Mary Madelynn Nayga and Jose Perico Esguerra
Theoretical Physics Group
National Institute of Physics
University of the Philippines Diliman

I.
Introduction
II. Lévy path integral and fractional Schrödinger equation
III. Path integration via summation of perturbation expansions
IV. Dirac delta potential
V. Infinite square well with delta - perturbation
VI. Conclusions and possible work externsions
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•
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 spacefractional quantum mechanics (Stickler, 2013)
•
Methods of solving SFSE
 piece-wise solution approach
 momentum representation method
 Lévy path integral approach
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•
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
4 January 6, 2014 7th Jagna International Workshop

Lévy path integral and fractional Schrödinger equation
Propagator: fractional path integral measure:
January 6, 2014 7th Jagna International Workshop
(1)
5
(2)

Lévy path integral and fractional Schrödinger equation
Levy probability distribution function in terms of Fox’s H function
(3)
Fox’s H function is defined as
(4)
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Lévy path integral and fractional Schrödinger equation
1D space-fractional Schrödinger equation:
Reisz fractional derivative operator:
(5)
(6)
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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)
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Path integration via summation of perturbation expansions
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Introduce time-ordering operator,
•
Consider delta perturbations
(8)
(9)
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Path integration via summation of perturbation expansions
•
Energy-dependent Green’s function
• unperturbed system
• perturbed system
(10)
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(11)

Dirac delta potential
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Consider free particle V = 0 with delta perturbation
•
Propagator for a free particle (Laskin, 2000)
• Green’s function
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(10)
11
(11)

Dirac delta potential
Eigenenergies can be determined from:
Hence, we have the following
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(12)
12
(13)

Dirac delta potential
Solving for the energy yields where
β(m,n) is a Beta function ( Re(m),Re(n) > 0 )
This can be rewritten in the following manner
January 6, 2014 7th Jagna International Workshop
(12)
13
(13)

Dirac delta potential
Solving for the energy yields where
β(m,n) is a Beta function ( Re(m),Re(n) > 0 )
This can be rewritten in the following manner
January 6, 2014 7th Jagna International Workshop
(12)
14
(13)

•
Propagator for an infinite square well (Dong, 2013)
(12)
• Green’s function
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(13)

• Green’s function for the perturbed system
(14)
January 6, 2014 7th Jagna International Workshop 16

• 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
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