Abstract
We consider the dynamics of a quantum mechanical system which consists of some particles with large masses and some particles with small masses. As we increase the large masses to infinity we obtain the following results: The particles of smaller mass move adiabatically and determine an effective potential in which the heavier particles move semiclassically. Our methods can be applied to diatomic molecules with Coulomb forces.
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References
Birkhoff, G., Rota, G.-C.: Ordinary differential equations. London, Toronto, Waltham, Mass.: Blaisdell Publishing Company 1962
Born, M., Oppenheimer, R.: Zur Quantentheorie der Molekeln. Ann. Phys. (Leipzig)84, 457–484 (1927)
Combes, J. M.: On the Born-Oppenheimer approximation. In: International symposium on mathematical problems in theoretical physics (ed. H. Araki), pp. 467–471. Berlin, Heidelberg, New York: Springer 1975
Combes, J. M.: The Born-Oppenheimer approximation. In: The Schrödinger equation (eds. W. Thirring, P. Urban), pp. 139–159. Wien, New York: Springer 1977
Combes, J. M., Seiler, R.: Regularity and asymptotic properties of the discrete spectrum of electronic Hamiltonians. Int. J. Quantum Chem.14, 213–229 (1978)
Eyring, H.: The activated complex and the absolute rate of chemical reactions. Chem. Rev.17, 65–77 (1935)
Hagedorn, G. A.: Semiclassical quantum mechanics for coherent states. In: The proceedings of the international conference on mathematical physics (ed. K. Osterwalder). Berlin, Heidelberg, New York: Springer (to appear 1980)
Hagedorn, G. A.: Semiclassical quantum mechanics I: Theh → 0 limit for coherent states. Commun. Math. Phys.71, 77–93 (1980)
Hagedorn, G. A.: Semiclassical quantum mechanics II: the large mass asymptotics for coherent states. In: Trends in applications of pure mathematics to mechanics Vol. III (ed. R. J. Knops). London: Pitman Publishing Ltd. (to appear 1980)
Heller, E. J.: Classical S-matrix limit of wave-packet dynamics. J. Chem. Phys.65, 4979–4989 (1976)
Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys.35, 265–277 (1974)
Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn.5, 435–439 (1950)
Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966
London, F.: Über den Mechanismus der homoöpolaren Bindung. In: Probleme der modernen Physik (ed. P. Debye). Leipzig: S. Hirzel 1928
Nenciu, G.: On the adiabatic theorem of quantum mechanics. Preprint, Central Institute of Physics, Institute for Physics and Nuclear Engineering, Bucharest, Romania 1979
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. I: functional analysis. New York, London: Academic Press 1972
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. III: scattering theory. New York, London: Academic Press 1979
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. IV: analysis of operators. New York, London: Academic Press 1978
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Communicated by B. Simon
Supported in part by the National Science Foundation under Grant PHY 78-08066
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Hagedorn, G.A. A time dependent Born-Oppenheimer approximation. Commun.Math. Phys. 77, 1–19 (1980). https://doi.org/10.1007/BF01205036
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DOI: https://doi.org/10.1007/BF01205036