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Lévy path integral approach to the fractional Schrödinger
equation with δ-perturbed infinite square well
Mary Madelynn Nayga and Jose Perico EsguerraTheoretical Physics Group
National Institute of PhysicsUniversity of the Philippines Diliman
7th Jagna International Workshop 2
Outline
January 6, 2014
I. IntroductionII. Lévy path integral and fractional Schrödinger equationIII. Path integration via summation of perturbation expansionsIV. Dirac delta potentialV. Infinite square well with delta - perturbationVI. Conclusions and possible work externsions
7th Jagna International Workshop 3
Introduction
• Fractional quantum mechanics first introduced by Nick Laskin (2000) space-fractional Schrödinger equation (SFSE) containing the Reisz
fractional derivative operator path integral over Brownian motions to Lévy flights time-fractional Schrödinger equation (Mark Naber) containing the
Caputo fractional derivative operator space-time fractional Schrödinger equation (Wang and Xu)
• 1D Levy crystal – candidate for an experimental realization of space-fractional quantum mechanics (Stickler, 2013)
•Methods of solving SFSE piece-wise solution approach momentum representation method Lévy path integral approach
January 6, 2014
7th Jagna International Workshop 4
Introduction
• Objectives
use Lévy path integral method to SFSE with perturbative terms
follow Grosche’s perturbation expansion scheme and obtain energy-dependent Green’s function in the case of delta perturbations
solve for the eigenenergy of consider a delta-perturbed infinite square well
January 6, 2014
7th Jagna International Workshop 5
Lévy path integral and fractional Schrödinger equation
Propagator:
fractional path integral measure:
(1)
(2)
January 6, 2014
7th Jagna International Workshop 6
Lévy path integral and fractional Schrödinger equation
Levy probability distribution function in terms of Fox’s H function
Fox’s H function is defined as
(3)
(4)
January 6, 2014
7th Jagna International Workshop 7
Lévy path integral and fractional Schrödinger equation
1D space-fractional Schrödinger equation:
Reisz fractional derivative operator:
(5)
(6)
January 6, 2014
7th Jagna International Workshop 8
Path integration via summation of perturbation expansions
• Follow Grosche’s (1990, 1993) method for time-ordered perturbation expansions
• Assume a potential of the form
• Expand the propagator containing Ṽ(x) in a perturbation expansion about V(x)
(7)
January 6, 2014
7th Jagna International Workshop 9
Path integration via summation of perturbation expansions
• Introduce time-ordering operator,
• Consider delta perturbations
(8)
(9)
January 6, 2014
7th Jagna International Workshop 10
Path integration via summation of perturbation expansions
•Energy-dependent Green’s function• unperturbed system
• perturbed system
(10)
(11)
January 6, 2014
7th Jagna International Workshop 11
Dirac delta potential
• Consider free particle V = 0 with delta perturbation
• Propagator for a free particle (Laskin, 2000)
(10)
(11)
• Green’s function
January 6, 2014
7th Jagna International Workshop 12
Dirac delta potential
Eigenenergies can be determined from:
(12)
(13)
Hence, we have the following
January 6, 2014
7th Jagna International Workshop 13
Dirac delta potential
Solving for the energy yields
(12)
(13)
January 6, 2014
where β(m,n) is a Beta function ( Re(m),Re(n) > 0 )
This can be rewritten in the following manner
7th Jagna International Workshop 14
Dirac delta potential
Solving for the energy yields
(12)
(13)
January 6, 2014
where β(m,n) is a Beta function ( Re(m),Re(n) > 0 )
This can be rewritten in the following manner
7th Jagna International Workshop 15
Infinite square well with delta - perturbation
• Propagator for an infinite square well (Dong, 2013)
• Green’s function
(12)
(13)
January 6, 2014
7th Jagna International Workshop 16
Infinite square well with delta - perturbation
• Green’s function for the perturbed system
(14)
January 6, 2014
7th Jagna International Workshop 17
Summary
• present non-trivial way of solving the space fractional Schrodinger equation with delta perturbations
• expand Levy path integral for the fractional quantum propagator in a perturbation series
• obtain energy-dependent Green’s function for a delta-perturbed infinite square well
January 6, 2014