Solution & visualization in The Sturm Liouville problem

Transcription

1 JASS 009 Alexandra Zykova Solution & viualization in The Sturm Liouville roblem uervier: PhD Vadim Monakhov Deartment of Comutational Phyic SPSU

2 Outline. Introduction. The regular Sturm - Liouville roblem.. The Two Center Problem in quantum mechanic. 3. SLEIGN 3. Manual for rogram ackage SLEIGN 3. SLEIGN with BARSIC

3 , '' b a x x y x w x y x q x y x λ a, b oen interval {, q, w} - coefficient defined on the oen interval a, b λ -ectral arameter 3. Introduction. The regular Sturm - Liouville roblem.

4 Boundary condition: Regular R Singular S Searated Couled Regular boundary condition: earated boundary condition 0 ' 0 ' b y B b y B a y A a y A couled boundary condition ' ' b y a y b y a y 4

5 Eigen olution of the regular Sturm - Liouville roblem - { λ, y } λ eigenvalue, for which differential equation ha nontrivial olution ψ y eigenfunction, correond to eigenvalue, atifie boundary condition 5

6 The regular Sturm - Liouville roblem & Schrödinger roblem,, '' b a x x y w x x y x q x y x λ a, b i the integration interval and the boundary condition,, '' b a x x y x y x q x y λ 6 a, b i the integration interval and the boundary condition

7 . The Two Center Problem in quantum mechanic. Δψ r ; R h m e e E Z Z ψ r ; R r r 0 Z and Z - nuclear charge r r Z and ditance between electron and, R E ER - internuclear ditance - energy term Z 7

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9 Prolate heroidal coordinate ytem r r r r ζ, η, ϕ arctan y / x R R ζ, η - coordinate of charge Z ζ, η - coordinate of charge Z 9

10 ψ ψ j kqm ζ, η, ϕ ; R N R X ζ; R Y η; R e kqm mk mq imϕ j { k, q, m} k q m - et of quantum number - rincial quantum number - orbital quantum number - magnetic quantum number * ψ kqm ξ, η, ϕ; R ψ ' ' ' ξ, η, ϕ; R dv δ k q m V kk δ ' ' qq δ ' mm 0

11 0 ; ] [ ; 0 ; ] [ ; ~ R Y m b R Y d d d d R X m a R X d d d d mq mq mk mk η η η η λ η η η η ζ ζ ζ ζ λ ζ ζ ζ ζ R Z Z b R Z Z a R E j j, 0 > - energy arameter - charge arameter, a mk ζ λ λ, ~ b mq η λ λ,, b a mq mk η ζ λ λ, - earation contant

12 Quairadial equation d d m ζ Xmk ζ; R [ λ ζ aζ ] X dζ dζ ζ X mk ; R <, X Jaffé exanion mk ζ; R ζ 0, ζ [, mk ζ; R 0 t ζ / ζ - tranformation of variable ζ t,, 0, ζ [ ; t [0; - tranformation of equilibrium oint - tranformation of interval

13 m mk g e R X ; / Σ ζ ζ ζ ζ ζ σ ζ / m a δ - three - termed relation between coefficient g 0 g g g γ β α δ δ γ λ δ δ δ β α m m m m 3

14 Eigenvalue of the roblem are found from the condition of nullifying of the continued fraction. F, a, λ α 0γ β 0... α γ β β α γ 4 β β 4 O Coefficient of three termed relation converge for all It follow from the equation above. The ratio of the erie coefficient g g O 0 > 0 rovide the convergence of Jaffé exanion on the comlete interval [0; t or ζ [ ; 4

15 Quaiangular equation 0 ; ] [ ; ~ R Y m b R Y d d d d mq mq ζ η η η λ η η η η ; ; < ± η R Y mq 5

16 Baber Hae exanion 0 ; m m mq P c e R Y η η η - relation between three term coefficient c 0 c c c δ χ ρ ] [ 3 m m b m m m m b m δ λ χ ρ 6

17 The ratio of ucceeding erie coefficient at large indexe: 0 ~ c c Solution of can be found from the equation 0...,, 0 0 k k k b F ρδ ρ δ λ η,, b a q m mk η ζ λ λ Continuou fraction converge due to following limit: χ χ δ ρ 7

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26 3. SLEIGN General-uroe rogram for comuting the eigenvalue and eigenfunction of Sturm - Liouville roblem Program SLEIGN ha been develoed by Bailey, Gordon and Shamine, rogramming language FORTRAN Code in the NAG Library ha been develoed by Pryce and Marletta, rogramming language FORTRAN Program SLEDGE ha been develoed by Fulton and Prue, rogramming language FORTRAN Program SLEIGN ha been develoed by Bailey Everitt and Zettl, rogramming language FORTRAN 6

27 To meet the need of numerical comuting technique wa made the following aumtion:. The interval a, b of R may be bounded or unbounded., q and w are real-valued function on a, b 3., q and w iecewie continuou on a, b 4. and w trictly oitive on a, b Condition on the coefficient: Minimal condition: q, Smoothne condition:, w L a, x, w x > 0 b, ', q, w C a, b x, w x > 0 7

28 Endoint claification To claify endoint a and b, it i convenient to chooe a oint c a, b а i Regular R, if < a <, q, w - iecewie continuou on [ a, c] a > 0,w a > 0 А i Singular S, if a or ± c a R, but { x q x w x} dx a 8

29 The ingular endoint a i Limit Point LP if for ome real λ at leat one olution of differential equation atifie the condition c w y dx a <a The endoint a i Weakly Regular WR if & c a q w dx < 9

30 The ingular endoint i Limit-Circle Non-OcillatoryLCNO if for ome real value of ectral arameter λ ALL realvalued olution atify the condition y x, λ y x, λ c a w y < and ha at mot a finite number of zero in a, c ] The ingular endoint i Limit-Circle OcillatoryLCO if for ome real value of ectral arameter λ ALL realvalued olution atify the condition y x, λ y x, λ c a and ha an infinite number of zero in a, c ] w y < 30

31 SLP roblem are claified into variou clae baed on the claification of the endoint and on whether the boundary condition are earated S or couled C. We have the following categorie:. R/R, Searated. R/R, Couled 3. R/LCNO LCNO/R, Searated 4. R/LCNO LCNO/R, Couled 5. R/LCO LCO/R, Searated 6. R/LCO LCO/R, Couled 7. LCNO /LCO LCO/ LCNO LCO/ LCO, Searated 8. LCNO /LCO LCO/ LCNO LCO/ LCO, Couled 9. LP/R LP/LCNO LP/LCO R/LP LCNO/LP LCO/LP 0.LP/LP 3

32 The algorithm in SLEIGN Initial interval a, b i converted to interval -, in the SLEIGN ackage The comutation rocedure i imlemented by the ue of Prüfer Tranform. 3

33 Prüfer Tranform. y x ρ xin θ x y' x ρ xco θ x Differential equation for ρ & θ : θ' x x co ρ' x/ ρ x x θ x λw x q xin Boundary condition for θ : θ x λw x q xin θ xco θ x θ a arctg A θ b πn / A arctg B / B 33

34 The following diadvantage were found in the rogram ackage SLEIGN: The interface of the rogram i organized on the bae of conole dialog. Thi aroach coniderably increae the time for defining the roblem aramter. The rogram i untable toward the inut : if number are taken in the incorrect format ay, with comma intead of dot or letter are taking intead of number. In thi cae the rogram i terminated and it i required to tart work from the very beginning. Additional rogram MAKEPQW i required in order to create own examle. 34

35 Solution in BARSIC Buine And Reearch Scientific Interactive Calculator Numerical algorithm from SLEIGN remain unchanged. Subroutine of SLEIGN ackage rogramming language FORTRAN are comiled into dll file o file in cae of OS Linux and then they are called from BARSIC rogram. Additional function for calculation of firt and econd derivative were created It neceary to write them in FORTRAN when SLEIGN i ued directly 35

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42 The comarion between SLEIGN and well known mathematical ackage Mathematica, Male, COMSOL Multihyic how that SLEIGN i more efficient from the oint of view time of comutation and numerical error. Moreover, i not oible to olve roblem with couled condition in Male and Mathematica. 4

43 Reference and reource:. Werener O. Amerin, Andre M. Hinz, David B. Pearon Sturm Liouville Theory. Pat an Preent. J.D. Pryce Numerical olution of Sturm-Liouville Problem Oxford Univerity Pre; htt:// 4. htt:// 5. A. Devdariani, E. Dalimier Diole tranition-matrix element of the one-electron heterodiatomic quaimolecule 43

44 Thank you for attention! 44