Mary Madelynn. Nayga

Report
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 Diliman
Outline
I.
II.
III.
IV.
V.
VI.
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 externsions
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7th Jagna International Workshop
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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 spacefractional 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:
(1)
fractional path integral measure:
(2)
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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:
(5)
Reisz fractional derivative operator:
(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,
(8)
• Consider delta perturbations
(9)
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Path integration via summation of perturbation
expansions
•Energy-dependent Green’s function
• unperturbed system
(10)
• perturbed system
(11)
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Dirac delta potential
• Consider free particle V = 0 with delta perturbation
•Propagator for a free particle (Laskin, 2000)
(10)
• Green’s function
(11)
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Dirac delta potential
Eigenenergies can be determined from:
(12)
Hence, we have the following
(13)
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Dirac delta potential
Solving for the energy yields
(12)
where β(m,n) is a Beta function ( Re(m),Re(n) > 0 )
This can be rewritten in the following manner
(13)
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Dirac delta potential
Solving for the energy yields
(12)
where β(m,n) is a Beta function ( Re(m),Re(n) > 0 )
This can be rewritten in the following manner
(13)
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Infinite square well with delta - perturbation
• Propagator for an infinite square well (Dong, 2013)
(12)
• Green’s function
(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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