Abstract
Symbolic-numeric solving of the boundary value problem for the Schrödinger equation in cylindrical coordinates is given. This problem describes the impurity states of a quantum wire or a hydrogen-like atom in a strong homogeneous magnetic field. It is solved by applying the Kantorovich method that reduces the problem to the boundary-value problem for a set of ordinary differential equations with respect to the longitudinal variables. The effective potentials of these equations are given by integrals over the transverse variable. The integrands are products of the transverse basis functions depending on the longitudinal variable as a parameter and their first derivatives. To solve the problem at high magnetic quantum numbers |m| and study its solutions we present an algorithm implemented in Maple that allows to obtain analytic expressions for the effective potentials and for the transverse dipole moment matrix elements. The efficiency and accuracy of the derived algorithm and that of Kantorovich numerical scheme are confirmed by calculating eigenenergies and eigenfunctions, dipole moments and decay rates of low-excited Rydberg states at high |m|~200 of a hydrogen atom in the laboratory homogeneous magnetic field γ~2.35×10− 5(B~6T).
This work was partially supported by the RFBR Grants Nos. 10-02-00200 and 11-01-00523.
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Gusev, A., Vinitsky, S., Chuluunbaatar, O., Gerdt, V., Le Hai, L., Rostovtsev, V. (2012). Symbolic-Numerical Calculations of High-|m| Rydberg States and Decay Rates in Strong Magnetic Fields. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2012. Lecture Notes in Computer Science, vol 7442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32973-9_13
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