APPENDIX Fo the lectues of Claude Cohen-Tannoudji on Atom-Atom Inteactions in Ultacold Quantum Gases
Pupose of this Appendix Demonstate the othonomalization elation(ϕ ϕ = δ k k δ δ )k - The wave function u ( ) ϕ ( ) = Y ( θϕ, ) l m k ll mm k π descibes, in the angula momentum epesentation, a paticle of mass μ, with enegy E=ħ k /μ, in a cental potential V() - The adial wave function u () is a egula solution of () d μ () ()()() + + k V u 0 V V tot = = + l tot d (μ )k u 0 = 0 which behaves, fo, as: )sin/(u k lπ (+ δ k - Thee ae othe (non egula) solutions behaving, fo, as: )exp/()exp( )l u i ( k lπ ( ) = i ik ) l (A.) (A.) (A.3) (A.4) (A.5) (A.6)
( Calculate the Geen function of: = p /μ + V )H with outgoing and ingoing asymptotic behavio ), ( H) G ( ) = δ ( ) E = k /E (μ - Show that: ), (exp()*(,)(,)()( μ G ( ) = i δ Y θ ϕ Y θ ϕ u u k whee > ( < ) is the lagest (smallest) of and - Intoducing the Heaviside function: θ = + if > ( ) = 0 if < (A.8) can also be witten: ), (exp()*(,)(,) μ G ( ) = i δ Y θ ϕ Y θ ϕ k ()()()()l θ ( ) u u + θ ( ) u u < > )l 3 Calculate the asymptotic behavio of these Geen functions and demonstate Equation (.39) of Lectue (A.7) (A.8) (A.9) (A.0) 3
Wonskian Theoem The calculations pesented in this Appendix use the Wonskian theoem (see demonstation in Ref. Chapte III-8) - Conside the D second ode diffeential equation: )()( + F y )y = 0(Equation (A.4) is of this type with: ) μ ( = k V tot( )F )() - Let and y be solutions of this equation coesponding to ()(),y diffeent functions F and F espectively. The wonskian of y and y is by definition:(,)()()()(y y = y y y y )W (- One can show that:,)(,)(,)w y y = a W y y W y y = b = a b ()()()()b = F F y y d( a (A.) (A.) (A.3) (A.4) 4
Demonstation of (A.) We conside diffeent values k and k of k. Accoding to (A.): )( F )F = k (k (A.5) (A.4) then gives the scala poduct of y = u and y = u ()() (,)b b y y d = W y y (A.6) a a,(,)k k If we take a = 0 W y y 0 because of (A.4) = = a (), If we take b = R vey lage compaed to the ange of V we can use the asymptotic behavio (A.5) of u and u )() ()()()()R u u d( u u u u = 0 (A.7) k k = R ),()Using (A.5) and putting δ k δ δ δ we get:( 0 ()()u u d k l nr = k = l,l in ( ) π δ δ k + k R s+ + = + k + k si ( ) + δ δ k k R + k k (A.8) 5
- When R, the fist tem of the ight side of (A.8) vanishes as a distibution, because it is a apidly oscillating function of k +k (k and k being both positive k +k cannot vanish) - The second tem becomes impotant when k -k is close to zeo (we have then δ -δ =0) -Using: we get: i =sπ nlimrx x ()R δ x π (δ u u d = k k 0 l )k (- We then have, accoding to (A.): )()*3 ( )( ) (,)(,)()()d ϕ ϕ = dωy θ ϕ Y θ ϕ u u d k l m k l m k l π * (= δ k which demonstates (A.). = δ δ ll mm )k δ δ ll mm π ()= δ k k (A.9) (A.0) (A.) 6
Demonstation of (A.8) Let us apply E-H to the ight side of (A.8). Using (A.0) and: ) μ ()L H = Δ + V = V μ (μ we get, using (A.): ),(exp()*(,)(,) E H G = i δ Y θ ϕ Y θ ϕ k () ()()()()l F + θ ( ) u u + θ ( ) u u (A.3) To calculate the second line of (A.3), we use: ( ) ( ) ( θ ) = θ ( ) = δ ( ) )()() δ ( ) f = f δ ( ) + f δ ( ) ( (A.) (A.4) The second ode deivative (of the )second (line )of (A.3) (gives )3 types of tems: (popotional )/to θ and θ, to δ and to δ 7
()()/()( - The tems ae multiplied by ) θ F + u () which vanishes because u is a solution of (A.3). ()The same agument applies fo the tems θ which ae ()/ multiplied by F + ( 0() u = ) )- The tems popotional to δ / cancel out(- The only tems suviving (in the )second line of (A.3) ae those popotional to δ which gives fo this )()/()()/(line:(), ( u u ) u ( u ) δ - We ecognize in the backet of (A.5) the Wonskian of u and We can thus use (A.4) with F = F since u and u coespond. to the same value of k (A.5) - Equation (A.4) shows that the Wonskian is independant of when F = F We can thus calculate it fo vey lage values of whee we.know the asymptotic behavio (A.5) and (A.6) of u and u u 8
- The calculation of the Wonskian appeaing in (A.5) is staightfowad using (A.5) and (A.6) and gives:,)exp(+ W u u = k iδ l ) - Inseting (A.6) into (A.5) and then in (A.3) gives: ),*(()(,)(, (( E H) G ( ) = δ Y θ ϕ Y θ ϕ ) - We can then use the closue elation fo the spheical hamonics (A.6) (A.7) (see Ref. 3, Complement (AVI):*Y,)(,)(coscos)θ ϕ Y θ ϕ = δ θ θ δ ϕ ϕ to obtain: H G = ),(( ) ( ) ()(coscos)() δ δ θ θ δ ϕ ϕ ( )E = δ which demonstates (A.8). () (A.8) (A.9) 9
Asymptotic behavio of G + Fo vey lage, only the fist tem of the backet of (A.0) is non zeo and we )get:,* (,)(,)()(+ μ iδ + )l G ( ) e(y θ ϕ Y θ ϕ u u k Accoding to (A.6), we have e),* ()(,)(,)(ik + μ l iδ )l G ( ) i e(y θ ϕ Y θ ϕ u k On the othe hand, fom Eq. (.46) of lectue and (A.), we have: *()() ()exp()()()u l ϕ = i iδ Y n Y n n = n = kn l k π Using (A.3), we can ewite (A.3) as: ()*, μ π ( G + ( ) ϕ kn )ik which demonstates Eq. (.39) of lectue. e (A.30) (A.3) (A.3) (A.33) 0