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Mary Madelynn Nayga and Jose Perico Esguerra Theoretical Physics Group

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. Introduction

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Mary Madelynn Nayga and Jose Perico Esguerra Theoretical Physics Group

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  1. 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

  2. Outline 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 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 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 7th Jagna International Workshop

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

  6. 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) 7th Jagna International Workshop

  7. Lévy path integral and fractional Schrödinger equation 1D space-fractional Schrödinger equation: (5) Reisz fractional derivative operator: (6) 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) 7th Jagna International Workshop

  9. Path integration via summation of perturbation expansions • Introduce time-ordering operator, (8) • Consider delta perturbations (9) 7th Jagna International Workshop

  10. Path integration via summation of perturbation expansions • Energy-dependent Green’s function • unperturbed system • perturbed system (10) (11) 7th Jagna International Workshop

  11. Dirac delta potential • Consider free particle V = 0 with delta perturbation • Propagator for a free particle (Laskin, 2000) (10) • Green’s function (11) 7th Jagna International Workshop

  12. Dirac delta potential • Eigenenergies can be determined from: (12) • Hence, we have the following (13) 7th Jagna International Workshop

  13. 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) 7th Jagna International Workshop

  14. 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) 7th Jagna International Workshop

  15. Infinite square well with delta- perturbation • Propagator for an infinite square well (Dong, 2013) (12) • Green’s function (13) 7th Jagna International Workshop

  16. Infinite square well with delta- perturbation • Green’s function for the perturbed system (14) 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 7th Jagna International Workshop

  18. References 7th Jagna International Workshop

  19. References 7th Jagna International Workshop

  20. The end. Thank you. 7th Jagna International Workshop

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