POSITIVE-NEGATIVE ENERGY PARTNER STATES AND CONJUGATE QUANTUM POLYNOMIALS FOR A LINEAR HARMONIC OSCILLATOR

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1 Fuametal Joural of Mathematic a Mathematical Sciece Vol., Iue, 5, Page Thi paper i available olie at Publihe olie March 3, 5 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND CONJUGATE QUANTUM POLYNOMIALS FOR A LINEAR HARMONIC OSCILLATOR Departmet of Phyic a Material Sciece Maeo Uiverity P.O. Private Bag, Maeo, Keya ojakeyo4@yahoo.co.uk Abtract Factorizatio i ormal orer form provie poitive eergy olutio, while the alterative ati-ormal orer form provie egative eergy olutio of the time-iepeet Schroeiger equatio for a liear harmoic ocillator. The poitive a egative eergy tate pace are relate by quatum cojugatio effecte through ig-reveral of Plack quatizatio cotat ħ. Photo occupyig egative eergy tate have the ame iplacemet xˆ a Hamiltoia H, but oppoite mometum pˆ compare to photo occupyig the correpoig poitive eergy parter tate. Emiio of poitive eergy photo from a poitive eergy quatum tate i equivalet to aborptio of egative eergy photo ito a egative eergy quatum tate, leaig to tate lowerig, while aborptio of poitive eergy photo ito a poitive eergy quatum tate i equivalet to emiio of egative eergy photo from a egative eergy quatum tate, leaig to tate raiig. The complete poitiveegative eergy pectrum of the quatize ocillator may the be Keywor a phrae: poitive-egative eergy parter tate, quatize ocillator polyomial, quatum cojugatio. Mathematic Subject Claificatio: 33C45, 34A34, 34A5, 34C5, 3D, 3G. Receive Jauary 3, 5; Accepte February 6, 5 5 Fuametal Reearch a Developmet Iteratioal

2 56 iterprete a a photo-atiphoto ytem. We have icovere ew ħ- epeet quatize ocillator polyomial a their quatum cojugate efiig poitive a egative eergy tate eigefuctio. Thee polyomial atify correpoig eco orer oriary ifferetial equatio.. Itrouctio The oe-imeioal time-iepeet Schroeiger equatio for a liear harmoic ocillator of ma m, agular frequecy ω, total eergy E a iplacemet x i obtaie a ħ m x + mω x ψ( x) = Eψ( x) (a) which o itroucig parameter efie by = mωx (b) for eae of phyical iterpretatio take the form E ħ + ψ( ) = ψ( ). (c) ω It ha alway bee aume that the Schroeiger equatio (a) applie oly to oegative eergy, E, tate ψ ( x). Iee, taar metho geerally applie to olve equatio (a) or it alterative form (c) yiel oly the expecte poitive eergy pectrum. We oberve that operator orerig i a effective factorizatio proceure well evelope i the preet author book [] provie for both poitive a egative eergy olutio, eve though oly the familiar poitive eergy olutio bae o the ormal orer form have bee compreheively preete i the book. We evelop the olutio proceure for the ati-ormal orer form yielig the largely igore egative eergy pectrum i the preet paper. Negative eergy tate are geerally kow to exit a are aociate with ati-matter withi the geeral framework of relativitic quatum mechaic a quatum fiel theory [3-7]. What ha remaie ukow i that egative eergy tate alo exit i o-relativitic quatum mechaic, which we etablih here through olutio of the Schroeiger equatio (a).

3 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 57.. Factorizatio To evelop the factorizatio proceure, we coier the Schroeiger equatio i the form (c). Notig that the operator ħ +, we apply a effective factorizatio proceure [] to expre equatio (c) i two alterative factorize form ħ + ħ E + ψ = ω ħ ψ, (a) ħ E + ħ + ψ = + ħ ψ ω (b) which iffer by operator orerig. To etermie the ature of the operator orerig, we ivie equatio (a)-(b) by a itrouce operator aˆ = = ħ + ; aˆ ħ + (c) to rewrite equatio (a)-(b) i the correpoig form ˆ ˆ E a aψ = ħ ψ, () ħω ˆ ˆ E a a ψ = + ħ ψ. (e) ħω Subtractig equatio () from equatio (e) a roppig the arbitrary ψ give the commutatio bracket [ ˆ, aˆ ] = ħ. It will become clear later that the operator â a a (f) â efie i equatio (c) are aihilatio a creatio operator, repectively, of the quatize ocillator. We have eliberately exclue Plack quatizatio cotat ħ from the efiitio of a ˆ, aˆ for coitecy i the applicatio of cojugatio proceure evelope i Sectio 4. Accorig to taar operator orerig efiitio [], equatio () a it origial expreio (a) are i ormal orer form, while equatio (e) a it origial

4 58 expreio (b) are i ati-ormal orer form. Sice both form arie from factorizatio of the ame equatio (c), they repreet alterative olutio of the origial Schroeiger equatio (a). We obtai thee olutio below, tartig with the ormal orer form, which provie the familiar poitive eergy tate i Sectio, followe by the ati-ormal orer form, which provie the geerally igore egative eergy tate i Sectio 3. A quatum cojugatio theory for traformig the poitive eergy tate ito their egative eergy parter tate, leaig to the icovery of quatum cojugate ocillator polyomial, i evelope i Sectio 4.. Normal Orer Form: Quatize Ocillator Polyomial a Poitive Eergy Spectrum The ormal orer form i equatio (a) i a eigevalue equatio with E eigevalue ħ. It ha a lower bou of zero eigevalue obtaie a ω E ω ħ E ω ħ = E = ħω, (3a) where E eote the lowet value of E obtaie at zero eigevalue. The E correpoig lowet orer eigefuctio ψ ( ) at zero eigevalue ħ = ω i etermie through equatio (a) uer the coitio (3a) accorig to E ; ( ) ( ) = E = ħ ω ψ = ψ ħ + ħ + ψ( ) =. (3b) Applyig Hermitia cojugatio of the operator reverig the ig of ħ accorig to (ee Sectio 4) i equatio (3b) give ħ + a ħ + by : ħ ħ = ħ + ħ + (3c) ψ ħ + ħ + = (3) which o multiplyig from the left by the (ħ -ig revere) Hermitia cojugate

5 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 59 ψ ( ) of the lowet orer eigefuctio ψ ( ) take the form = ( ) ( ) ψ ħ + ħ + ψ = ħ + ψ( ). (3e) The baic equatio for the lowet orer eigefuctio ψ ( ) the follow from equatio (3e) i the form with a imple olutio ħ + ψ ( ) = (4a) ψ ( ) = e ħ (4b) otig that the itegratio cotat evaluate at = i ψ ( ) =. Eigefuctio ψ ( ) of geeral orer are geerate through repeate applicatio of the cojugate operator ħ + o the lowet orer eigefuctio ψ ( ) accorig to ( ) ψ = ħ + ψ( ) ; =,,, 3,..., (4c) which o ubtitutig ψ ( ) from equatio (4b) a evaluatig for =, give the firt two lower orer eigefuctio i the form, ψ ( ) = e ħ ψ ( ) = ψ ( ). (4) To evaluate higher orer eigefuctio ψ ( ),, we erive a implifyig formula for ay fuctio f ( ), φ( ) i the form a apply the geeral relatio f ħ + f φ = f ħ + φ ħ φ (4e) ( ) = ħ + ψ ( ) ; =,,, 3,... (4f) ψ +

6 6 which follow eaily from equatio (4c) by ettig +. For ψ ( ), equatio (4f) give ψ( ) = ħ + ψ( ) (5a) which o ubtitutig ψ ( ) = ψ( ) from equatio (4) a applyig the formula (4e) with f =, φ = ψ ( ), the uig equatio (4f) i the fial tep give ψ ( ) = ( ψ ( ) ħ ψ ( )). (5b) Proceeig i the ame maer for ψ3 ( ) = ( ) ; 4( ) ħ + ψ ψ = ħ + ψ3( ) (5c) eaily give the form ψ ( ) = ( ψ ( ) ħ ψ ( )) ; ψ ( ) = ( ψ ( ) 3ħψ ( )). (5) We arrive at the importat geeral reult that higher orer eigefuctio are obtaie i the form of a recurrece relatio ψ+ ( ) = ( ψ ( ) ħ ψ ( )), =,,, 3,...,. (5e) Settig =,,, 3,... i equatio (5e) a ubtitutig lower orer eigefuctio a appropriate, recallig ψ ( ) from equatio (4b) or (4), we obtai the geeral eigefuctio ψ ( ) i the form ψ ( ) = H (, ħ ) e ħ, =,,, 3,..., (6a) where H (, ħ) i a polyomial epeig explicitly o the parameter a Plack quatizatio cotat ħ. We hall call H (, ħ) the quatize ocillator polyomial. Uig equatio (4b) i equatio (4c) a ubtitutig the reult o the l.h.. of equatio (6a) provie the geeral relatio for geeratig the quatize ocillator polyomial i the form (, ) H ħ = e ħ ħ + e ħ, =,,,...,. (6b)

7 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 6 Uig equatio (4b) together with it (ħ -ig revere) Hermitia cojugate ψ ( ) = e ħ (6c) i equatio (6b) efie the quatize ocillator polyomial i term of the lowet orer eigefuctio accorig to H (, ħ ) = ψ( ) ħ + ψ( ). (6) Explicit form of H (, ħ) are eaily obtaie uig a recurrece relatio erive i the ext ubectio. H.. Recurrece relatio for (, ħ) Settig + i equatio (6b) a iertig e ħ e ħ = a appropriate, the uig equatio (6b) give the relatio H ( ) ( ), ħ e ħ ħ H, ħ e ħ + = + (7a) which i eaily evaluate to obtai the firt recurrece relatio for the polyomial H + (, ħ) i the form H H = H ħ, H m = H m (, ħ), m =, +. (7b) + Settig = i equatio (6b) give H =. (7c) Settig =,,, 3 i equatio (7b) the provie the firt five quatize ocillator polyomial a 3 4 = ; H = ; H = 4 ħ ; H3 = 8 ħ; H 4 = 6 48ħ + ħ H (7) takig the geeral expaio m= m! ( ħ) ( ) ( ) ( ) ( H m), ħ = (7e) m! m!

8 6 which clearly iplay the explicit epeece of the polyomial o the quatizatio cotat ħ, thu uggetig the referece quatize ocillator polyomial. The ymbol i the ummatio mea that m ru over iteger value up to the iteger part of, e.g., = 3, m =,. The quatize ocillator polyomial take the ame form a the Hermite polyomial [, 8-], but iffer oly o the ħ factor i the variou term. Settig ħ = i equatio (7e) give the correpoig Hermite polyomial H ( ) i the geeral expaio form [9] H! ( ) ( ) ( ) ( ) ( = m) (7f) m! m! m= which woul arie if we efie the parameter i the imeiole form mω = x itea of the form i equatio (b). ħ ψ Subtitutig + + =, ( ) = H (, ħ) e ħ ; ψ ( ) = H (, ħ) e ħ ; ψ ( ) H ( ħ) e m ħ ito equatio (5e) give the eco recurrece relatio for the quatize ocillator polyomial i the form ( ħ) = H (, ħ) ħh (, ). (7f) H +, ħ Comparig the firt recurrece relatio (7b) a the eco recurrece relatio (7f) eaily provie the thir recurrece relatio for the quatize ocillator polyomial i the form H = H, H = H (, ħ). (7g) m m Applyig ħ o equatio (7g) give H H ħ = ħ. (8a)

9 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 63 Uig equatio (7e) together with the reult of ettig + i equatio (7g) give H H ħ = H (8b) which we ubtitute ito equatio (8a) to obtai the ifferetial equatio for the quatize ocillator polyomial i the form H H ħ + H, H H (, ħ) = = (8c) which iffer from the familiar Hermite ifferetial equatio [, 8-] oly ue to the factor ħ. Settig ħ = reuce equatio (8c) to the Hermite ifferetial equatio... Poitive eergy pectrum Subtitutig H (, ħ) = ψ ( ) e ħ (9a) from equatio (6a) ito equatio (8c) a reorgaizig give the fial reult ħ + ψ ( ) = ħ( + ) ψ ( ) (9b) which cofirm that the eigefuctio ψ ( ) atify the origial Schroeiger equatio i the form (c), with pecifyig quatizatio. E ω Comparig equatio (c) a (9b), otig takig the correpoig icrete form ħ ( + ) E E give E = ħ ( + ) (9c) ω which provie the poitive eergy pectrum for the quatize liear harmoic ocillator i the uual form E = ħ ω +, =,,, 3,...,. (9) Thi i the poitive eergy pectrum ariig from the olutio of the Schroeiger

10 64 equatio factorize i the ormal orer form (a)..3. Algebraic operatio with a ˆ, aˆ H Applyig the operator ħ ħ (, ) e a evaluatig give ħ + o the geeral eigefuctio ψ ( ) = ħ H + ψ = ħ e ħ (a) which o uig the thir recurrece relatio (7f) a the ubtitutig, ħ = ψ ( ) H ( ) e ħ (b) take the fial form ħ + ψ( ) = ħψ ( ). (c) Thi operatio with ħ + i eetially the revere of the operatio with it cojugate ħ + o ψ ( ) obtaie earlier i equatio (4f). Brigig the two together give the complete pair of revere algebraic operatio o the geeral eigefuctio accorig to It follow that ( ) ( ) ; ħ + ψ = ψ+ ħ + ψ ( ) = ħψ ( ). () ħ + ( ) ( ) ; ψ = ψ ħ + ψ ( ) = ħ( + ) ψ( ). (e) + Succeive operatio i ormal a ati-ormal orer the eaily give eigevalue equatio ħ + ħ + ψ ( ) = ħψ ( ); ħ + ħ + ψ ( ) = ħ( + ) ψ ( ). (f)

11 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 65 Diviig thee through by a itroucig the aihilatio a creatio operator â a â a efie i equatio (c) give aˆ aˆ ψ ( ) = ħ ψ ( ) ; aa ˆ ˆ ψ ( ) = ħ( + ) ψ( ). (a) Effective operatio with â a â coitet with the et of equatio (), (f) a (a) take the algebraic form ( ψ = ψ( ) ) a ˆ ψ = ψ+ ; aˆ ψ = ħ ψ. (b) Notice that ubtitutig equatio (c) for â a â eparately ito equatio () woul leave factor a a appropriate o the r.h.. of each equatio i (b), which woul effectively yiel the reult of ucceive operatio i equatio (a). Hece, the effective algebraic operatio with the aihilatio a creatio operator a ˆ, aˆ take the form i equatio (b). A uual, equatio (b) reveal that the creatio operator â i a raiig operator, while the aihilatio operator â i a lowerig operator for the poitive eergy eigefuctio ψ ( ) obtaie a olutio of the ormal orer form (a). The form of the algebraic operatio i equatio (b) i ifferet from the Dirac form i taar quatum mechaic textbook [8], but it i more ueful i the phyical iterpretatio of operatio with the creatio a aihilatio operator. The traitio to the familiar Dirac algebraic form i preete i []. Firt we ote a importat feature that i the baic operator algebraic relatio obtaie i equatio (b), the quatity ħ i equal to the well kow quatize orbital agular mometum. Secoly, accorig to equatio (b), the operatio of a creatio operator â o ψ ( ) repreet photo aborptio proce at the -th quatum tate, where the umber of photo aborbe a the quatum tate i raie to the ( + ) -th tate ecribe by ψ + ( ) remai hie, i.e., aborbe photo are iteral a are ot exterally obervable. O the other ha, the operatio of a aihilatio operator â o ψ ( ) repreet photo emiio proce at the -th quatum tate, where the umber of photo emitte () a the quatum tate i lowere to the ( ) -th tate ecribe by ψ ( ) i reveale, i.e., emitte photo are exteral a are irectly obervable.

12 66 3. Ati-ormal Orer Form: Cojugate Quatize Ocillator Polyomial a Negative Eergy Spectrum The ati-ormal orer form i equatio (b) i a eigevalue equatio with E eigevalue + ħ. It ha a upper bou of zero eigevalue obtaie a ω E ω + ħ E ω + ħ = E = ħω, (a) where E eote the highet value of E obtaie at zero eigevalue i the atiormal orer form. The egative value E = ħω repreet a egative eergy at the highet level i the ati-ormal orer eergy pectrum. The correpoig highet orer egative eergy eigefuctio (highet orer ati-eigefuctio) ψ ( ) at zero E eigevalue + ħ = i etermie through equatio (b) uer the coitio ω (a) accorig to E ; ( ) ( ) = E = ħ ω ψ = ψ ħ + ħ + ψ( ) =. (b) Applyig ħ -ig revere Hermitia cojugatio of the operator ħ + accorig to ħ + a ħ + = ħ + (c) i equatio (b) give ψ ħ + ħ + = () which o multiplyig from the left by the (ħ -ig revere) Hermitia cojugate ψ ( ) of the highet orer ati-eigefuctio ψ ( ) take the form

13 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 67 ψ = ( ) ( ) ħ + ħ + ψ = ħ + ψ( ). (e) The baic equatio for the highet orer ati-eigefuctio ψ ( ) the follow from equatio (3e) i the form with a imple olutio ħ + ψ ( ) = (3a) ψ ( ) = e ħ (3b) otig that the itegratio cotat evaluate at = i ψ ( ) =. Ati-eigefuctio ψ ( ) of geeral orer are geerate through repeate applicatio of the cojugate operator ħ + o the highet orer ati- eigefuctio ψ ( ) accorig to ( ) ψ = ħ + ψ( ) ; =,,, 3,,..., (3c) which o ubtitutig ψ ( ) from equatio (3b) a evaluatig for =, give the firt two highet orer ati-eigefuctio i the form, ψ ( ) = e ħ ψ ( ) = ψ ( ). (3) To evaluate lower orer ati-eigefuctio ψ ( ),, we erive a implifyig formula a i equatio (4e) for ay fuctio f ( ), φ( ) i the form a apply the geeral relatio f ħ + fφ = f ħ + φ + ħ φ (3e) ( ) = ħ + ψ ( ), =,,, 3,... (3f) ψ + which follow eaily from equatio (3c) by ettig +.

14 68 For everal value i equatio (3f), we apply the formula (3e) a procee i the ame maer elaborate i the previou ectio to obtai geeral reult for lower orer ati-eigefuctio i the form of a recurrece relatio ψ+ ( ) = ( ψ ( ) + ħ ψ ( )), =,,, 3,...,. (4) Settig =,,, 3,... i equatio (4) a ubtitutig higher orer atieigefuctio a appropriate, recallig ψ ( ) from equatio (3b) or (3), we obtai the geeral ati-eigefuctio ψ ( ) i the form ψ ( ) (, ) = H ħ e ħ, =,,, 3,..., (5a) where H (, ħ) i the cojugate quatize ocillator polyomial. Uig equatio (3b) i equatio (3c) a ubtitutig the reult o the l.h.. of equatio (5a) provie the geeral relatio for geeratig the cojugate quatize ocillator polyomial i the form (, ) = H ħ e ħ ħ + e ħ, =,,,...,. (5b) Uig equatio (3b) together with it (ħ -ig revere) Hermitia cojugate ψ ( ) = e ħ (5c) i equatio (5b) efie the cojugate quatize ocillator polyomial i term of the highet orer ati-eigefuctio accorig to H (, ħ ) = ψ ( ) ħ + ψ( ), (5) Explicit form of H (, ħ) are eaily obtaie uig a recurrece relatio erive i the ext ubectio. 3.. Recurrece relatio for (, ħ) H Settig + i equatio (5b) a iertig e ħ e ħ = a appropriate, the applyig equatio (5b) give the relatio

15 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 69 H ( ) ( ), ħ = e ħ ħ H, ħ e ħ + + (6a) which i eaily evaluate to obtai the firt recurrece relatio for the polyomial H (, ħ) i the form H H + = H + ħ, H m = H m(, ħ), m =, +. (6b) Settig = i equatio (5b) give H =. (6c) Settig =,,, 3,... i equatio (6b) the provie the firt five cojugate quatize ocillator polyomial a H = ; H = ; H = 4 + ħ; 3 4 H 3 = 8 + ħ ; H 4 = ħ + ħ (6) takig the geeral expaio H! ( ħ) ( ) ( ) ( ) (, ħ = m) (6e) m! m! m= which are evietly relate to the correpoig quatize ocillator polyomial i equatio (7)-(7e) through ħ -ig revere cojugatio ( ħ ħ), leaig u to refer to the polyomial H (, ħ) a the cojugate quatize ocillator polyomial. They eem to be a ew et of polyomial (pecial fuctio), havig arie here for the firt time i the olutio of the ati-ormal orer form (b) of the factorize Schroeiger equatio which ha ever bee achieve before i both phyic a mathematic. Settig ħ = i equatio (7) give the correpoig ħ - iepeet cae H ( ), which we may call cojugate Hermite polyomial, ariig from the olutio of the ati-ormal orer equatio if we efie the parameter i mω the imeiole form = x itea of the form i equatio (b). ħ Subtitutig m

16 7 ψ + + =, ( ) = H (, ħ) e ħ ; ψ ( ) = H (, ħ) e ħ ; ψ ( ) H ( ħ) e ħ ito equatio (4) give the eco recurrece relatio for the cojugate quatize ocillator polyomial i the form ( ħ) = H (, ħ) ħh (, ). (6f) H +, + ħ Comparig the firt recurrece relatio (6b) a the eco recurrece relatio (6f) eaily provie the thir recurrece relatio for the cojugate quatize ocillator polyomial i the form H = H, H m = H m(, ħ). (6g) Applyig ħ o equatio (6g) give H H ħ = ħ. (7a) Uig equatio (6f) together with the reult of ettig + i equatio (6g) give H H ħ = H (7b) which we ubtitute ito equatio (7a) to obtai the ifferetial equatio for the cojugate quatize ocillator polyomial i the form H H ħ + H, H H (, ħ) = = (7c) which i clearly relate to the ifferetial equatio for the quatize ocillator polyomial obtaie earlier i equatio (8c) by ħ-ig revere cojugatio. It i a ew eco orer oriary ifferetial equatio, which emerge through the olutio of the Schroeiger for a liear harmoic ocillator factorize i ati-ormal orer form (b). We are ot aware of it equivalet i the curret phyic a mathematic literature, i cotrat to it ħ -ig revere cojugate ifferetial equatio (8c), which i equivalet to the Hermite ifferetial equatio for ħ =. Settig ħ = reuce equatio (7c) to the ifferetial equatio for the ħ -

17 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 7 iepeet cojugate Hermite polyomial H ( ) (ew i phyic a mathematic) takig the form H H mω ħ = : + H, H H ( ) ; x. = = = (7) ħ A oberve earlier, H ( ), which are obtaie here by ettig ħ = i equatio (5b) givig = : ( ) = ħ H + e e, =,,,..., (7e) arie i the olutio of the ati-ormal orer equatio (b) with parameter efie i the imeiole form give above i equatio (7). Thi i the uual efiitio of the parameter which woul lea to olutio of the ormal orer form (a) i term of the familiar Hermite polyomial. The olutio i term of the Hermite polyomial, icluig Hermite ifferetial equatio, are eaily obtaie by ettig mω ħ = i all the expreio i the previou ectio, but ow efiig = x. ħ 3.. Negative eergy pectrum Subtitutig H (, ħ) = ψ ( ) e ħ (8a) from equatio (5a) ito equatio (7c) a reorgaizig give the fial reult ħ + ψ( ) = ħ( + ) ψ ( ) (8b) which cofirm that the ati-eigefuctio ψ ( ) atify the origial Schroeiger E equatio i the form (c), with takig the correpoig icrete form ω ħ ( + ) pecifyig quatizatio i the egative eergy ector. Comparig equatio (c) a (8b), otig E E give E = ħ ( + ) (8c) ω

18 7 which provie the egative eergy pectrum for the quatize liear harmoic ocillator i the ufamiliar form E = ħ ω +, =,,, 3,...,. (8) Thi i the egative eergy pectrum which arie from the olutio of the Schroeiger equatio factorize i the ati-ormal orer form (b) Algebraic operatio with a ˆ, aˆ Applyig the operator ħ + o the geeral ati-eigefuctio ψ ( ) = H (, ħ) e ħ a evaluatig give H ħ + ψ = ħ e ħ (9a) which o uig the thir recurrece relatio (6f) a the ubtitutig ψ ( ) H ( ħ) e ħ (9b) =, take the fial form ħ + ψ ( ) = ħψ ( ). (9c) Thi operatio with ħ + i eetially the revere of the operatio with it cojugate ħ + o ψ ( ) obtaie earlier i equatio (3f). Brigig the two together give the complete pair of revere algebraic operatio o the geeral atieigefuctio accorig to It follow that ( ) ( ) ; ħ + ψ = ψ+ ħ + ψ ( ) = ħψ ( ). (9) ( ) ( ) ; ħ + ψ = ψ ħ + ψ+ ( ) = ħ( + ) ψ ( ). (9e)

19 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 73 Succeive operatio i ati-ormal a ormal orer the eaily give eigevalue equatio ħ + ħ + ψ ( ) = ħψ( ); ħ + ħ + ψ ( ) = ħ( + ) ψ ( ). (9f) Diviig thee through by a itroucig the aihilatio a creatio operator â a â a efie i equatio (c) give aˆ aˆ ψ ( ) = ħ ψ ( ) ; aˆ aˆ ψ( ) = ħ( + ) ψ ( ). (a) Effective operatio with â a â coitet with the et of equatio (9), (9f) a (a) take the algebraic form ( ψ = ψ( ) ) aˆ ψ = ψ+ ; aˆ ψ = ħ ψ. (b) A importat phyical iterpretatio ow emerge. Accorig to equatio (b), the operatio of a creatio operator â o a ati-eigefuctio ψ ( ) ecribig egative eergy tate ow repreet egative eergy photo emiio proce at the -th egative eergy quatum tate. Thi emiio of egative eergy photo from the -th egative eergy tate raie it to the higher ( ) -th egative eergy tate. The emiio of egative eergy photo from a egative eergy tate i equivalet to aborptio of poitive eergy photo ito a poitive eergy tate, both procee thu leaig to excitatio to a higher level withi the correpoig eergy tate pace, poitive or egative. I thi repect, the creatio operator â maitai it role a a tate raiig operator for both poitive a egative eergy tate, cauig the tate raiig effect through poitive eergy photo aborptio accorig to equatio (b) i the poitive eergy tate pace a through egative eergy photo emiio accorig to equatio (b) i the egative eergy tate pace. O the other ha, equatio (b) reveal that the operatio of a aihilatio operator â o a ati-eigefuctio ψ ( ) ecribig egative eergy tate ow repreet egative eergy photo aborptio proce at the -th egative eergy quatum tate. Thi aborptio of egative eergy photo ito the -th egative eergy tate lower it to the lower ( + ) -th egative eergy tate. The aborptio of

20 74 egative eergy photo ito a egative eergy tate i equivalet to emiio of poitive eergy photo from a poitive eergy tate, both procee thu leaig to e-excitatio to a lower level withi the correpoig eergy tate pace, poitive or egative. I thi repect, the aihilatio operator â maitai it role a a tate lowerig operator for both poitive a egative eergy tate, cauig the tate lowerig effect through poitive eergy photo emiio accorig to equatio (b) i the poitive eergy tate pace a through egative eergy photo aborptio accorig to equatio (b) i the egative eergy tate pace. 4. Quatum Cojugatio To etermie the phyical coectio betwee the poitive a egative eergy tate pace, we tart by recogizig that the oly fuametal quatum mechaical parameter efiig the baic operator ħ + a ħ + or a ˆ, aˆ ariig from the factorizatio i equatio (a)-(b) i the Plack quatizatio cotat ħ. We therefore evelop a appropriate cojugatio rule bae o the ig reveral of Plack quatizatio cotat ħ. We call the cojugatio rule effecte by ħ -ig reveral the quatum cojugatio, i cotrat to the uual mathematical complex cojugatio effecte by ig reveral of the imagiary umber i =. We apply the quatum cojugatio accorig to the rule ħ ħ. () Hece, we efie quatum cojugatio a the reveral of the ig of Plack quatizatio cotat ħ accorig to ħ ħ everywhere i operator, eigefuctio a relate quatitie to obtai the correpoig quatum cojugate. I thi theory of quatum cojugatio, we efie quatum Hermitia cojugatio of a matrix by takig the quatum cojugatio of every etry a the takig the trapoe. We hall geerally treat quatum cojugate a quatum hermitia cojugate, eote by upercript. The uual mathematical rule of Hermitia cojugatio the apply, except ow we replace mathematical complex cojugatio ( i i) with quatum cojugatio ( ħ ħ). Quatum cojugatio eem atural a itictly ifferet from the uual mathematical complex cojugatio, ice it ivolve the chage of ig of a

21 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 75 fuametal phyical parameter ħ a applie to both real a complex quatitie. Ay operator, eigefuctio or quatity which i iepeet of ħ or epe o ħ j oly through ħ, j =, ±, ±, ± 3,... i ot chage by quatum cojugatio a i ai to be quatum Hermitia. Sice we carry out quatum cojugatio eparately, a quatum Hermitia quatity may till be a mathematically complex quatity with a complex cojugate. We etablih below that quatum cojugatio applie fully i the quatum theory of a liear harmoic ocillator, traformig the poitive eergy tate ito their parter egative eergy tate, thu proviig a proceure for traformig photo quatum tate ito their parter ati-photo quatum tate. 4.. Quatum cojugatio of poitive eergy eigefuctio For the lowet orer poitive eergy eigefuctio ψ ( ) = e ħ, we apply the rule i equatio () to obtai the quatum cojugate a ψ ( ) = e ħ. (a) Recogizig thi reult to be the highet orer ati-eigefuctio ψ ( ) of the egative eergy pectrum, we expre equatio (a) a ψ ( ) = ψ ( ). (b) The geeral poitive eergy eigefuctio ψ ( ) i geerate from ψ ( ) a expree i term of the quatize ocillator polyomial H (, ħ) accorig to ψ ( ) = ħ + ψ ( ) = H (, ħ) ψ ( ) which o takig the quatum cojugatio a uig equatio (b) become ψ ( ) = ħ + ψ( ) = H (, ħ) ψ( ) (c) from which follow the quatum cojugatio of the quatize ocillator polyomial i the form

22 76 H (, ħ ) ψ( ) = ħ + ψ( ). () Multiplyig equatio () by the quatum cojugate ψ ( ) of ψ ( ) from the left a uig give ψ = ( ) ψ ( ) (e) H (, ħ ) = ψ ( ) ħ + ψ( ) (f) which o recallig that the r.h.. provie the cojugate quatize ocillator polyomial H (, ħ) a efie i equatio (5) i the egative eergy tate pace take the form H (, ħ) = H (, ħ). (g) Subtitutig thi reult ito the lat tep of equatio (c) a itroucig the geeral egative eergy ati-eigefuctio provie the quatum cojugatio reult ( ) = H (, ħ ) ψ ( ) ψ The quatum cojugate of the poitive eergy pectrum E = ħ ω + i obtaie a ψ ( ) = ψ ( ). (h) E = = ω ħ ω + ħ + (3a) which o itroucig the egative eergy pectrum E = ħ ω + become

23 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 77 E E =. (3b) The quatum cojugate of the quatize ocillator polyomial ifferetial equatio i obtaie a H H ħ + H ; H H (, ħ) = = H H ħ + H ; H H (, ħ) = = (3c) which o ubtitutig equatio (g) take the fial form H ħ H + H = ; H = H (, ħ). (3) We recogize thi a the ifferetial equatio for the cojugate quatize ocillator polyomial obtaie earlier. 4.. Quatum cojugatio of the baic operator For the baic operator cojugatio rule i equatio () to obtai ħ + a ħ +, we apply the quatum ; ħ + = ħ + ħ + ħ +. (4a) Thee baic operator are quatum cojugate. It alo follow that the aihilatio a creatio operator efie by aˆ = = ħ + aˆ ; ħ + (c) are quatum cojugate accorig to ( ˆ ) ˆ a = a ; ( ˆ a ) = aˆ. (4b) We ue equatio (c) to obtai = ( aˆ + aˆ ); = ( aˆ aˆ ) ħ (4c)

24 78 which o applyig equatio () a (4b) are etablihe a quatum Hermitia accorig to = ; =. (4) Let u ow itrouce the poitio cooriate a liear mometum operator xˆ = x; pˆ = iħ (5a) x to expre the aihilatio a creatio operator i (c) i the form givig = pˆ pˆ aˆ = i + xˆ ; aˆ i + mω mω mω mω x ˆ (5b) mω x ˆ = ( aˆ + aˆ ); pˆ = i ( aˆ + aˆ ). (5c) mω Takig quatum cojugatio of equatio (5c) a applyig equatio (4b) give aˆ = xˆ; pˆ = pˆ (5) which how that the iplacemet operator xˆ i quatum Hermitia, while the liear mometum operator pˆ i quatum ati-hermitia. Not that the quatum cojugatio of pˆ a efie i equatio (5c) ivolve oly the operatio ħ ħ, leavig the imagiary umber itact. We oberve that a combie quatum a mathematical complex cojugatio ( ħ ħ, i i) woul give Hermitia form pˆ pˆ, but we are ot purig the combie cojugatio operatio here, ice it phyical meaig i the cotext of the traformatio of poitive eergy tate ito egative eergy parter tate may ot be clear. For the Hamiltoia H pˆ ˆ = + ω (5e) m m x quatum cojugatio give ( pˆ ) pˆ ˆ ˆ H = + mω x = + mω x (5f) m m

25 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 79 which o ubtitutig equatio (5e) how that the Hamiltoia i quatum Hermitia accorig to H = H. (5g) 4.3. Quatum cojugatio of the baic operator algebraic operatio Takig the quatum cojugatio of the poitive eergy algebraic operatio give aψ ( ) = ħ ψ ( ) ; aˆ ψ ( ) = ψ ( ) ˆ + ( aψ ( ) ) = ( ħ ψ ( )) aˆ ψ ( ) = ħψ ( ), (6a) ˆ ( aψ ( ) ) ( ψ ( ) ) aˆ ψ ( ) = ( ). (6b) ˆ + ψ+ The Hamiltoia of the quatize liear Harmoic ocillator i expree i term of the aihilatio a creatio operator i the form H = ω( aˆ aˆ + aa ˆ ˆ ); H = H (6c) which i eaily cofirme to be quatum Hermitia ( H = H ). The quatum Hermitia property mea that the Hamiltoia i the ame i both poitive a egative eergy tate pace. I the poitive eergy tate pace, the Hamiltoia act o the eigefuctio ψ ( ) accorig to equatio (b) to prouce the poitive eergy pectrum i the form ( ) ( ) ( ) ( ) Hψ = ω aˆ aˆ + aa ˆ ˆ ψ = Eψ ; E = ħ ω + (6) while i the egative eergy tate pace, the Hamiltoia act o the atieigefuctio ψ ( ) accorig to equatio (b) to prouce the egative eergy pectrum i the form Hψ ( ) ( ˆ ˆ ˆ ˆ ) ( ) ( ); = ω a a + aa ψ = Eψ E = ħ ω +. (6e) The commutator [ ˆ, aˆ ] a act withi the poitive a egative eergy tate pace

26 8 accorig to [ aˆ, aˆ ] ψ ( ) = ħ ( + ) ψ ( ) ; [ aˆ, aˆ ] ψ ( ) = ħ{ ( + )} ψ ( ) (6f) from which it follow that the commutatio bracket remai the ame i the quatum cojugatio traformatio relatig the poitive a egative eergy tate pace, i.e., the commutatio bracket i quatum Hermitia accorig to ([ ˆ, a ]) = [ aˆ, a ] = ħ. a ˆ ˆ (6g) 5. Geeral Iterpretatio All the reult of the quatum cojugatio how that the poitive eergy eigefuctio, quatize ocillator polyomial, eergy pectrum, baic operator a operator algebraic operatio traform ito their egative eergy parter. Thi lea to a fuametal feature of quatum yamic that the geeral quatum tate pace of a ocillator i compoe of two cojugate tate pace, amely, the poitive eergy tate pace a the egative eergy tate pace. The two quatum tate pace are relate by quatum cojugatio traformatio effecte through ig reveral of Plack quatizatio cotat ħ (i.e., ħ ħ ). Poitive a egative eergy tate are therefore iterprete a quatum cojugatio parter. Each quatum tate pace i pecifie by it phyical elemet, eetially coitig of tate eigefuctio a quatum operator which provie iformatio o the yamic through algebraic operatio withi the tate pace. I geeral, if the phyical elemet of the poitive eergy tate pace are eote by q, the the correpoig phyical elemet of the egative eergy tate pace eote by q are obtaie through quatum cojugatio accorig to ħ ħ: q = q. (7a) A quatum tate phyical elemet which oe ot chage uer the quatum cojugatio operatio accorig to q = q q = q (7b) i quatum Hermitia. Such quatum Hermitia elemet take the ame form i both poitive a egative eergy tate pace a they may be calle uiveral phyical elemet. I aitio, phyical elemet uch a aihilatio a creatio operator

27 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 8 which uer quatum cojugatio imply iterchage role are alo uiveral elemet, ice they maitai their form withi the two cojugate tate pace. The egative eergy tate ati-eigefuctio ψ ( ), aihilatio operator a a creatio operator a are obtaie from the correpoig poitive eergy eigefuctio ψ ( ), aihilatio operator â a creatio operator â through quatum cojugatio accorig to ψ ( ) = ψ ( ) ; a = aˆ ; aˆ aˆ. (8a) = The correpoig eergy eigevalue a quatize ocillator polyomial are obtaie a E = ħω ; E E ħ + = = ω + ; H (, ħ) H (, ħ). (8b) = The aihilatio a creatio operator act o repective eigefuctio withi the poitive a egative eergy tate pace accorig to aψ ( ) = ħ ψ ( ) ; aˆ ψ ( ) = ψ ( ), (8c) ˆ + aψ ( ) = ħ ψ ( ) a ψ ( ) = ψ ( ) (8) ; + which are relate by quatum cojugatio. Accorig to equatio (8c)-(8), the actio of the aihilatio operator â withi the poitive eergy tate pace caue emiio of poitive eergy photo from the tate ψ ( ), lowerig it to the tate ψ ( ), while the actio of the aihilatio operator a withi the egative eergy tate pace caue emiio of egative eergy photo from the tate ψ ( ), raiig it to the tate ψ ( ). O the other ha, the actio of the creatio operator â withi the poitive eergy tate pace caue aborptio of poitive eergy photo ito the tate ψ ( ), raiig it to the tate ψ +( ), while the actio of the creatio operator a withi the egative eergy tate pace caue aborptio of egative eergy photo ito the tate ψ ( ), lowerig it to the tate ψ +( ). Emiio of poitive eergy photo from a poitive eergy tate, leaig to poitive eergy tate lowerig i thu ee to be the revere proce relative to the

28 8 emiio of egative eergy photo from a egative eergy tate, leaig to egative eergy tate raiig. Likewie, aborptio of poitive eergy photo ito a poitive eergy tate, leaig to poitive eergy tate raiig i ee to be the revere proce relative to the aborptio of egative eergy photo from a egative eergy tate, leaig to egative eergy tate lowerig. Thee revere role of tate lowerig raiig or raiig lowerig i ue to the iterchage of creatio a aihilatio operator uer quatum cojugatio traformatio from poitive to egative eergy tate pace, which i eviet i equatio (8a). It follow from the above that the aihilatio a creatio operator a ˆ, aˆ are uiveral operator, actig i both poitive a egative eergy tate pace. Iee, they atify uiveral commutatio property Subtitutig a ˆ ˆ (8e) [, a ] = ([ aˆ, a ]) = [ aˆ, a ] = ħ. a = aˆ, a = aˆ from equatio (8a) ito equatio (8) give their actio withi the egative eergy tate pace accorig to a ψ ( ) = ħ ψ ( ) ; aˆ ψ ( ) = ψ ( ). (8f) ˆ + It i clear from equatio (8c) a (8f) that poitive eergy photo emiio from a poitive eergy tate a egative eergy photo aborptio ito a egative eergy tate ue to the actio of the aihilatio operator â lea to tate lowerig i both poitive a egative eergy tate pace, while poitive eergy photo aborptio ito a poitive eergy tate a egative eergy photo emiio from a egative eergy tate ue to the actio of the creatio operator â lea to tate raiig i both poitive a egative eergy tate pace. We arrive at the fuametal quatum mechaical feature of the yamic of a liear harmoic ocillator that emiio of poitive eergy photo from a poitive eergy quatum tate i equivalet to aborptio of egative eergy photo ito a egative eergy quatum tate, leaig to tate lowerig, while aborptio of poitive eergy photo ito a poitive eergy quatum tate i equivalet to emiio of egative eergy photo from a egative eergy quatum tate, leaig to tate raiig. The quatum Hermitia property of the iplacemet xˆ a Hamiltoia H accorig to x = xˆ = xˆ; H = H = H (8g)

29 POSITIVE-NEGATIVE ENERGY PARTNER STATES AND 83 a the quatum ati-hermitia property of the mometum pˆ accorig to p = pˆ = pˆ (8h) lea to the importat iterpretatio that photo occupyig egative eergy tate ( ψ ( ) ) have the ame iplacemet xˆ a Hamiltoia H, but oppoite mometum pˆ compare to photo occupyig the correpoig poitive eergy parter tate ( ψ ( ) ). The Hamiltoia H ha poitive eergy eigevalue pectrum E = ħ ω + i the poitive eergy tate pace a egative eergy eigevalue pectrum E = ħ ω + i the egative eergy tate pace. 6. Cocluio We have etablihe that the full eergy pectrum of a o-relativitic quatize liear harmoic ocillator i compoe of poitive a egative eergy tate relate by quatum cojugatio effecte through ig-reveral of Plack quatizatio cotat ħ. Photo occupyig egative eergy tate have the ame iplacemet xˆ a Hamiltoia H, but oppoite mometum pˆ compare to photo occupyig the correpoig poitive eergy parter tate. Emiio of poitive eergy photo from a poitive eergy quatum tate i equivalet to aborptio of egative eergy photo ito a egative eergy quatum tate, leaig to tate lowerig, while aborptio of poitive eergy photo ito a poitive eergy quatum tate i equivalet to emiio of egative eergy photo from a egative eergy quatum tate, leaig to tate raiig. Thee fuametal quatum feature are imilar to the well etablihe Dirac particle-hole or particle-atiparticle theory i relativitic quatum mechaic. We have itrouce a ew pair of quatum cojugate ħ - epeet polyomial which pecify the eigefuctio a ati-eigefuctio i the poitive a egative eergy tate pace. Ackowlegemet I thak Maeo Uiverity for proviig facilitie a coucive work eviromet urig the preparatio of the maucript.

30 84 Referece [] J. Akeyo Omolo, Parametric Procee a Quatum State of Light, Lambert Acaemic Publihig (LAP Iteratioal), Berli, Germay, 4. [] L. Mael a E. Wolf, Optical Coherece a Quatum Optic, Cambrige Uiverity Pre, Cambrige 995. [3] P. A. M. Dirac, Proceeig of the Royal Society (Loo) A 8 (98), 35. [4] P. A. M. Dirac, The Priciple of Quatum Mechaic, Oxfor Uiverity Pre, Oxfor, 958. [5] M. Sach, Aale Foatio Loui e Broglie 3 (5), 38. [6] L. H. For, It. J. Mo. Phy. A 5 (), 355; arxiv: [quat-ph]. [7] L. H. For a T. A. Roma, Phy. Rev. D 87 (3), 85; arxiv:3.859 [grqc]. [8] J. J. Sakurai, Moer Quatum Mechaic, The Bejami/Cummig Publihig Compay, Ic., Memlo Park, Califoria, 985. [9] G. B. Arfke a H. J. Weber, Mathematical Metho for Phyicit, Acaemic Pre, Ic., Sa Diego, Califoria, 995. [] G. Stepheo a P. M. Ramore, Avace Mathematical Metho for Egieerig a Sciece Stuet, Cambrige Uiverity Pre, Cambrige, 99.