Solution of the Dirac equation with position-dependent mass in the Coulomb field

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1 Soutio of the Dirac equatio with positio-depedet mass i the Couomb fied A. D. Ahaidari Physics Departmet, Kig Fahd Uiversity of Petroeum & Mieras, Box 547, Dhahra 36, Saudi Arabia e-mai: haidari@maiaps.org We obtai exact soutio of the Dirac equatio for a charged partice with positiodepedet mass i the Couomb fied. The effective mass of the spior has a reativistic compoet which is proportioa to the square of the Compto waveegth ad varies as /r. It is suggested that this mode coud be used as a too i the reormaizatio of utravioet divergeces i fied theory. The discrete eergy spectrum ad spior wavefuctio are obtaied expicity. PACS umbers: 3.65.Pm, 3.65.Ge,..Gh I. INTRODUCTION Systems with spatiay depedet effective mass were foud to be very usefu i studyig the physica properties of various microstructures. Specia appicatios i codesed matter are foud i the ivestigatio of eectroic properties of semicoductors [], quatum wes ad quatum dots [], 3 He custers [3], quatum iquids [4], graded aoys ad semicoductor heterostructures [5], etc. These appicatios stimuated the deveopmet of various methods ad techiques for studyig systems with mass that depeds o positio. Recety, severa cotributios have emerged that give soutios of the wave equatio for such systems. The oe dimesioa Schrödiger equatio with smooth mass ad potetia steps was soved exacty by Dekar et a. [6]. The formaism of supersymmetric quatum mechaics was exteded to icude positio depedet mass [7]. Shape ivariace was aso addressed i this settig ad the eergy spectra were obtaied for severa exampes. A cass of soutios was obtaied expicity for such systems with equi-spaced spectra [8]. Coordiate trasformatios i supersymmetric quatum mechaics were used to geerate isospectra potetias for systems with positio-depedet mass [9]. The orderig ambiguity of the mass ad mometum operators ad its effect o the exact soutios was addressed by de Souza ad Ameida [] where severa exampes are cosidered. so(,) Lie agebra as a spectrum geeratig agebra ad as a potetia agebra was used to obtai exact soutios of the effective mass wave equatio []. Poit caoica trasformatio (PCT) was used to obtai the eergy spectra ad wavefuctios for a arge cass of probems i oe ad three dimesios []. A cass of quasi-exacty sovabe probems with effective mass was preseted by Koç et a. [3] where the wavefuctios are obtaied i terms of orthogoa poyomias satisfyig mass depedet recurrece reatio. The agebraic method utiizig a give reaizatio of the geerators of the su(,) agebra was used i [4] to geerate exacty sovabe systems with positio-depedet mass. The oreativistic Gree s fuctio was formay addressed i [5] by usig path itegra formuatio to reate the costat mass Gree s fuctio to that of positio-depedet mass. Quite recety, the PCT method deveoped i [] was exteded to give a expicit costructio of the two-poit Gree s fuctio for various potetia casses of systems with positio-depedet mass [6]. The reativistic extesio of these formuatios, o the other had, remais udeveoped. Oe of the advatages of such a extesio is the eimiatio of the

2 orderig ambiguity of the mass ad mometum operators which is preset i the oreativistic kietic eergy P. This issue was addressed by Cavacate et a. [7]. m I this artice we obtai a soutio of the three dimesioa Dirac equatio with the Couomb potetia for a partice with positio-depedet mass. The discrete eergy spectrum ad spior wavefuctio are obtaied. I atomic uits (ħ = m = ), we take the foowig sphericay symmetric siguar mass distributio mr () = ƛ r (.) where ƛ is the Compto waveegth ħ mc = c, ad is a rea scae parameter with iverse egth dimesio. The rest mass of the partice ( m = ) is obtaied either as the asymptotic imit (r ), or the oreativistic imit ( ƛ ) of m(r). Cosequety, a possibe iterpretatio for this siguar mass term might be foud i reativistic quatum fied theory. It coud be cosidered as a mode cotributig to the reormaizatio of utravioet divergeces which occur at high eergies (equivaety, sma distaces r ). I such a mode, these divergeces are reormaized ito the siguar mass term where stads for the reormaizatio scae. It shoud aso be oted that this positiodepedet mass term has a reativistic origi as we sice it is proportioa to the Compto waveegth which vaishes as c (equivaety, ƛ ). I the foowig sectio, we set up the three dimesioa Dirac equatio for a spior with positio-depedet mass iteractig with the eectromagetic 4-potetia ( A, A ). We impose spherica symmetry ad cosider the specia case where the space compoet of the eectromagetic potetia vaishes (i.e., A = ). The time compoet, o the other had, is take as the Couomb potetia. That is A = ƛ Vr (), where Vr () = Zrad Z is the partice charge i uits of e. The probem is the reduced to sovig the radia compoet of the Dirac equatio. A goba uitary trasformatio is appied to this equatio to separate the variabes such that the resutig secod order radia differetia equatio for the two spior compoets become Schrödiger-ike. This makes the soutio of the reativistic probem easiy attaiabe by simpe ad straightforward correspodece with the we-kow exacty sovabe oreativistic probem. The correspodece resuts i a map amog the reativistic ad oreativistic parameters. Usig this map ad the kow oreativistic eergy spectrum oe ca easiy ad directy obtai the reativistic spectrum. Moreover, the two radia compoets of the spior wavefuctio are obtaied from the oreativistic wavefuctio usig the same parameter map. II. ENERGY SPECTRUM AND WAVEFUNCTION Dirac equatio is a reativisticay ivariat first order differetia equatio i 4- dimesioa space-time for a four-compoet wavefuctio ( spior ) ψ. For a free structureess partice it reads ( iħ Mc) ψ =, where M is the mass of the partice ad c the speed of ight. The summatio covetio over repeated idices is used. That 3 is, = =. =.. ct { } 3 = are four costat square, ν ν ν ν = = η, where matrices satisfyig the aticommutatio reatio { }

3 η is the metric of Mikowski space-time which is equa to diag(,,, ). A fourdimesioa matrix represetatio that satisfies this reatio is as foows: I σ =, = (.) I σ where I is the uit matrix ad σ are the three hermitia Paui matrices. The mass M is geeray space-time depedet i which case it shoud trasform as a scaar fuctio uder the actio of the Loretz trasformatio. I atomic uits (ħ = m = e = ), = c ƛ ad Dirac equatio reads ( i m ) ψ = ƛ, where m = M m = mtr (, ). Next, we et the Dirac spior be charged ad couped to the 4-compoet eectromagetic potetia A = ( A, A). Gauge ivariat coupig, which is accompished by the miima substitutio, trasforms the free Dirac equatio to ia i ( ia ) m ƛ ψ = which, whe writte i detais, reads as foows i m ƛ ψ = t ( iα α A A β) ψ ƛ Hψ (.) ƛ where H is the Hamitoia, α ad β are the hermitia matrices defied as = = I σ α ad β = = (.3) σ I Substitutig these i Eq. (.) gives the foowig matrix represetatio of the Hamitoia m ƛA ƛiσ ƛσ A H = (.4) iσ σ A m A ƛ ƛ ƛ Thus the eigevaue wave equatio reads ( H εψ ) =, where ε is the reativistic eergy which is rea. Now, we choose A = ad impose spherica symmetry by takig A = ƛ Vr () ad m = m(r). I this case, the aguar variabes coud be separated ad we ca write the spior wavefuctio as [8] j igr [ ( ) r] χ m ψ = j (.5) [ f( r) r] σ rˆ χ m where f ad g are rea radia fuctios, ˆr is the radia uit vector, ad the aguar wavefuctio for the two-compoet spior is writte as m/ ± m / Y j χ m( Ω ) =, for j = ± ½ (.6) m / ± m / Y m / Y ± is the spherica harmoic fuctio. I Eqs. (.5) ad (.6), the etter m stads for the itegers i the rage,,..., ad shoud ot be cofused with the positiodepedet mass m(r). Spherica symmetry gives iσ ( r ) ψ( r, Ω ) = ( κψ ) ( r, Ω), where κ is the spi-orbit quatum umber defied as κ =± ( j ½) =±, ±,... for = j ± ½. Usig this we obtai the foowig usefu reatios j df κ j ( σ )( σ rfr ˆ) () χm = F χ m dr r (.7) j df κ j ( σ ) Fr () χ ( ˆ m = F σ r) χ m dr r 3

4 Empoyig these i the wave equatio ( H εψ ) = resuts i the foowig matrix equatio for the two radia spior compoets mr () ƛ V() r ε ƛ κ d () ( r gr dr ) = (.8) κ d ƛ( r ) mr () ƛ V() r ε dr f() r Now, we speciaize further to the Couomb potetia Vr () = Zr ad take m(r) as the positio depedet mass give by Eq. (.). This maps Eq. (.8) ito the foowig Z ƛ r ε ƛ κ d ( r g dr ) κ d Z = (.9) ƛ( r ) ƛ dr r ε f which resuts i two couped first order differetia equatios for the two radia spior compoets f ad g. Eimiatig oe compoet i favor of the other gives a secod order differetia equatio. This equatio is ot Schrödiger-ike (i.e., it cotais first order derivatives). To obtai a Schrödiger-ike equatio we proceed as foows. A i goba uitary trasformatio U ( η) = exp( ƛ ησ ) is appied to the radia Dirac equatio (.9), where η is a rea costat parameter ad σ is the Paui matrix ( i i ). The Schrödiger-ike requiremet dictates that the parameter η satisfies the costrait: C Sκ ƛ =± Z (.) where S = si( ƛ η), C = cos( ƛ η) ad π ƛ η π. The soutio of this costrait gives two ages whose cosies are ( ) ( ) C = κ ƛ Z τ κ κ ± Z > ƛ ƛ (.) where τ is a ± sig which is chose such that the resutig vaue is positive. We choose ot to fix the sig i Eq. (.) but cotiue the deveopmet with the two possibiities. Cosequety, oe shoud distiguish betwee two vaues for the aguar parameters C ad S (e.g., C ± ad S ± ). However, for ecoomy ad simpicity of otatio we wi maitai the same symbos. Equatio (.9) is ow trasformed ito the foowig ( ) Z S C ε ± ƛ d () r ƛ( r φ r ƛ dr ) S = (.) d ƛ( ) ( ) Z r C ε ƛ dr r φ () r ƛ κ where = τ κ ƛ ( Z ) ad κ η η φ ƛ ƛ cos si g Uψ = = η η (.3) φ ƛ ƛ si cos f Equatio (.) gives oe spior compoet i terms of the other as foows ± ƛ S d φ = ± φ (.4) C ε ƛ r dr Whereas, the resutig Schrödiger-ike wave equatio becomes d ( ± ) Zε ε () r φ ± = dr r r (.5) ƛ 4

5 Comparig this equatio with that of the we-kow oreativistic Couomb probem with costat mass d ( ) Z E Φ () r = dr r r (.6) gives, by correspodece, the foowig two maps betwee the parameters of the two probems: for φ : or, Z Zε, E ( ε ) ƛ (.7a) for φ : or, Z Zε, E ( ε ) ƛ (.7b) The first (secod) choice for i each oe of these two maps is for positive (egative) vaues of κ, respectivey. Usig these parameter maps i the we-kow oreativistic eergy spectrum, E = Z ( ), gives the foowig reativistic spectrum Z Z Z ( ) ƛ ( ) ( ) ε = ƛ ƛ ± (.8) where stads for either oe of the four possibe aterative vaues i (.7) associated idepedety with φ ± ad ± κ. Oe ca easiy verify that i the oreormaizatio imit ( ) the famiiar reativistic spectrum for the Couomb probem with costat mass is recovered. Moreover, i the oreativistic imit ( ƛ ), the oreativistic spectrum is obtaied. It is aso iterestig to ote that i the absece of the Couomb iteractio (Z = ) the positio-depedet mass term is equivaet to a scaar potetia V () r r s = whose eergy spectrum is obtaied as ε =± ( ) obtaied usig the same parameter map (.7) ad the oreativistic wavefuctio ρ () r ρ ( ) ( )( ρ r) e r Φ = Γ Γ L ( ρ r) (.9) ƛ. Now, the radia compoets of the spior wavefuctio is where Z. The resut of this map, uder the compatibiity reatio (.4), is the foowig ( ) ω Γ C ε ( r ) ( ), ( ) r ω ω e L ω r κ > Γ C φ () r = ( ) ω Γ C ε ω r ( ω ) ( ), ( ) C r e L ω r κ < Γ (.) ω ( ) Γ C ε ( r ) ( ), ( ) r ω e L ω C ω r κ > Γ φ () r = ω ( ) Γ Cε ω r ( ω ) ( ), ( ) C r e L ω r κ < Γ (.) where ω = Zε ( ) ad C is give by Eq. (.) i which the ± sig goes with φ ±, respectivey. The two owest eergy states, where ε =± C, are ψ = ( φ ) for φ κ > ad ψ = ( ) for κ <. ρ = ( ) 5

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