A fractional Schrödinger equation and its solution. (English) Zbl 1197.81126

Summary: This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order \(\alpha\). We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
70H03 Lagrange’s equations
49S05 Variational principles of physics
Full Text: DOI


[1] Schrödinger, E.: Quantisierung als Eigenwert Problem (Erste Mitteilung). Ann. Phys. 79, 734 (1926) · JFM 52.0966.03
[2] Nelson, E.: Quantum Fluctuations. Princeton University Press, Princeton (1985) · Zbl 0563.60001
[3] Nelson, E.: Derivation of Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079 (1966)
[4] Hall, M.J.W., Reginatto, M.: Schrödinger equation from an exact uncertainty principle. J. Phys. A 35, 3289 (2002) · Zbl 1045.81003
[5] Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) · Zbl 0292.26011
[6] Miller, K.S., Ross, B.: An Introduction to the Fractional Integrals and Derivatives-Theory and Applications. Wiley, New York (1993) · Zbl 0789.26002
[7] Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002
[8] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[9] Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005) · Zbl 1083.37002
[10] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[11] Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Reading (2006)
[12] Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153 (2001) · Zbl 1054.35156
[13] Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890 (1996)
[14] Klimek, M.: Fractional sequential mechanics-models with symmetric fractional derivative. Czechoslov. J. Phys. 51, 1348 (2001) · Zbl 1064.70507
[15] Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368 (2002) · Zbl 1070.49013
[16] Muslih, S.I., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. J. Math. Anal. Appl. 304, 599 (2005) · Zbl 1149.70320
[17] Baleanu, D., Muslih, S.I.: Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scr. 72(2–3), 119 (2005) · Zbl 1122.70360
[18] Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891 (2007) · Zbl 1104.70012
[19] Dong, J.P., Xu, M.Y.: Some solutions to the space fractional Schrödinger equation using momentum representation method. J. Math. Phys. 48, 072105 (2007) · Zbl 1144.81341
[20] Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45, 3339 (2004) · Zbl 1071.81035
[21] Wang, S.W., Xu, M.Y.: Generalized fractional Schrödinger equation with space-time fractional derivatives. J. Math. Phys. 48, 043502 (2007) · Zbl 1137.81328
[22] Laskin, N.: Lévy flights over quantum paths. Commun. Nonlinear. Sci. Numer. Simul. 12, 2 (2007) · Zbl 1101.81079
[23] Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62, 3135 (2000) · Zbl 0948.81595
[24] Muslih, S.I., Agrawal, O.P., Baleanu, D.: A fractional Dirac equation and its solution. J. Phys. A, Math. Gen. 45(3), 055203 (2010) · Zbl 1185.81078
[25] Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading (1980) · Zbl 0491.70001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.