Geodesic
Quantum Coulomb problem in a Gaussian time-dependent electric field within the path integral formalism
Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 1, pp. 111-120 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We solve the Coulomb problem in nonrelativistic quantum mechanics with a charge depending on a parameter that plays the role of time. The choice of this dependence is needed, for example, after certain spatio–temporal transformations when dealing with the interaction of a “small” system (quantum sub-system) with a “large” system, e.g., a bath. These spatio–temporal transformations, combined with path integral, allow us to find the Feynman propagator of the quantum subsystem. To test our approach, we derive the pure Coulomb Green's function as a limit of our result.
Keywords: path integral, Green's function, modified Coulomb potential.
Mots-clés : perturbation series 
@article{TMF_2023_215_1_a5,
     author = {N. Bedida and S. Fadhel and M. Difallah and M. Meftah},
     title = {Quantum {Coulomb} problem in {a~Gaussian} time-dependent electric field within the~path integral formalism},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {111--120},
     year = {2023},
     volume = {215},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a5/}
}
TY  - JOUR
AU  - N. Bedida
AU  - S. Fadhel
AU  - M. Difallah
AU  - M. Meftah
TI  - Quantum Coulomb problem in a Gaussian time-dependent electric field within the path integral formalism
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 111
EP  - 120
VL  - 215
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a5/
LA  - ru
ID  - TMF_2023_215_1_a5
ER  - 
%0 Journal Article
%A N. Bedida
%A S. Fadhel
%A M. Difallah
%A M. Meftah
%T Quantum Coulomb problem in a Gaussian time-dependent electric field within the path integral formalism
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 111-120
%V 215
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a5/
%G ru
%F TMF_2023_215_1_a5
N. Bedida; S. Fadhel; M. Difallah; M. Meftah. Quantum Coulomb problem in a Gaussian time-dependent electric field within the path integral formalism. Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 1, pp. 111-120. http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a5/

[1] P. A. M. Dirak, Printsipy kvantovoi mekhaniki, Nauka, M., 1979 | MR | MR | Zbl

[2] H. R. Griem, A. C. Kolb, R. Y. Chen, “Stark broadening of hydrogen lines in a plasma”, Phys. Rev., 116:1 (1959), 4–15 | DOI

[3] N. G. Van Kampen, Stokhasticheskie protsessy v fizike i khimii, Vysshaya shkola, M., 1990 | MR | MR

[4] J. W. Dufty, private communication, 2022

[5] L. Chetouani, L. Guechi, T. F. Hammann, “Generalized canonical transformations and path integrals”, Phys. Rev. A, 40:3 (1989), 1157–1164 | DOI | MR

[6] J. G. Hartley, J. R. Ray, “Solutions to the time-dependent Schrödinger equation”, Phys. Rev. A, 25:4 (1982), 2388–2390 | DOI | MR

[7] A. K. Dhara, S. V. Lawande, “Time-dependent invariants and the Feynman propagator”, Phys. Rev. A, 30:1 (1984), 560–567 | DOI | MR

[8] A. Mustafazadeh, On a class of quantum canonical transformations and the time-dependent harmonic oscillator, arXiv: quant-ph/9612038

[9] C. Grosche, “Path integrals for potential problems with $\delta$-function perturbation”, J. Phys. A: Math. Gen., 23:22 (1990), 5205–5234 ; “$\delta$-Function perturbations and Neumann boundary conditions by path integration”, 28:3 (1995), L99–Ll05 | DOI | MR | DOI | MR

[10] D.-H. Lin, “Green's function for the relativistic Coulomb system via sum over perturbation series”, J. Phys. A: Math. Gen., 31:37 (1998), 7577–7584 | DOI | MR

[11] K. V. Bhagwat, S. V. Lawande, “Path integral treatment of Coulomb potential by exact summation of a perturbation series”, Phys. Lett. A, 135:8–9 (1989), 417–420 | DOI

[12] D. C. Khandekar, S. V. Lawande, K. V. Bhagwat, Path-integral Methods and Their Applications, World Sci., Singapore, 1993 | MR

[13] C. Grosche, “Path integration via summation of perturbation expansions and applications to totally reflecting boundaries, and potential steps”, Phys. Rev. Lett., 71:1 (1993), 1–4 | DOI

[14] R. P. Feinman, A. Khibbs, Kvantovaya mekhanika i integraly po traektoriyam, Mir, M., 1968 | MR | Zbl

[15] L. S. Schulman, Techniques and Application of Path Integration, Wiley, New York, 1981 | MR

[16] D. A. McQuarrie, Statistical Mechanics, Harper and Row, New York, 1976

[17] R. Roychoudhury, Y. P. Varshni, M. Sengupta, “Family of exact solutions for the Coulomb potential perturbed by a polynomial in $r$”, Phys. Rev. A, 42:1 (1990), 184–192 | DOI | MR

[18] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, ryadov i proizvedenii, BKhV-Peterburg, SPb., 2011 | MR | MR | Zbl | Zbl