged (decreasing with r fast enough) potential at suciently small momenta. It can be used as a denition of the scattering length a even when potentials are not hardcore. Perhaps these expressions are also used to dene the scattering length in general in atomic physics; depending on the scattering physics, f(ω) derived from the S-matrix may give a negative, or arbitrarily large value for a, and this is controlled by the detuning ν from the Feshbach resonance. For a, σ a 2 (twice the classical result due to diraction eects) Scattering on a quantum well; resonant scattering and unitarity limit A quantum well potential represents attractive interactions between particles in the center of mass reference frame, and can be idealized by V (r) = V 0 for r < a and V (r) = 0 for r > a (with V 0 > 0). Wavenumber of a free particle with momentum will be α = α while inside the quantum well, where α 0 = 2mV 0 (we set = ). After solving for the phase shifts δ l the following approximate results for the scattering cross-section are obtained. In the limit a : ( σ 4πa 2 tan(α ) ( a) 4πa 2 tan(α ) 0a) α a α 0 a

4 4 The scattering cross-section is mostly independent of in this limit, but by changing V 0 the quantity α 0 a can approach an odd-integer multiple of π/2 and give rise to a resonance. The expression for σ has to be rederived then, whenever the energy of an incomming particle is close to the energy of a bound (or metastable) state in the quantum well. For α 0 a π/2 or so (the partial wave l = 0 in resonance with a bound state at zero energy): σ 4π 2 + α0 2 cot(α 0a) 4π 2 This is the situation in which the scattering cross-section acquires a signicant momentum dependence, and loses dependence on the microscopic scales that characterize the scattering potential. The microscopic details of the interactions between scattering particles cease to matter at the resonance, and a degree of universality is obtained, called the unitarity limit. The forward S-matrix in the unitarity limit is: if(0) = Im{f(0)} = σ 4π S = ( + 2if(0)) δ(0) = δ(0) Note that the scattering amplitude f(0) is purely imaginary at the resonance, reecting the tendency of the attractive potential to bind the scattering particle. This is formally shown by calculating the l = 0 phase shift δ l, which turns out to be π/2 exactly at the resonance (if other values of l do not lie exactly at the resonance, their contribution to σ will be negligible for 0 compared to the contribution of the l = 0 wave). If one insisted to dene the scattering length even in the vicinity of the resonance, which is not necessarily a natural thing to do, then one could use the expressions for the hard-sphere scattering as denitions of the scattering length a and nd that it diverges (ν = /a 0): f(θ) ν i i This is measured in ultra-cold atom experiments as a function of magnetic eld H by tting to the following empirical formula: a = ( ν = a bg H ) w H H 0 where a bg is the bacground (far-o resonance) scattering length, and H w the resonance width (both measured experimentally; H 0 is a eld that characterizes the microscopic aspects of fermion interactions). The width of the resonance is characterized by a parameter γ ( F r 0 ), where r 0 is an eective range (of the pairing interactions?). Scattering in one dimension In one spatial dimension the previous denitions of the scattering cross-section mae no sense, but it is still possible to dene the S-matrix and scattering amplitude. Consider a potential barrier V (x) = V 0 for 0 < x < a, and V (x) = 0 otherwise (tae V 0 > 0). An incomming plane wave partially reects and partially passes through the barrier, which we can write asymptotically: { } ψ(x) = e ix f+ e + ix, x f e ix, x The solution of the Schrodinger equation gives: f + = 4α e i(α )a ( + α ) 2 ( α ) 2 e 2iα a

5 ( 2 α 2 ) ( e 2iα a ) f = ( + α ) 2 ( α ) 2 e 2iα a where α = 2m(E V 0 ) and E = 2 /2m. When α (low energies - barrier bottom), or α (energies close to barrier top, but only in the limit V 0 so that ), and sin(α a) 0, a universal regime is obtained: f +, f corresponding to full bacward scattering. At high energies α, so that f + e i(α )a and f 0, meaning that only forward scattering occurs albeit with a phase shift (α )a a amv 0 /.