Green Function on a Quantum Disk for the Helmholtz Problem |

B. Benali
^{ a}, B. Boudjedaa^{ b} and M.T. Meftah^{c}^{a}Mathematical Department, Biskra University, 07000 and El-Oued University, 39000, Algeria
^{b}Mathematical Department, Jijel University, 18000, Algeria
^{c}Physics Department, LRPPS Laboratory, UKM Ouargla, 30000 Algeria |

Received: March 9, 2013; In final form: May 17, 2013 |

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In this work, we present a new result which concerns the derivation of the Green function relative to the time-independent SchrÃ¶dinger equation in two-dimensional space. The system considered in this work is a quantal particle that moves in an axi-symmetric potential. At first, we have assumed that the potential V(r) to be equal to a constant V_{0} inside a disk (radius a) and to be equal to zero outside the disk. We have used, to derive the Green function, the continuity of the solution and of its first derivative, at r=a (at the edge). Secondly, we have assumed that the potential V(r) is equal to zero inside the disk and is equal to V_{0} outside the disk (the inverted potential). Here, also we have used the continuity of the solution and its derivative to obtain the associate Green function showing the discrete spectra of the Hamiltonian. |

DOI: 10.12693/APhysPolA.124.636 PACS numbers: 03.75.Lm, 03.65.Db, 02.30.Gp, 02.30.Sa |