δ perturbed infinite square well Mary Madelynn Nayga and Jose Perico Esguerra Theoretical Physics Group National Institute of Physics University of the Philippines Diliman Outline January 6 2014 ID: 289450
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Slide1
Lévy path integral approach to the fractional Schrödinger equation with δ-perturbed infinite square well
Mary Madelynn Nayga and Jose Perico Esguerra
Theoretical Physics Group
National Institute of Physics
University of the Philippines DilimanSlide2
Outline
January 6, 2014
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7th Jagna International Workshop
Introduction
Lévy
path integral and fractional Schrödinger
equation
Path integration via summation of perturbation
expansions
Dirac delta
potential
Infinite square well with delta - perturbation
Conclusions and possible work
externsionsSlide3
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 January 6, 20144
7th Jagna International WorkshopSlide5
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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7th Jagna International WorkshopSlide8
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, 201487th Jagna International WorkshopSlide9
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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7th Jagna International Workshop
where
β
(
m,n
)
is a Beta function (
Re(m),Re(n) > 0 )This can be rewritten in the following mannerSlide14
Dirac delta potential
Solving for the energy yields
(12)
(13)
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7th Jagna International Workshop
where
β
(
m,n
)
is a Beta function (
Re(m),Re(n) > 0 )This can be rewritten in the following mannerSlide15
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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