Green Function on a Quantum Disk for the Helmholtz Problem

B. Benali a, B. Boudjedaa b and M.T. Meftahc aMathematical Department, Biskra University, 07000 and El-Oued University, 39000, Algeria bMathematical Department, Jijel University, 18000, Algeria cPhysics 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 Schrödinger 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 V0 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 V0 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