International Science Index


1846

An Expansion Method for Schrödinger Equation of Quantum Billiards with Arbitrary Shapes

Abstract:A numerical method for solving the time-independent Schrödinger equation of a particle moving freely in a three-dimensional axisymmetric region is developed. The boundary of the region is defined by an arbitrary analytic function. The method uses a coordinate transformation and an expansion in eigenfunctions. The effectiveness is checked and confirmed by applying the method to a particular example, which is a prolate spheroid.
References:
[1] M. Sieber and F. Steiner, Classical and quantum mechanics of a strongly chaotic billiard system, Physica D, vol.44, pp. 248-266, August 1990.
[2] M. V. Berry, Quantizing a classically ergodic system: Sinai-s billiard and the KKR method, Ann.Phys., vol. 131, pp. 163-216, 1981.
[3] D. A. McGrew and W. Bauer, Constraint operator solution to quantum billiard problem, Phys. Rev. E, vol. 54, pp. 5809-5818, November 1996.
[4] M. Sieber and F. Steiner, Quantum chaos in the hyperbola billiard, Phys. Lett. A, vol. 148, pp. 415-420, September 1990.
[5] M. Robnik, Quantizing a generic family of billiards with analytic boundaries, J. Phys. A: Math. Gen., vol. 17, pp. 1049-1074, 1984.
[6] T. Papenbrock, Numerical study of a three-dimensional generalized stadium billiard, Phys. Rev. E, vol. 61, pp. 4626-4628, April 2000.
[7] H. Primak and U. Smilansky, The quantum three-dimensional Sinai billiard - A semiclassical analysis, Phys. Rep., vol. 327, pp. 1-107, April 2000.
[8] W. A. Strauss, Partial Differential Equations, New York: John-Wiley, 1992.
[9] H. Tas┬©eli, İInci M. Erhan and O┬¿ . Ug╦ÿur, An eigenfunction expansion for the Schr┬¿odinger equation with arbitrary non-central potentials, J. Math. Chem., vol. 32, pp. 323-338, November 2002.
[10] M. Abramovitz and I. A. Stegun , Handbook of Mathematical Functions, New York:Dover, 1970.
[11] S. A. Moszkowski, Particle states in spheroidal nuclei, Phys Rev. vol. 99, pp. 803-809, 1955.