Geodesic
Approximate analytical method for solving the schrödinger equation with the Gaussian potential in the momentum representation
Problemy fiziki, matematiki i tehniki, no. 3 (2020), pp. 18-21.

Voir la notice de l'article provenant de la source Math-Net.Ru

The approximate analytical method is proposed for solving the Schrödinger equation with the Gaussian potential in the momentum representation. The idea of the method is to represent the desired wave function in the form of a superposition of wave functions of the three-dimensional harmonic oscillator in the momentum representation. Solutions are found in the case of coupled states.
Keywords: Schrödinger equation, wave function, Gaussian potential, harmonic oscillator, coupled states. 
@article{PFMT_2020_3_a2,
     author = {Yu. A. Grishechkin and A. V. Paulenko},
     title = {Approximate analytical method for solving the schr\"odinger equation with the {Gaussian} potential in the momentum representation},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {18--21},
     publisher = {mathdoc},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2020_3_a2/}
}
TY  - JOUR
AU  - Yu. A. Grishechkin
AU  - A. V. Paulenko
TI  - Approximate analytical method for solving the schrödinger equation with the Gaussian potential in the momentum representation
JO  - Problemy fiziki, matematiki i tehniki
PY  - 2020
SP  - 18
EP  - 21
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PFMT_2020_3_a2/
LA  - ru
ID  - PFMT_2020_3_a2
ER  - 
%0 Journal Article
%A Yu. A. Grishechkin
%A A. V. Paulenko
%T Approximate analytical method for solving the schrödinger equation with the Gaussian potential in the momentum representation
%J Problemy fiziki, matematiki i tehniki
%D 2020
%P 18-21
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PFMT_2020_3_a2/
%G ru
%F PFMT_2020_3_a2
Yu. A. Grishechkin; A. V. Paulenko. Approximate analytical method for solving the schrödinger equation with the Gaussian potential in the momentum representation. Problemy fiziki, matematiki i tehniki, no. 3 (2020), pp. 18-21. http://geodesic.mathdoc.fr/item/PFMT_2020_3_a2/

[1] Z. Flyugge, Zadachi po kvantovoi mekhanike, v 2 t., v. 1, 3-izd., LKI, M., 2010, 344 pp.

[2] G. Stephenson, “Eigenvalues of the Schrodinger equation with a Gaussian potential”, J. Phys. A: Math Gen., 10 (1977), L229–L232 | DOI | MR | Zbl

[3] C.S. Lai, “On the Schrodinger equation for the Gaussian potential $-A\mathrm{exp}(-\lambda r^2)$”, J. Phys. A: Math Gen., 16 (1983), L181–L185 | DOI | MR | Zbl

[4] R.E. Crandale, “Fast eigenvalue algorithm for central potentials”, J. Phys. A: Math Gen., 16 (1983), L395–L399 | DOI | MR

[5] A. Chatterjee, “$1 / N$ expansion for Gaussian potential”, J. Phys. A: Math. Gen., 18 (1985), 2403–2408 | DOI | MR

[6] S.S. Gomez, R.H. Romero, “Few-electron semiconductor quantum dots with Gaussian confinement”, Central Eur. J. Phys., 7 (2009), 12–21

[7] K. Koksal, “A simple analytical expression for bound state energies for an attractive Gaussian confining potential”, Phys. Scr., 86 (2012), 035006 | DOI | Zbl

[8] V.V. Kudryashov, A.V. Baran, “Reshenie radialnogo uravneniya Shredingera dlya potentsiala Gaussa v modifitsirovannom VKB-priblizhenii”, Doklady Natsionalnoi akademii nauk Belarusi, 57:5 (2013), 43–48 | Zbl

[9] H. Mutuk, “Asymptotic iteration and variational methods for Gaussian potential”, Pramana J. Phys., 2019, 66 | DOI

[10] Yu.A. Grishechkin, A.V. Pavlenko, V.N. Kapshai, “Ob odnom priblizhennom analiticheskom metode resheniya uravneniya Shredingera s gaussovym potentsialom”, Problemy fiziki, matematiki i tekhniki, 2019, no. 4 (41), 7–10

[11] A.A. Logunov, A.N. Tavkhelidze, “Quasi-Optical Approach in Quantum Field Theory”, Nuovo Cimento, 29:2 (1963), 380–399 | DOI | MR

[12] V.G. Kadyshevskii, R.M. Mir-Kasimov, N.B. Skachkov, “Trekhmernaya formulirovka relyativistskoi problemy dvukh tel”, EChAYa, 2:3 (1972), 635–690 | MR

[13] G. Arfken, Matematicheskie metody v fizike, Atomizdat, M., 1970, 712 pp.

[14] I.S. Gradshtein, I.M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvodnykh, Izd. 7-e, BKhV-Peterburg, SPb., 2011, 1232 pp.