A Fractional Schrödinger Equation and Its Solution

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 α. 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.

[1]  Erwin Schrödinger,et al.  Über das Verhältnis der Heisenberg‐Born‐Jordanschen Quantenmechanik zu der meinem , 1926 .

[2]  M. Naber.  Time fractional Schrödinger equation , 2004, math-ph/0410028.

[3]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[4]  Mingyu Xu,et al.  Generalized fractional Schrödinger equation with space-time fractional derivatives , 2007 .

[5]  Dumitru Baleanu,et al.  The Hamilton formalism with fractional derivatives , 2007 .

[6]  O. Agrawal,et al.  A fractional Dirac equation and its solution , 2010 .

[7]  Dumitru Baleanu,et al.  Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives , 2005 .

[8]  Dumitru Baleanu,et al.  Lagrangian Formulation of Classical Fields within Riemann-Liouville Fractional Derivatives , 2005 .

[9]  I. Podlubny.  Fractional differential equations , 1998 .

[10]  Edward Nelson.  Derivation of the Schrodinger equation from Newtonian mechanics , 1966 .

[11]  Malgorzata Klimek,et al.  Fractional sequential mechanics — models with symmetric fractional derivative , 2001 .

[12]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[13]  George M. Zaslavsky.  Hamiltonian Chaos and Fractional Dynamics , 2005 .

[14]  Nick Laskin.  Lévy flights over quantum paths , 2005 .

[15]  N. Laskin,et al.  Fractional quantum mechanics , 2008, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Om P. Agrawal,et al.  Formulation of Euler–Lagrange equations for fractional variational problems , 2002 .

[17]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[18]  Michael J. W. Hall,et al.  Schrödinger equation from an exact uncertainty principle , 2001 .

[19]  Mingyu Xu,et al.  Some solutions to the space fractional Schrödinger equation using momentum representation method , 2007 .

[20]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[21]  Riewe,et al.  Nonconservative Lagrangian and Hamiltonian mechanics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  J. W. Humberston.  Classical mechanics , 1980, Nature.