Summary
An algorithm for the solution of the Schrödinger equation in a discrete basis is illustrated with reference to the problem of quantization on spheres of any dimension (hyperquantization). It exploits the explicit construction of discrete analogs of spherical harmonics and leads to sparse matrix representations of the kinetic energy operator and a diagonal representation of the interaction potential. Applications are discussed for inelastic and reactive scattering.
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Lill JV, Parker GA, Light JC (1982) Chem Phys Lett 89:483; Heather RW, Light JC (1983) J Chem Phys 79:147. See Muckerman JT (1990) Chem Phys Lett 173:200 for recent developments
Aquilanti V, Cavalli S, Grossi G (1986) J Chem Phys 85:1362
Aquilanti V, Grossi G (1985) Lett Nuovo Cimento 42:157
Aquilanti V, Beneventi L, Grossi G, Vecchiocattivi F (1988) J Chem Phys 89:751
Goldberger ML, Watson KM (1964) Collision theory. Wiley, New York, p 882
Jacob M, Wick GC (1959) Ann Phys 7:404
Aquilanti V, Grossi G (1980) J Chem Phys 73:1165; Aquilanti V, Grossi G, Laganà A (1981) Nuovo Cimento 63b:7
Chang ES, Fano U (1972) Phys Rev A6:173
Vilenkin NY (1968) Special functions and the theory of group representations. Am Math Soc, Providence, R.I.
Smorodinskii YA, Suslov SK (1982) Sov J Nucl Phys 35:108; (1982) 36:623 and references therein; Smirnov YF, Suslov SK, Shirokov AM (1984) J Phys A17:2157; Atakishiyev NM, Suslov SK (1985) J Phys A18:1583; Pluhar Z, Smirnov YF, Tolstoy VN (1986) J Phys A19:21
Rose ME (1957) Elementary theory of angular momentum. Wiley, New York; Edmonds AR (1960) Angular momentum in quantum mechanics. Princeton, New Jersey; Brink DM, Satchler GR (1968) Angular momentum. Clarendon, Oxford; Varshalovic DA, Moskalev AN, Khersonskii VK (1988) Quantum theory of angular momentum. World Scientific, Singapore; Zare RN (1988) Angular momentum. Wiley, New York
Hildebrand FB (1956) Introduction to numerical analysis. McGraw-Hill, New York, p 287
Abramowitz M, Stegun IA (eds) (1965) Handbook of mathematical functions. Dover, New York, p 790
Smith FT (1960) Phys Rev 120:1058
Aquilanti V, Grossi G, Laganà A (1982) J Chem Phys 76:1587
Fano U (1981) Phys Rev A24:2402
Arthurs AM, Dalgarno A (1960) Proc Roy Soc A256:540
Alder K, Bohr A, Huus T, Mottelson B, Winther A (1956) Rev Mod Phys 28:432; Brussaard PJ, Tolhoek HA (1957) Physica 23:955; Schulten K, Gordon RG (1975) J Math Phys 16:1971
Khare V (1978) J Chem Phys 68:4361
Dickinson AS, Certain PR (1968) J Chem Phys 49:4209
Aquilanti V, Grossi G, Laganà A (1982) Chem Phys Lett 93:174; Aquilanti V, Cavalli S, Laganá A (1982) Chem Phys Lett 93:179; Aquilanti V, Cavalli S (1987) Chem Phys Lett 141:309; Aquilanti V, Cavalli S, Grossi G, Anderson RW (1990) J Chem Soc Faraday Trans 86:1681
Walker RB, Light JC (1975) Chem Phys 7:84
Launay JM (1976) J Phys B9:1823
Erdelyi A (1953) Higher trascendental functions, vol 2. McGraw-Hill, New York; Karlin S, McGregor JL (1961) Scripta Math 26:33
Kil'dyushov MS (1972) Sov J Nucl Phys 15:113
Aquilanti V, Cavalli S, Grossi G, Anderson RW (in preparation)
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Aquilanti, V., Cavalli, S. & Grossi, G. Discrete analogs of spherical harmonics and their use in quantum mechanics: The hyperquantization algorithm. Theoret. Chim. Acta 79, 283–296 (1991). https://doi.org/10.1007/BF01113697
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DOI: https://doi.org/10.1007/BF01113697