Composite Hermite and Anti-Hermite Polynomials

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1 Avaces i Pure Mathematics Publishe Olie December 5 i SciRes. Composite Hermite a Ati-Hermite Polyomials Joseph Akeyo Omolo Departmet of Physics a Materials Sciece Maseo Uiversity Maseo Keya Receive 6 February 5; accepte 7 December 5; publishe December 5 Copyright 5 by author a Scietific Research Publishig Ic. This work is licese uer the Creative Commos Attributio Iteratioal Licese (CC BY). Abstract The Weber-Hermite ifferetial equatio obtaie as the imesioless form of the statioary Schroeiger equatio for a liear harmoic oscillator i quatum mechaics has bee epresse i a geeralize form through itrouctio of a costat cojugatio parameter accorig to the trasformatio where the cojugatio parameter is set to uity ( = ) at the e of the evaluatios. Factorizatio i ormal orer form yiels -epeet composite eigefuctios Hermite polyomials a correspoig positive eigevalues while factorizatio i the ati-ormal orer form yiels the parter composite ati-eigefuctios ati-hermite polyomials a egative eigevalues. The two sets of solutios are relate by a -sig reversal cojugatio rule. Settig = provies the staar Hermite polyomials a their parter ati- Hermite polyomials. The ati-hermite polyomials satisfy a ew ifferetial equatio which is iterprete as the cojugate of the staar Hermite ifferetial equatio. Keywors Weber-Hermite Differetial Equatio Eigefuctios Ati-Eigefuctios Hermite Ati-Hermite Positive-Negative Eigevalues. Itrouctio The Weber-Hermite ifferetial equatio arises as the imesioless form of the oe-imesioal statioary Schroeiger equatio for a liear harmoic oscillator of mass m agular frequecy ω total eergy E a isplacemet obtaie i quatum mechaics i the form []-[4] + mω ψ ( ) = Eψ ( ). (a) m How to cite this paper: Akeyo Omolo J. (5) Composite Hermite a Ati-Hermite Polyomials. Avaces i Pure Mathematics

2 Itroucig parameters s a λ efie by mω E s = ; λ = ω we easily trasform Equatio (a) ito the imesioless form + ψ = s λψ s s s which we call the Weber-Hermite ifferetial equatio sice its geeral solutios are the Weber-Hermite fuctios compose of the Hermite polyomials []-[4]. It is coveiet to replace to epress Equatio (c) i the familiar mathematical form (b) (c) s ; ψ s y () + = λ y y. (e) We provie cojugate pairs of solutios of this equatio through factorizatio. We efie a cojugatio parameter a evelop the factorizatio proceure i Sectio. Normal-orer solutios i terms of composite Hermite polyomials their recurrece relatios positive eigevalues a ifferetial equatio are presete i Sectio 3. while the composite ati-hermite polyomials their recurrece relatios egative eigevalues a ifferetial equatio arisig from the ati-ormal orer solutios are cotaie i Sectio 3.. Factorizatio a the Cojugatio Parameter Factorizatio is a powerful techique for solvig seco-orer oriary ifferetial equatios. A importat feature of factorizatio is factor orerig i the resultig prouct of factors especially if the factors are operators []. To take accout of operator factor orerig i geeral form we itrouce a costat parameter which is set to uity ( = ) at the e of the evaluatios accorig to a trasformatio rule (a) to epress the Weber-Hermite Equatio (e) i the geeral form + y ( ) = λ y( ) (b) which is the same as Equatio (e) for =. Eve though the mai motivatio for itroucig the parameter is to accout for operator orerig it turs out that plays a fuametal role as a cojugatio parameter which provies a cojugatio rule relatig the two alterate ormal a ati-ormal orer factorize forms of Equatio (b). The geeral solutios of the ormal or ati-ormal orer forms are cojugate polyomials relate by the -cojugatio rule. Notig that the operator + takes the form of a ifferece of two squares we apply a effective factorizatio proceure [] to epress Equatio (b) i two alterately orere forms + + y = λ y (3a) + + y = λ + y (3b) The operators are relate by -sig reversal cojugatio rule 88

3 givig (3c) + + (3) The operators are sai to be -sig reversal cojugates satisfyig cojugatio rule (3c) accorig to otatio : + = + ; + = + (3e) where we have aopte the usual Hermitia cojugatio otatio usig the symbol to apply i geeral. For operators or eigefuctios epressible i matri form the Hermitia cojugatio uer the -sig reversal cojugatio is effecte by applyig the cojugatio rule (3c) to every elemet a the takig the traspose. We ote that i a case where = i which woul arise from a equivalet mathematical operatio = = i i ; i = the -cojugatio woul costitute the familiar Hermitia cojugatio rule which justifies the use of the Hermitia cojugatio otatio aopte here. We observe that the mathematical operatio i Equatio (3f) applies to the factorizatio of a seco orer operator of the form f +. Accorig to the cojugatio rule i Equatio (3c) the factorize forms (3a) a (3b) are -sig reversal cojugates. Subtractig Equatio (3a) from Equatio (3b) usig the cojugatio relatio (3e) a roppig the arbitrary fuctio y( ) we obtai the commutatio relatio For reasos which may become clear below we recogize + + = + + =. (3g) + as a lowerig operator a + = + as a raisig operator. I this respect the factorize form (3a) is sai to be i ormal orer while the form (3b) is i ati-ormal orer.. Geeral Solutio Sice Equatios (3a) a (3b) are relate by the -cojugatio rule (3c) their geeral solutios are -sig reversal cojugates. The ormal orer form (3a) yiels the staar eigefuctios Hermite polyomials a the correspoig positive eigevalues while the ati-ormal orer form (3b) yiels ati-eigefuctios ati-hermite polyomials a the correspoig egative eigevalues... Normal-Orer Form: Eigefuctios Hermite Polyomials a Positive Eigevalues We start by cosierig that the ormal orer form (3a) is a eigevalue equatio with eigevalue λ. It has a lower bou of zero eigevalue obtaie as (3f) λ λ = λ = (4a) where λ eotes the lowest value of λ obtaie at zero eigevalue. The correspoig lowest orer eigefuctio y ( ) at zero eigevalue ( λ = ) is etermie through Equatio (3a) uer the coitio (4a) accorig to ; y ( ) y ( ) ( ) λ λ = = = + + y =. (4b) 89

4 Applyig Hermitia cojugatio of the operators + a + accorig to Equatio (3e) we epress Equatio (4b) i the form ( ) + + y = (4c) which o multiplyig from the left by the ε-sig reversal cojugate y ( ) y ( ) takes the form of the lowest orer eigefuctio y + + y = + y =. (4) The basic equatio for the lowest orer eigefuctio y the follows from Equatio (4) i the form + y ( ) = (5a) with a simple solutio ( ) = y e (5b) otig that the itegratio costat evaluate at y =. y of geeral orer are geerate through repeate applicatio of the cojugate operator Eigefuctios + = + = is o the lowest orer eigefuctio y which o substitutig accorig to y ( ) = + y( ) = + y( ); = 3 (5c) y from Equatio (5b) a evaluatig for = give the first two lower orer eigefuctios i the form To evaluate higher orer eigefuctios ( ) f ( ) φ ( ) i the form ( ) = e ( ) = ( ) y y y. (5) y we erive a simplifyig formula for ay fuctios f + fφ = f + φ φ (5e) a the apply the geeral relatio y+ ( ) = + y( ) = 3 (5f) which follows easily from Equatio (5c) by settig +. y Equatio (5f) gives For which o substitutig y( ) = y( ) φ = y ( ) the usig Equatio (5f) i the fial step gives y( ) = ( y( ) ε y( )) y( ) = + y( ) (6a) from Equatio (5) a applyig the formula (5e) with f =. (6b) Proceeig i the same maer for y ( 3 ) ( ); 4( ) 3( ) y y = + = + y (6c) 8

5 easily gives the forms ( ) = ( ( ) ( )); ( ) = ( ( ) 3 ( )) y y y y y y. (6) We arrive at the importat geeral result that higher orer eigefuctios are obtaie i the form of a recurrece relatio ( ε ) y = y y = 3. (6e) + Settig = 3 i Equatio (6e) a substitutig lower orer eigefuctios as appropriate recallig y i the form y ( ) from Equatio (5b) or (5) we obtai the geeral eigefuctio y = H e = 3 (7a) where H ( ) below we shall call H ( ) the composite Hermite polyomials the geeral eigefuctios y ( ) is a polyomial epeig eplicitly o the parameter. For reasos which will be clear are calle the composite Weber-Hermite fuctios. Usig Equatio (5b) i Equatio (5c) a substitutig the result o the l.h.s. of Equatio (7a) provies the geeral relatio for geeratig the composite Hermite polyomials i the form H = e + e =. (7b) Usig Equatio (5b) together with its -sig reversal cojugate y = e (7c) i Equatio (7b) efies the composite Hermite polyomials i terms of the lowest orer eigefuctio accorig to Eplicit forms of H ( ) H ( ) = y( ) + y( ). (7) are easily obtaie usig a recurrece relatio erive i the et subsectio.... Recurrece Relatios a Differetial Equatio for H ( ) Settig + i Equatio (7b) a isertig the relatio e e = as appropriate the usig Equatio (7b) gives = + H+ e H e (8a) which is easily evaluate to obtai the first recurrece relatio for the polyomials H ( ) Settig = i Equatio (7b) gives i the form H H = H + H m H m( ) m = = +. (8b) H =. (8c) Settig = 3 i Equatio (8b) the provies the first five composite Hermite polyomials as takig the geeral epasio H = ; H = ; H = 4 ; H = 8 ; H = (8) ! ( ) m= ( ) m ( m ) H = ; H =. (8e) m! m! 8

6 The symbol i the summatio meas that m rus over iteger values up to the iteger part of e.g. = 3 m =. The geeral form i Equatio (8e) clearly isplays the eplicit epeece of the polyomials o the parameter which provies the justificatio for callig H ( ) the composite Hermite polyomials sice the polyomials become the staar Hermite polyomials after settig = while settig trasforms the polyomials to their cojugatio parters. Substitutig ( ) ( ) e ; ( ) ( ) e ; ( ) ( ) e + = + = = y H y H y H ito Equatio (6e) gives the seco recurrece relatio for the composite Hermite polyomials i the form ( ) = ( ) ε ( ) H H H. (8f) + Comparig the first recurrece relatio (8b) a the seco recurrece relatio (8f) easily provies the thir recurrece relatio for the composite Hermite polyomials i the form H m m = H H = H. (8g) Applyig o Equatio (8g) gives H H =. (9a) Usig Equatio (8e) together with the result of settig + i Equatio (8g) gives H H = H (9b) which we substitute ito Equatio (9a) to obtai the ifferetial equatio for the composite Hermite polyomials i the form H H + H = H = H( ) (9c) which iffers from the familiar Hermite ifferetial equatio []-[] oly by the factor o the seco orer erivative term. Settig = reuces Equatio (9c) to the Hermite ifferetial equatio.... Positive Eigevalue Spectrum Substitutig = from Equatio (7a) ito Equatio (9c) a reorgaizig gives the fial result which cofirms that the eigefuctios y ( ) spoig iscrete form ( + ). Comparig Equatios (e) a (b) otig which correspo to the eigefuctios ( ) H y e (a) + y( ) = ( + ) y( ) (b) s satisfy the origial Equatio (e) with λ takig the corre- λ λ gives the positive eigevalue spectrum λ = + ; = 3 (c) y. 8

7 ..3. The Hermite Polyomials We ow set = i Equatios (7a) a (c) to obtai the staar eigefuctios a correspoig positive eigevalues satisfyig λ y = H e ; = + ; = 3 (a) + = λ y y. (b) The eigefuctios y ( ) are the staar Weber-Hermite fuctios [6]. Settig = i Equatios (8e) (8b) (8f) a (8g) gives the staar Hermite polyomials H their recurrece relatios i the familiar form [5]-[]! ( ) ( ) a m ( m ) ; (c) m= m! m! H = H = H H H = H ; = H ; H = H H + () + The first five Hermite polyomials are the same as Equatio (8) with =. Fially we set = i Equatio (9c) to obtai the staar Hermite ifferetial Equatio [5]-[] H H + H = (e).. Ati-Normal Orer Form: Ati-Eigefuctios Ati-Hermite Polyomials a Negative Eigevalues The ati-ormal orer form (3b) is a eigevalue equatio with eigevalue λ +. It has a upper bou of zero eigevalue obtaie as λ+ λ + = λ = (a) where λ eotes the highest value of λ obtaie at zero eigevalue. The correspoig highest orer atieigefuctio y ( ) at zero eigevalue ( λ + = ) is etermie through Equatio (3b) uer the coitio (a) accorig to ; y ( ) y ( ) ( ) λ λ = = = + + y = (b) Applyig Hermitia cojugatio accorig to Equatio (3e) we epress Equatio (b) i the form ( ) + + y = (c) which o multiplyig from the left by the (ε-sig reversal) Hermitia cojugate y ( ) ati-eigefuctio y ( ) takes the fial form The basic equatio for the highest orer ati-eigefuctio y form with a simple solutio of the highest orer + y ( ) =. () the follows from Equatio () i the + y ( ) = (3a) 83

8 y = e (3b) otig that the itegratio costat evaluate at y =. y of geeral orer are geerate through repeate applicatio of the cojugate Ati-eigefuctios operator + = + = is o the highest orer ati-eigefuctio y which substitutig accorig to y ( ) = + y( ) = + y( ); = 3 (3c) y from Equatio (3b) a evaluatig for = give the first two highest orer ati-eigefuctios i the form To evaluate lower orer ati-eigefuctios ( ) f ( ) φ ( ) i the form a apply the geeral relatio ( ) = e ( ) = ( ) y y y. (3) which follows easily from Equatio (3c) by settig +. y Equatio (3f) gives For y we erive a simplifyig formula for ay fuctios f + fφ = f + φ + φ (3e) y+ ( ) = + y( ) = 3 (3f) which o substitutig y( ) = y( ) φ = y ( ) y( ) = ( y( ) + y( )) y( ) = + y( ) (4a) from Equatio (3) a applyig the formula (3e) with f = the usig Equatio (3f) i the fial step gives Proceeig i the same maer for. (4b) y( ) = + y ( ) 3 (4c) easily gives the importat geeral result that lower orer ati-eigefuctios are obtaie i the form of a recurrece relatio ( ) y = y + y = 3. (4) + Settig = 3 i Equatio (4) a substitutig higher orer ati-eigefuctios as appropriate recallig y ( ) from Equatio (3b) or (3) we obtai the geeral ati-eigefuctio y ( ) i the form where H ( ) y = H e = 3 (5a) are composite ati-hermite polyomials. Usig Equatio (3b) i Equatio (3c) a substitutig the result o the l.h.s. of Equatio (5a) provies the geeral relatio for geeratig the composite ati-hermite polyomials i the form H ( ) = e + e = (5b) 84

9 Usig Equatio (3b) together with its ( -sig reversal) Hermitia cojugate = ( ) y e (5c) i Equatio (5b) efies the composite ati-hermite polyomials i terms of the highest orer ati-eigefuctio accorig to Eplicit forms of H ( ) H ( ) = y ( ) + y( ). (5) are easily obtaie usig a recurrece relatio erive i the et subsectio.... Recurrece Relatios a Differetial Equatio for H ( ) Settig + i Equatio (5b) a isertig gives the relatio e e = as appropriate the usig Equatio (5b) = + H+ e H e (6a) which is easily evaluate to obtai the first recurrece relatio for the polyomials H ( ) Settig = i Equatio (5b) gives i the form H H = H + + H m H m( ) m = = +. (6b) H =. (6c) Settig = 3 i Equatio (6b) the provies the first five composite ati-hermite polyomials as takig the geeral epasio Substitutig H = ; H = ; H = 4 + ; H = 8 + ; H = (6) m! m= ( ) ( m ) H = ; H =. (6e) m! m! ( ) ( ) e ; ( ) ( ) e ; ( ) ( ) e + = + = = y H y H y H ito Equatio (4) gives the seco recurrece relatio for the composite ati-hermite polyomials i the form ( ) = ( ) + ( ) H H H. (6f) + Comparig the first recurrece relatio (6b) a the seco recurrece relatio (6f) easily provies the thir recurrece relatio for the composite ati-hermite polyomials i the form H m m = H H = H. (6g) Applyig o Equatio (6g) gives H H =. (7a) Usig Equatio (6f) together with the result of settig + i Equatio (6g) gives 85

10 H H H (7b) = which we substitute ito Equatio (7a) to obtai the ifferetial equatio for the composite Hermite polyomials i the form H H + H = H = H ( ) (7c) which is a ew ifferetial equatio. It is the cojugate of the composite Hermite ifferetial Equatio (9c). Applyig the cojugatio rule takes Equatio (7c) to Equatio (9c).... Negative Eigevalue Spectrum Substitutig = H y e (8a) from Equatio (5a) ito Equatio (7c) a reorgaizig gives the fial result which cofirms that the eigefuctios y ( ) spoig iscrete form ( + ). Comparig Equatios (e) a (8b) otig + y( ) = ε ( + ) y( ) (8b) s satisfy the origial Equatio (e) with λ takig the corre- λ λ gives the egative eigevalue spectrum which correspo to the ati-eigefuctios ( ) λ = + ; = 3 (8c) y...3. The Ati-Hermite Polyomials We ow set = i Equatios (5a) a (8c) to obtai the ati-eigefuctios a correspoig egative eigevalues satisfyig λ y = H e ; = + ; = 3 (9a) + = λ y y. (9b) The ati-eigefuctios y ( ) may be calle the ati-weber-hermite fuctios. Settig = i Equatios (6e) (6b) (6f) a (6g) gives the ati-hermite polyomials H their recurrece relatios i the ( ) The first five ati-hermite polyomials ( a! ( m ) ; m= m! m! (9c) H = H = H H H = H + ; = H H = H + H +. (9) + H = 3 ) are the same as Equatio (6) with =. Fially we set = i Equatio (7c) to obtai the ati-hermite ifferetial equatio H H + H =. (9e) We observe that the ati-eigefuctios y ( ) ati-hermite polyomials H ( ) a the correspoig 86

11 egative eigevalues H ( ) a positive eigevalues λ relate by the set to uity ( = ) at the e of the evaluatios. 3. Coclusio λ are -cojugatio parters of the eigefuctios y ( ) J. Akeyo Omolo Hermite polyomials cojugatio rule. The cojugatio parameter is We have establishe that the Weber-Hermite ifferetial equatio which is the imesioless form of the statioary Schroeiger equatio for a liear harmoic oscillator has two sets of solutios characterize by positive a egative eigevalues. Factorizatio i the ormal orer form yiels the staar eigefuctios Hermite polyomials a the correspoig positive eigevalues while factorizatio i the ati-ormal orer form yiels the parter ati-eigefuctios ati-hermite polyomials a the correspoig egative eigevalues. The two sets of solutios are relate by a fuametal cojugatio rule. Ackowlegemets I thak Maseo Uiversity a Techical Uiversity of Keya for proviig facilities a coucive work eviromet urig the preparatio of the mauscript. Refereces [] Akeyo Omolo J. (4) Parametric Processes a Quatum States of Light. Lambert Acaemic Publishig (LAP Iteratioal) Berli Germay. [] Sakurai J.J. (985) Moer Quatum Mechaics. The Bejamis/Cummigs Publishig Compay Ic. Memlo Park. [3] Merzbacher E. (97) Quatum Mechaics. Wiley New York. [4] Schiff L. (968) Quatum Mechaics. McGraw-Hill New York. [5] Arfke G.B. a Weber H.J. (995) Mathematical Methos for Physicists. Acaemic Press Ic. Sa Diego. [6] Stepheso G. a Ramore P.M. (99) Avace Mathematical Methos for Egieerig a Sciece Stuets. Cambrige Uiversity Press Cambrige. [7] Lebeev N.N. (965 97) Special Fuctios a Their Applicatios (Traslate by Silverma R.A.). Pretice-Hall Eglewoo Cliffs; Paperback Dover New York. [8] Magus W. Oberhetiger F. a Soi R.P. (966) Formulas a Theorems for the Special Fuctios of Mathematical Physics. Spriger New York. [9] Raiville E.D. (96) Special Fuctios. Macmilla New York. [] Seo I.N. (98) Special Fuctios of Mathematical Physics a Chemistry. Logma New York. 87