Green function of quantum particle moving in two-dimensional annular potential

Brahim Benali ; Said Douis ; Mohammed Tayeb Meftah

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 5, pp. 598-610.

Résumé

In this work, we present a new result which concerns the obtainment of the Green function relative to the time-independent Schrodinger equation in two dimensional space. The system considered in this work is a particle that have an energy E and moves in an axi-symmetrical potential. Precisely, we have assumed that the potential ($V(r)$), in which the particle moves, to be equal to zero inside an annular region (radius b) and to be equal a positive constant ($V_{0}$) in a crown of internal radius b and external radius a ($b$) and equal zero outside the crown ($r>a$). We have explored the bounded states regime for which ($E$). We have used, to obtain the Green function, the continuity of the solution and of its derivative at ($r=b$) and ($r=a$): We have obtained the associate Green function and the discrete spectra of the Hamiltonian in the region ($r$).

Détail

Keywords: quantum mechanics, Schrodinger equation, Green's function, bounded states.

Bibliographie

[1] G.Green, An Essay on the application of Mathematical analysis to the theory of electricity and magnetism, in privately at the author's expense, Nottingham, 1828; G. Green, Cambridge Edited by N. M. Ferrers, Gonville and Caius College, M.A., 1871

[2] H. Von Helmholtz, Die Lehre von den Tonempfindungenals Physiologische Grundlagefur die Theorie der Musik, Sexhste Ausgabe Mit Dem Bildnis des Verfasffers und 66 Abbildungen, Springer Fachmedien Wiesbaden Gmbh, 1913

[3] Y.A.Melnikov, “Green's function of a thin circular plate with elastically supported edge”, Eng. Anal. Bound. Elem., 25 (2001), 669–676 | DOI | Zbl

[4] E.Pan, “General time-dependent Green's functions of line forces in a two-dimensional, anisotropic, elastic, and infinite solid”, Eng. Anal. Bound. Elem., 124:201, 174–184 | DOI | MR | Zbl

[5] J.Kidawa-Kukla, “Application of the Green's function method to the problem of thermally induced vibration of a circular plate”, Scientific Research of the Institute of Mathematics and Computer Science, 9 (2010), 53–60

[6] J.Kidawa-Kukla, “Temperature distribution in an annular plate with a moving discrete heat generation source”, Scientific Research of the Institute of Mathematics and Computer Science, 8 (2009), 77–84

[7] S.Kukla, U.Siedlecka, I.Zamorska, “Green's functions for interior and exterior Helmholtz problems”, Scientific Research of the Institute of Mathematics and Computer Science, 11 (2012), 53–62 | DOI

[8] S.K.Adhikari, “Quantum scattering in two dimensions”, American Journal of Physics, 54 (1986), 362 | DOI | MR

[9] R.Layeghnejad, M.Zare, R.Moazzemi, “Dirac particle in a spherical scalar potential well”, Phys. Rev. D, 84 (2011), 125026 | DOI

[10] N.Yoshio, “Finite depth spherical well model for excited states of ultra small semiconductor particles: an application”, J. Phys. Chem., 95 (1991), 5054–5058 | DOI

[11] I.M.Nemenman, A.S.Vernon JrSilbergleit, A.S.Silbergleit, “Explicit Green's function of a boundary value problem for a sphere and trapped flux analysis in Gravity Probe B experiment”, J. Appl. Phys., 86 (1999), 614–624 | DOI

[12] C.G.Neumann, “Revision einigerallgemeiner Satzeaus der Theorie des Newton'schen Potentiales”, Math. Ann., 3 (1871), 424–434 | DOI | MR

[13] Y.Nosaka, “Finite depth spherical well model for excited states of ultrasmall semiconductor particles: an application”, J. Chem. Phys., 95 (1991), 5054–5058 | DOI

[14] Y.Tago, K.Narasimha Swamy, “Mean spherical model for the square-well potential”, Phys. Lett. A, 45 (1973), 37–38 | DOI

[15] R.Jean-Pierre, “A very elementary proof of the Malgrange-Ehrenpreis theorem”, The American Mathematical Monthly, 98 (1991), 518–523 | DOI | MR | Zbl

[16] M.Krasnov, A.Kiselev, G.Makarenko, Problems and Exercices in Integral equations, Edition Mir, M., 1971 | MR