Functional Integral Method for Potential Scattering Amplitude in Quantum Mechanics
Main Article Content
Abstract
The functional integral method can be used in quantum mechanics to find the scattering amplitude for particles in the external field. We obtained the potential scattering amplitude from the complete Green function in the corresponding external field through solving the Schrodinger equation, after being separated from the poles on the mass shell, which takes the form of an eikonal (Glauber) representation in the high energy region and the small scattering angles. Considering specific external potentials such as the Yukawa or Gaussian potential, we found the corresponding differential scattering cross-sections.
Keywords:
Eikonal scattering theory, effective theory of quantum gravity, quasi-potential equation and modified perturbation theory.
References
We are grateful to thank Profs. Ha Huy Bang, Dang Van Soa and Tran Minh Hieu for his suggestion of the problem and many useful comments. This work was supported funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 103.01-2021.02. DTH is supported in part by the project 911 of Hanoi University of Science - VNU HN.
References
[1] B. M. Barbashov, Functional Intergrals in Quantum Electrodynamics and Infrared Asymptotics of Green Function, Soviet Physics JEFT, Vol. 21, No. 2, 1965, pp. 402-410,
http://jetp.ras.ru/cgi-bin/dn/e_021_02_0402.pdf (accessed on: February 6th, 2022).
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Theoret. and Math. Phys., Vol. 3, No. 3, 1970, pp. 555-562, https://doi.org/10.1103/PhysRev.186.1656.
[3] H. M. Fried, Basics of Functional Methods and Eikonal Models, World Scientific Publishing, ISBN 10: 2863320858, ISBN 13: 9782863320853, 31 Dec 1990, 320 pages.
[4] B. M. Barbashov, V. V. Nesterenko, Lectures on Eikonal Approximation for High Energy Scattering, Dubna, Preprint, JINR.USSR, 1978.
[5] R. J. Glauber, Lectures on Theoretical Physics, In: W. E. Brittin and L. G. Dunham, Eds., Interscience, New York, Vol. 1, 1959, pp. 315-414,
https://archive.org/stream/in.ernet.dli.2015.177153/2015.177153.Lectures-In-Theoretical-Physics-Vol-1_djvu.txt (accessed on: February 6th, 2022).
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Vol. 4, 1973, pp. 623-661.
[8] M. L. Goldberger, K. M. Watson, Collision Theory, Wiley J. & Son Inc., 1964, ISBN-10: 0486435075, ISBN-13: 978-0486435077.
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[10] Ng. S. Han, V. V. Nesterenko, High Energry Scattering of the Composite Particle in the Functional Approach, Journal of Theor. and Math. Phys, Vol. 24, No. 2, 1975, pp. 768-775, https://doi.org/10.1007/BF01029059.
[11] Ng. S. Han, V. V. Nesterenko, Bremsstrahlung Approximation for Inclusive Processes, Theor Math Phys, Vol. 27, 1976, pp. 323-327, https://doi.org/10.1007/BF01036548.
[12] Ng. S. Han, V. N. Pervushin, High Energy Scattering of Particles with Anomalous Magnetic Moment in Quantum Field Theory, Journal of Theor. And Math. Phys, Vol. 29, No. 2, 1976, pp. 1003-1011, https://doi.org/10.1007/BF01108503.
[13] A. I. Baz’, Y. B. Zel’dovich, A. M. Perelomov, Scattering, Reactions and Decays in a Non Relativistic Quantum Mechanics, Nauka, Moscow 1971, pp. 225-231, http://archive.org/details/in.ernet.dlis/in.ernet.dli.2015.177153/ Page/ n153/mode/2up (accessed on: February 6th, 2022).
[14] N. S. Han, E. Ponna, Straight-line Path Approximation for Studying Planckian-Energy Scattering in Quantum Gravity, Il Nuovo Cimento A, Vol.110, 1997, pp. 459-473, https://doi.org/10.1007/BF03035893,
[15] N. S. Han, Straight-Line Path Approximation for the High-Energy Elastic and Inelastic Scattering in Quantum Gravity, European Physical Journal C, Vol. 16, No. 3, 2000, pp. 547-553, https://doi.org/10.1007/s100520000328.
[16] N. S. Han, N. N. Xuan, Planckian Scattering Beyond Eikonal Scattering in Quantum Gravity, European Physical Journal C, Vol. 24, No. 1, 2002, pp. 643-651, http://doi.org/10.48550/arXiv.0804.3432.
[17] N. S. Han, L. H. Yen, N. N. Xuan, High Energy Scattering in the Quasi-Potential Approach, International Journal of Modern Physics A, Vol. 27, No. 1, 2012, pp. 1250004-1250023, http://doi.org/10.1142/S0217751X12500042.
[18] N. S. Han, L. A. Dung, N. N. Xuan, V. T. Thang, High Energy Scattering of Dirac Particles on Smooth Potentials, International Journal of Modern Physics A, Vol. 31, No. 23, 2016, pp. 1650126-1650144, http://doi.org/10.1142/S0217751X16501268.
[19] N. S. Han, D. T. Ha, N. N. Xuan, The Contribution of Effective Quantum Gravity to High Energy Scattering in the Framework of Modified Perturbation Theory and One Loop Approximation, Eur. Phys. J. C, Vol. 79, Iss. 10, 2019, pp.835-848, https://doi.org/10.1140 /epjc/s 10052-019-7355-6.
References
[1] B. M. Barbashov, Functional Intergrals in Quantum Electrodynamics and Infrared Asymptotics of Green Function, Soviet Physics JEFT, Vol. 21, No. 2, 1965, pp. 402-410,
http://jetp.ras.ru/cgi-bin/dn/e_021_02_0402.pdf (accessed on: February 6th, 2022).
[2] B. M. Barbashov, S. P. Kuleshov, V. A. Matveev, V. N. Pervushin, A. N. Sissakian, A. N. T. Eikonal Approximation in Quantum Field Theory, Teor. Mat. Fiz., Vol. 3, 1970, pp. 342-352,
Theoret. and Math. Phys., Vol. 3, No. 3, 1970, pp. 555-562, https://doi.org/10.1103/PhysRev.186.1656.
[3] H. M. Fried, Basics of Functional Methods and Eikonal Models, World Scientific Publishing, ISBN 10: 2863320858, ISBN 13: 9782863320853, 31 Dec 1990, 320 pages.
[4] B. M. Barbashov, V. V. Nesterenko, Lectures on Eikonal Approximation for High Energy Scattering, Dubna, Preprint, JINR.USSR, 1978.
[5] R. J. Glauber, Lectures on Theoretical Physics, In: W. E. Brittin and L. G. Dunham, Eds., Interscience, New York, Vol. 1, 1959, pp. 315-414,
https://archive.org/stream/in.ernet.dli.2015.177153/2015.177153.Lectures-In-Theoretical-Physics-Vol-1_djvu.txt (accessed on: February 6th, 2022).
[6] A. N. Sissakian, A. N. Tavkhelidze, Straight-Line Paths Approximation for Studying High-Energy Elastic and Inelastic Hadron Collisions in Quantum Field Theory, Phys.Lett., Vol. 33B, Iss. 7, 1907, pp. 484-488, http://doi.org/10/1016/0370-2693(70)90225-X.
[7] B. M. Barbashov, D. I. Blokhintsev, V. V. Nestere, V. N. Pervushin, Optical Model of Strong Interactions and Eikonal Approximation in Scattering Theory, Soviet Journal Particles and Nuclei, (Fiz El. Cht Atom Yad),
Vol. 4, 1973, pp. 623-661.
[8] M. L. Goldberger, K. M. Watson, Collision Theory, Wiley J. & Son Inc., 1964, ISBN-10: 0486435075, ISBN-13: 978-0486435077.
[9] V. N. Pervushin, Method of Functional Integration and an Eikonal Approximation for Potential Scattering Amplitudes, Theor Math Phys, Vol. 4, No. 1, 1970, pp. 643-651, https://doi.org/10.1007/BF01246663.
[10] Ng. S. Han, V. V. Nesterenko, High Energry Scattering of the Composite Particle in the Functional Approach, Journal of Theor. and Math. Phys, Vol. 24, No. 2, 1975, pp. 768-775, https://doi.org/10.1007/BF01029059.
[11] Ng. S. Han, V. V. Nesterenko, Bremsstrahlung Approximation for Inclusive Processes, Theor Math Phys, Vol. 27, 1976, pp. 323-327, https://doi.org/10.1007/BF01036548.
[12] Ng. S. Han, V. N. Pervushin, High Energy Scattering of Particles with Anomalous Magnetic Moment in Quantum Field Theory, Journal of Theor. And Math. Phys, Vol. 29, No. 2, 1976, pp. 1003-1011, https://doi.org/10.1007/BF01108503.
[13] A. I. Baz’, Y. B. Zel’dovich, A. M. Perelomov, Scattering, Reactions and Decays in a Non Relativistic Quantum Mechanics, Nauka, Moscow 1971, pp. 225-231, http://archive.org/details/in.ernet.dlis/in.ernet.dli.2015.177153/ Page/ n153/mode/2up (accessed on: February 6th, 2022).
[14] N. S. Han, E. Ponna, Straight-line Path Approximation for Studying Planckian-Energy Scattering in Quantum Gravity, Il Nuovo Cimento A, Vol.110, 1997, pp. 459-473, https://doi.org/10.1007/BF03035893,
[15] N. S. Han, Straight-Line Path Approximation for the High-Energy Elastic and Inelastic Scattering in Quantum Gravity, European Physical Journal C, Vol. 16, No. 3, 2000, pp. 547-553, https://doi.org/10.1007/s100520000328.
[16] N. S. Han, N. N. Xuan, Planckian Scattering Beyond Eikonal Scattering in Quantum Gravity, European Physical Journal C, Vol. 24, No. 1, 2002, pp. 643-651, http://doi.org/10.48550/arXiv.0804.3432.
[17] N. S. Han, L. H. Yen, N. N. Xuan, High Energy Scattering in the Quasi-Potential Approach, International Journal of Modern Physics A, Vol. 27, No. 1, 2012, pp. 1250004-1250023, http://doi.org/10.1142/S0217751X12500042.
[18] N. S. Han, L. A. Dung, N. N. Xuan, V. T. Thang, High Energy Scattering of Dirac Particles on Smooth Potentials, International Journal of Modern Physics A, Vol. 31, No. 23, 2016, pp. 1650126-1650144, http://doi.org/10.1142/S0217751X16501268.
[19] N. S. Han, D. T. Ha, N. N. Xuan, The Contribution of Effective Quantum Gravity to High Energy Scattering in the Framework of Modified Perturbation Theory and One Loop Approximation, Eur. Phys. J. C, Vol. 79, Iss. 10, 2019, pp.835-848, https://doi.org/10.1140 /epjc/s 10052-019-7355-6.