{"title":"An Expansion Method for Schr\u00f6dinger Equation of Quantum Billiards with Arbitrary Shapes","authors":"\u0130nci M. Erhan","volume":5,"journal":"International Journal of Physical and Mathematical Sciences","pagesStart":255,"pagesEnd":259,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/1846","abstract":"A numerical method for solving the time-independent Schr\u00f6dinger equation of a particle moving freely in a three-dimensional\r\naxisymmetric region is developed. The boundary of the region\r\nis defined by an arbitrary analytic function. The method uses a\r\ncoordinate transformation and an expansion in eigenfunctions. The\r\neffectiveness is checked and confirmed by applying the method to a\r\nparticular example, which is a prolate spheroid.","references":"[1] M. Sieber and F. Steiner, Classical and quantum mechanics of a strongly\r\nchaotic billiard system, Physica D, vol.44, pp. 248-266, August 1990.\r\n[2] M. V. Berry, Quantizing a classically ergodic system: Sinai-s billiard and\r\nthe KKR method, Ann.Phys., vol. 131, pp. 163-216, 1981.\r\n[3] D. A. McGrew and W. Bauer, Constraint operator solution to quantum\r\nbilliard problem, Phys. Rev. E, vol. 54, pp. 5809-5818, November 1996.\r\n[4] M. Sieber and F. Steiner, Quantum chaos in the hyperbola billiard, Phys.\r\nLett. A, vol. 148, pp. 415-420, September 1990.\r\n[5] M. Robnik, Quantizing a generic family of billiards with analytic boundaries,\r\nJ. Phys. A: Math. Gen., vol. 17, pp. 1049-1074, 1984.\r\n[6] T. Papenbrock, Numerical study of a three-dimensional generalized\r\nstadium billiard, Phys. Rev. E, vol. 61, pp. 4626-4628, April 2000.\r\n[7] H. Primak and U. Smilansky, The quantum three-dimensional Sinai\r\nbilliard - A semiclassical analysis, Phys. Rep., vol. 327, pp. 1-107,\r\nApril 2000.\r\n[8] W. A. Strauss, Partial Differential Equations, New York:\r\nJohn-Wiley, 1992.\r\n[9] H. Tas\u252c\u00a9eli, \u0130Inci M. Erhan and O\u252c\u00bf . Ug\u2566\u00ffur, An eigenfunction expansion for\r\nthe Schr\u252c\u00bfodinger equation with arbitrary non-central potentials, J. Math.\r\nChem., vol. 32, pp. 323-338, November 2002.\r\n[10] M. Abramovitz and I. A. Stegun , Handbook of Mathematical Functions,\r\nNew York:Dover, 1970.\r\n[11] S. A. Moszkowski, Particle states in spheroidal nuclei, Phys Rev. vol.\r\n99, pp. 803-809, 1955.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 5, 2007"}