The J-Matrix Method Episode 1 docx

[...]... 57, 212 9 (19 72) 3 C Schwartz, Ann Phys (NY) 16 , 36 (19 60) 4 E J Heller and H A Yamani, Phys Rev A 9, 12 01 (19 74); H A Yamani and L Fishman, J Math Phys 16 , 410 (19 75) 5 A M Lane and R G Thomas, Rev Mod Phys 30, 257 (19 58); A M Lane and D Robson, Phys Rev 17 8, 17 15 (19 69) 6 E J Heller, Phys Rev A 12 , 12 22 (19 75) 7 H A Yamani and M S Abdelmonem, J Phys A 26, L 118 3 (19 93); 27, 5345 (19 94); 28, 2709 (19 96);... first, the N − 1 conditions arising from m = 0, 1, , N − 2; second, the case for m = N − 1; third, the condition for m = N; and last, the remaining set of conditions arising from m = N + 1, N + 2, , ∞ The first case leads to the N − 1 equations N 1 N Jmn + Vmn an = 0, m = 0, 1, , N − 2 (20a) n=0 In case two, we have the equation N 1 N JN 1, n + VN 1, n an + JN 1, N n=0 cN sN t = −JN 1, N N +1 N... (19 96); 29, 69 91 (19 96) 8 E J Heller and H A Yamani, Phys Rev A 9, 12 09 (19 74) 9 I Bray and A T Stelbovics, Phys Rev Lett 69, 53 (19 92) 10 J T Broad and W P Reinhardt, J Phys B 9, 14 91 (19 76) 11 H A Yamani and M S Abdelmonem, J Phys B, 30, 16 33 (19 97); 30, 3743 (19 97) 12 P Horodecki, Phys Rev A 62, 052 716 (2000) 13 A D Alhaidari, H A Yamani, and M S Abdelmonem, Phys Rev A 63, 062708 (20 01) 14 H A Yamani,... 1) θ φn (r ) = φn (r ), n +1 n +1 n=N (19 b) ∞ cn cos(n + 1) θ φn (r ) = − φn (r ) n +1 n +1 n=N (19 c) The two forms (17 ) and (18 ) for ψ E are, of course, equivalent, but Eq (18 ) is more convenient We now proceed to verify that the (N +1) unknowns {an , t} are sufficient to determine an exact solution the Hamiltonian H0 + V N Equation (15 ) imposes a restriction on ψ E for each m, m = 0, 1, , ∞ We group these... +1 N +1 (20b) In case three, V N is no longer operative We consequently get J N,N 1 a N 1 + J N,N cN c N +1 sN s N +1 + J N,N +1 + J N,N +1 t = − J N,N N +1 N +2 N +1 N +2 which, upon using the three-term recursion relation satisfied by both cn and sn , reduces to JN,N 1 a N 1 − JN,N 1 c N 1 N t = JN,N 1 s N 1 N (20c) New L 2 Approach to Quantum Scattering: Theory 9 So far we have (N + 1) equations... |S + tC ( 21) if m ≥ N + 1 Equation ( 21) follows from the fact that V N is defined to be zero in this region of Hilbert space, and because (H0 − E) is tridiagonal in the basis {φn }, ˜ and therefore does not connect the N terms in the expansion of ⌽ or the first N ˜ ˜ terms in the expansion of S and C with φm for m ≥ N + 1 Furthermore, for each m ≥ N + 1 the right-hand side of Eq ( 21) leads to the three-term... pre-diagonalizing the inner N × N matrix H0 + V N − E nm 10 E.J Heller, H.A Yamani with the energy-independent transformation ⌫, where ˜ ⌫ H0 + V N − E ⌫ nm = (E n − E) δnm (23) Augmenting ⌫ to be the (N + 1) × (N + 1) matrix ⌫A = ⌫ 0 0 1 (24) and applying it to Eq (22), we obtain t = tan δ = sin Nθ N + ν(E) JN,N 1 sin(N + 1) θ (N + 1) cos Nθ N + ν(E) JN,N 1 cos(N + 1) θ (N + 1) , (25) N 1 where ν(E) = m=0 ⌫2 1, m... channels) are ∞ Sα = n=Nα sin(n + 1) θα (α) φn (r ) n +1 ∞ Cα = − n=Nα θα = cos 1 cos(n + 1) θα (α) φn (r ), n +1 2 kα − λ2 α 4 2 kα + (35) λ2 α 4 The internal function ⌽α is given by Nc Nα 1 ⌽α = α (α) a α n χ α φn (36) α =1 n=0 α The aα n ’s are the expansion coefficients to be determined The number of these N coefficients is α c =1 Nα The remaining unknowns Rαα are of course the elements of a reactance matrix... Quantum Scattering: Theory 15 ⌫ N 1, n = φ N 1 | ψn Note that ⌫ N 1, n and γn are both the components of the wave function ψn at the boundary of their respective inner spaces Recent work seems to indicate that the R-matrix method works best using eigenfunctions of the scattering H0 as a basis [10 ] In this basis, H0 is of course diagonal, and may be treated exactly by the addition of the Buttle correction... is the sine-like expansion χ E of Eq (10 ), and C is the ˜ cosine-like solution of Eq (13 ) The unknown coefficient t, then, corresponds to the ˜ tangent of the phase shift caused by V N Since the an ’s are yet to be determined, we ˜ ˜ can absorb the first N terms in the expansion of S and C into an ’s, writing ˜ ψ E = ⌽ + S + tC, (18 ) where N 1 an φn (r ), ⌽(r ) = n=0 ∞ S(r ) = n=N ∞ C(r ) = n=N (19 a) . provided by the J-matrix method is the explicit result for the exact S-matrix of a truncated potential: S(E) = (c N 1 −is N 1 ) + g N 1, N 1 (E)J N 1, N (E)(c N − is N ) (c N 1 +is N 1 ) + g N 1, N 1 (E)J N 1, N (E)(c N +. Scattering: Theory, Phys. Rev. A 9, 12 01 12 08, (19 74). Copyright (19 74) by the American Physical Society. http://link.aps.org/abstract/PRA/v9/p12 01 A.D. Alhaidari et al. (eds.), The J-Matrix Method, 3 17 257 (19 58); A. M. Lane and D. Robson, Phys. Rev. 17 8, 17 15 (19 69) 6. E. J. Heller, Phys. Rev. A 12 , 12 22 (19 75) 7. H. A. Yamani and M. S. Abdelmonem, J. Phys. A 26, L 118 3 (19 93); 27, 5345 (19 94);