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

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Outline

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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

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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

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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

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Lévy path integral and fractional Schrödinger equation

Propagator:

fractional path integral measure:

(1)

(2)

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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)

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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

• 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)

(11)

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Dirac delta potential

• Consider free particle V = 0 with delta perturbation

• Propagator for a free particle (Laskin, 2000)

(10)

(11)

• Green’s function

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Dirac delta potential

Eigenenergies can be determined from:

(12)

(13)

Hence, we have the following

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Dirac delta potential

Solving for the energy yields

(12)

(13)

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where β(m,n) is a Beta function ( Re(m),Re(n) > 0 )

This can be rewritten in the following manner

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Dirac delta potential

Solving for the energy yields

(12)

(13)

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where β(m,n) is a Beta function ( Re(m),Re(n) > 0 )

This can be rewritten in the following manner

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Infinite square well with delta - perturbation

• Propagator for an infinite square well (Dong, 2013)

• Green’s function

(12)

(13)

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Infinite square well with delta - perturbation

• Green’s function for the perturbed system

(14)

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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

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References

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References

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The end.

Thank you.

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