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Continuum Mechanics and Thermodynamics

, Volume 31, Issue 3, pp 639–667 | Cite as

A new class of exact solutions of the Schrödinger equation

  • E. E. PerepelkinEmail author
  • B. I. Sadovnikov
  • N. G. Inozemtseva
  • A. A. Tarelkin
Original Article

Abstract

The aim of this paper is to find the exact solutions of the Schrödinger equation. As is known, the Schrödinger equation can be reduced to the continuum equation. In this paper, using the nonlinear Legendre transform the equation of continuity is linearized. Particular solutions of such a linear equation are found in the paper, and an inverse Legendre transform is considered for them with subsequent construction of solutions of the Schrödinger equation. Examples of the classical and quantum systems are considered.

Keywords

Exact solution Schrödinger equation Legendre transform Nonlinear partial differential equation 

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Notes

References

  1. 1.
    Styer, D.F., Balkin, M.S., Becker, K.M., Burns, M.R., Dudley, C.E., Forth, S.T., Gaumer, J.S., Kramer, M.A., Oertel, D.C., Park, L.H., Rinkoski, M.T., Smith, C.T., Wotherspoon, T.D.: Nine formulations of quantum mechanics. Am. J. Phys. 70(3), 288–297 (2002)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics, vol. 3, p. 677. Pergamon Press, Oxford (1977)Google Scholar
  3. 3.
    Perepelkin, E.E., Sadovnikov, B.I., Inozemtseva, N.G.: The properties of the first equation of the Vlasov chain of equations. J. Stat. Mech. 2015, P05019 (2015).  https://doi.org/10.1088/1742-5468/2015/05/P05019 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Perepelkin, E.E., Sadovnikov, B.I., Inozemtseva, N.G.: \(\Psi \)-model of micro- and macrosystems. Ann. Phys. 383, 511–544 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Scott, T.C., Zhang, W.: Efficient hybrid-symbolic methods for quantum mechanical calculations. Comput. Phys. Commun. 191, 221–234 (2015)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Busch, T.: Two cold atoms in a harmonic trap. Found. Phys. 27(4), 549–559 (1998).  https://doi.org/10.1023/A:1018705520999 CrossRefGoogle Scholar
  7. 7.
    Simpao, V.A.: Real wave function from Generalised Hamiltonian Schrodinger Equation in quantum phase space via HOA (Heaviside Operational Ansatz): exact analytical results. J. Math. Chem. 52(4), 1137–1155 (2014).  https://doi.org/10.1007/s10910-014-0332-2. ISSN 0259-9791MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Biedenharn, L.C., Rinker, G.A., Solem, J.C.: A solvable approximate model for the response of atoms subjected to strong oscillatory electric fields. J. Opt. Soc. Am. B 6(2), 221–227 (1989)ADSCrossRefGoogle Scholar
  9. 9.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables I and II. Phys. Rev. 85, 166–193 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bohm, D., Hiley, B.J., Kaloyerou, P.N.: An ontological basis for the quantum theory. Phys. Rep. 144, 321–375 (1987)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bohm, D., Hiley, B.J.: The Undivided Universe: An Ontological Interpretation of Quantum Theory. Routledge, London (1993)Google Scholar
  12. 12.
    de Broglie, L.: Une interpretation causale et non lineaire de la mecanique ondulatoire: la theorie de la double solution, Gauthiers-Villiars, Paris (1956). Elsevier, Amsterdam (1960). (English translation)Google Scholar
  13. 13.
    Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  14. 14.
    Dawson, J.: On landau damping. Phys. Fluids 4, 869 (1961)ADSCrossRefGoogle Scholar
  15. 15.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics: Partial Differential Equation, vol. 2. Wiley, New York (1962)zbMATHGoogle Scholar
  16. 16.
    Zhidkov, E.P., Perepelkin, E.E.: An analytical approach for quasi-linear equation in secondary order. CMAM 1(3), 285–297 (2001)zbMATHGoogle Scholar
  17. 17.
    Perepelkin, E.E., Sadobnikov, B.I., Inozemtseva, N.G.: Solutions of nonlinear equations of divergence type in domains having corner points. J. Elliptic Parabol. Equ. 4, 107–139 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Al-Salam, W.A.: Operational representations for Laguerre and other polynomials. Duke Math J. 31(1), 127–142 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Koepf, W.: Identities for families of orthogonal polynomials and special functions. Integral Transforms Spec. Funct. 5, 69–102 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • E. E. Perepelkin
    • 1
    Email author
  • B. I. Sadovnikov
    • 1
  • N. G. Inozemtseva
    • 2
  • A. A. Tarelkin
    • 1
  1. 1.Faculty of PhysicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Dubna State UniversityMoscow Region, MoscowRussia

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