By using two-component approach to the one-dimensional effective mass Dirac
equation bound states are investigated under the effect of two new
non-PT-symmetric, and non-Hermitian, exponential type potentials. It is
observed that the Dirac equation can be mapped into a Schr\"{o}dinger-like
equation by rescaling one of the two Dirac wave functions in the case of the
position dependent mass. The energy levels, and the corresponding Dirac
eigenfunctions are found analytically.Comment: 10 page

Arda, Altug

Sever, Ramazan

Chinese Physics Letters

English

arXiv

By using a two-component approach to the one-dimensional effective mass Dirac equation, bound states are investigated under the effect of two new non-PT-symmetric and non-Hermitian exponential type potentials. It is observed that the Dirac equation can be mapped into a Schrodinger-like equation by rescaling one of the two Dirac wave functions in the case of the position-dependent mass. The energy levels and the corresponding Dirac eigenfunctions are found analytically

Arda, ALTUĞ

Hacettepe  University Reserach Information System

Exact Solutions of Effective Mass Dirac Equation with Non-PT-Symmetric and Non-Hermitian Exponential-type Potentials

OpenMETU (Middle East Technical University)

arXiv:0908.1290v1  [quant-ph]  10 Aug 2009Exact Solutions of Effective Mass Dirac Equation withnon-PT -Symmetric and non-Hermitian Exponential-typePotentialsAltuğ Arda1, ∗ and Ramazan Sever2, †1Department of Physics Education, Hacettepe University, 06800, Ankara,Turkey2Department of Physics, Middle East Technical University, 06800, Ankara,Turkey(Dated: November 17, 2018)AbstractBy using two-component approach to the one-dimensional effective mass Dirac equation boundstates are investigated under the effect of two new non-PT -symmetric, and non-Hermitian, expo-nential type potentials. It is observed that the Dirac equation can be mapped into a Schrödinger-likeequation by rescaling one of the two Dirac wave functions in the case of the position dependentmass. The energy levels, and the corresponding Dirac eigenfunctions are found analytically.Keywords: Dirac equation, Position-Dependent Mass, non-PT -symmetric potentialPACS numbers: 03.65.-w; 03.65.Ge; 12.39.Fd∗E-mail: arda@hacettepe.edu.tr†E-mail: sever@metu.edu.tr1The investigation of the quantum systems having so-called non-Hermitian Hamiltonianshas been a great interest because of its theoretical contributions to quantum mechanics [1-15]. The study of such quantum systems has been received many applications, especially inquantum field theories [16-18], and nuclear theory [19]. The solutions of the non-relativisticand relativistic equations with non-Hermitian Hamiltonians having real or complex energyspectra have also been studied by many authors by using different methods [3, 6, 11, 20].Non-Hermitian Hamiltonians satisfy the condition given by ÔHÔ−1 = ÔHÔ = H , where theoperator Ô ≡ PT is an operator combined with parity, and time-reversal transformations,respectively. So they are related with PT -symmetry [8]. If a potential V (x) has the conditionwritten as V (−x) = V ⋆(x) under the transformation of x→ −x, and i→ −i, then it is saidthat the potential is a PT -symmetric.The relativistic wave equations, especially Dirac equation, give numerous important re-sults, and explanations in the view of quantum mechanics. One of them is the spin-orbitcoupling, and related topics, such as the Hall effect, spin torque, Zitterbewegung, etc [21-23]. Many authors have also been studied the solution of the Dirac equation with differentpotentials in different theoretical backgrounds [24, 25].Recently, the generalization of the solutions of the relativistic and non-relativistic waveequations with PT -symmetric potentials in the case of the constant mass to the case ofthe position-dependent mass become very attractive research topic. Many authors havestudied the effects of the position-dependent mass on the solutions of the above equations[26-36]. The new formalism based on the position-dependent mass is a useful ground tostudy the properties of some physical systems, such as quantum dots [37], semiconductorheterostructures [38], quantum liquids [39].In this letter, we intend to study the effects of the position-dependent mass on the solu-tions of the one-dimensional Dirac equation by using the map between the mass, and po-tential functions without the spin-effects [43]. Our aim, by taking two examples of the massdistributions, is to show that the two-component approach [40, 41] to the one-dimensionalDirac equation can be used in the position-dependent mass formalism. We obtain two newnon-PT -symmetric, and non-Hermitian potential functions produced by using the map be-tween the mass, and potential functions.2The relativistic Dirac equation in the absence of an external potential can be written as(iγµ∂∂xµ−m)Ψ = 0 , (1)where m is the rest mass of the particle, and γµ (µ = 0, 1, 2, 3) are gamma matrices (h̄ = c =1). The Dirac equation for a particle moving in an external potential V (x) in one-dimensioncan be written as [40][α . p+ βm(x)− (E − V (x))]ψ(x) = 0 , (2)where E is the relativistic energy of the particle, p is the momentum operator, m(x) isthe mass function and α, β are 2 × 2 matrices which are set to Pauli matrices σ3 and σ1,respectively. The Dirac wave function, ψ(x), is decomposed into an upper ψ+(x), and lowercomponent ψ−(x), so thatψ(x) =(ψ+(x)ψ−(x)), (3)We have the following set of two coupled differential equations by inserting Eq. (3) intoEq. (2)dψ+(x)dx+ [m(x) + (E − V (x))]ψ−(x) = 0 , (4)dψ−(x)dx+ [m(x)− (E − V (x))]ψ+(x) = 0 . (5)We define the wave functions f(x) and g(x) in two-component approach which makespossible to obtain the solutions easily without the spin effects [40, 41] as(f(x)g(x))=(1 i1 −i)(ψ+(x)ψ−(x)), (6)and insert into Eqs. (4) and (5), we havedf(x)dx− i[E − V (x)]f(x) + im(x)g(x) = 0 , (7)dg(x)dx+ i[E − V (x)]g(x)− im(x)f(x) = 0 . (8)By using the transformation written in terms of mass function m(x) in Eq. (7)f(x) =√m(x) φ(x) , (9)3one getsidφ(x)dx+[im′(x)2m(x)+ (E − V (x))]φ(x)−√m(x) g(x) = 0 . (10)where prime denotes the derivative to the spatial coordinate. To obtain a Schrödinger-likeequation for φ(x), one sets the potential term V (x) in Eq. (10) asV (x) = im′(x)2m(x), (11)which givesd2φ(x)dx2− Veff(x)φ(x) = −E2φ(x) . (12)This is the Schrödinger equation having the energy eigenvalue E2 under the effect ofthe potential Veff(x) = m2(x). By choosing a suitable mass function, one can constructa potential V (x) from Eq. (11). By solving the Schrödinger-like equation given by Eq.(12), one obtains the energy spectra and corresponding spinors of the Dirac equation. TheSchrödinger and Schrödinger-like equations can be solved by using the calculation procedure,which suggests a hypergemetric type equation in the following form [42]d2ψ(z)dz2+τ̃(z)σ(z)dψ(z)dz+σ̃(z)σ2(z)ψ(z) = 0 , (13)where σ(z), σ̃(z) are polynomials at most second degree and τ̃(z) is a first degree polynomial.By using the transformation ψ(z) = χ(z)y(z), one getsσ(z)d2y(z)dz2+ τ(z)dy(z)dz+ λy(z) = 0 , (14)The first part of the total wave function χ(z) is defined as [42]χ′(z)χ(z)=π(z)σ(z), (15)and the other part of the solution y(z) is given by the Rodrigues relationyn(z) =anρ(z)dndzn[σn(z) ρ(z)] , (16)where an is a normalization constant, and ρ(z) is the weight function obtained from thefollowing relation [42]ddz[σ(z) ρ(z)] = τ(z) ρ(z). (17)4The polynomial π(z) in Eq. (15), and the parameter λ required for the method are givenbyπ(z) =σ′(z)− τ̃(z)2±√√√√(σ′(z)− τ̃(z)2)2− σ̃(z) + kσ(z) . (18)λ = k + π′(z) . (19)The discriminant of the expression under the square root in the polynomial π(z) in Eq. (18)must be zero, which defines the constant k. Thus, a new eigenvalue equation becomesλ = λn = −nτ ′(z)−n(n− 1)2σ′′(z) . (20)where τ(z) = τ̃(z) + 2π(z) has a negative derivative. The energy eigenvalues are obtainedfrom the Eqs. (19) and (20).We prefer the following mass function as a first casem(x) = m0(1 + e− δx) , (21)which gives an exponential type, non-PT -symmetric, non-Hermitian potentialV (x) = −iδ2e− δx1 + e− δx. (22)By using the mass function, Eq. (12) givesd2φ(x)dx2+{E2 −m20 − 2m20e− δx −m20e−2 δx}φ(x) = 0 , (23)By inserting the new variable z−1 = eδx, one obtains the following equationd2φ(z)dz2+1zdφ(z)dz+1z2{Ẽ2 − α2 − 2α2z − α2z2}φ(z) = 0 . (24)where Ẽ2 = E2/δ2, and α2 = m20/δ2. Comparing Eqs. (24) and Eq. (13), and using theparameters −a1 = −α2, −a2 = −2α2, and −ǫ = Ẽ2 − α2, we obtainτ̃(z) = 1 ; σ(z) = z ; σ̃(z) = −a1z2 − a2z − ǫ . (25)To obtain the polynomial π(z), one inserts these polynomials into Eq. (18), with σ ′(z) =1, then we getπ(z) = ±√a1z2 + (k + a2)z + ǫ , (26)5We obtain two solutions for k as k1,2 = −a2±2√a1ǫ , from the condition that the discriminantof the expression under the square root has to be zero. Eq. (18) givesπ(z) =±[√a1 z +√ǫ ], k → k1 ,±[√a1 z −√ǫ ], k → k2 .We find the polynomial τ(z) by choosing k2 asτ(z) = τ̃(z) + 2π(z) = 1 + 2√ǫ − 2√a1 z , (27)which gives τ ′(z) = −2√a1 < 0. Thus, an energy eigenvalue equation is obtained from Eqs.(19) and (20) as(2n+ 1)√a1 + 2√a1ǫ + a2 = 0 , (28)E = ±√m20 −δ24[2n+ 1 +2m0δ]2. (29)It can be seen that the energy levels of particle and antiparticle are symmetric aboutzero. We observe that it is obtained zero energy in the case of the limit δ → 0 which is theenergy level of the following equation derived from Eq. (23)d2φ(x)dx2+ ω20φ(x) = 0 , (30)where ω20 = E2 −m20 . The Dirac Hamiltonian with a general scalar potential, V ′(x), givesalways zero-energy solutions in the ultrarelativistic limit [43], where the upper and lowercomponents given in Eq. (2) are written asψ+(x) ∼ e−(mx+h(x)) , (31)ψ−(x) ∼ e+(mx+h(x)) . (32)with h(x) =∫ x V ′(y)dy , m denotes the rest mass of particle. The normalization conditionsfor zero-energy eigenstates are discussed in Ref. [43]. The solutions of Eq. (30) have thesame form with the ones given in Eqs. (31) and (32) for the limit δ → 0 , i.e., h(x) = 0 form→ 2m0 . One has to find first the weight function ρ(z) to obtain the Dirac wave functionφ(z). Using Eq. (17), we getρ(z) = e−2√a1 z z2√ǫ (33)6and from Eq. (16), we obtainyn(z) =ane−2√a1 z z2√ǫdndzn[e−2√a1 z zn+2√ǫ]. (34)From the last equation, we can write yn(z) in terms of the generalized Laguerre polynomialsas yn(z) ≃ L(2√ǫ )n (z) [43]. The other part of the solution is obtained from Eq. (15) asχ(z) = e−√a1 z z√ǫ . (35)Finally, we obtain the unnormalized wave function asφn(z) = e−√a1 z z√ǫ L(2√ǫ )n (z) . (36)Thus, we write the upper component from Eq. (9) asfn(z) =√m0(1 + z) e−√a1 z z√ǫ L(2√ǫ )n (z) (37)and the lower component from Eq. (8) asgn(z) =e−√a1 z z√ǫ L(2√ǫ )n (z)√m0(1 + z){− iδz[√ǫz−√a1 −L(2√ǫ )n−1 (z)L(2√ǫ )n (z)+ E]}. (38)It is seen that the mass of the Dirac particle contributes to the eigenfunctions, and theupper component depends on the energy of the particle.We choose the second mass distributionm(x) =m01 + e− δx, (39)which gives following exponential type, non-PT -symmetric and non-Hermitian potentialfrom Eq. (11) asV (x) =iδ211 + eδx. (40)It has the form of the Woods-Saxon potential for iδ/2 → −V0 [45]. Substituting Eq. (39)into Eq. (12), and using the new variable z = 1/(1 + e− δx), we obtaind2φ(z)dz2+1− 2zz(1− z)dφ(z)dz−{ Ẽ2z2(1− z)2+α2(1− z)2}φ(z) = 0 , (41)where −Ẽ2 = E2/δ2. Comparing the Eqs. (41) and Eq. (13), we getτ̃ (z) = 1− 2z ; σ(z) = z(1 − z) ; σ̃(z) = −Ẽ2 − α2z2 . (42)7Following the same procedure as in last section, we get the energy spectraE = ±14√√√√√8m20 − [√δ2 + 4m20 − δ(2n+ 1)]2 −( 4m20√δ2 + 4m20 − δ(2n+ 1))2. (43)and the corresponding wave functions are written in terms of the Jacobi polynomialsφn(z) = zẼ (1− z)−M P (2Ẽ ,−2M)n (1− 2z) . (44)The lower and upper components gn(z) and fn(z) are written by using Eqs. (44) and (8)respectively asfn(z) =√m0 zẼ+1/2 (1− z)−M P (2Ẽ ,−2M)n (1− 2z) , (45)andgn(z) = zẼ−1/2 (1− z)−M P (2Ẽ ,−2M)n (1− 2z){ iδ√m0[12(1− z) + Ẽ(1− z)+ Mz − z(1− z)(n + 1 + 2Ẽ − 2M)P(1+2Ẽ , 1−2M)n−1 (1− 2z)P(2Ẽ ,−2M)n (1− 2z)]+E√m0}. (46)where M =√Ẽ2 + α2 .The energy levels of the particles and antiparticles are symmetric about zero in Eq. (43).The non-PT -symmetric and non-Hermitian potential given by Eq. (40) has a real energyspectrum under the condition that 8m20 > [√δ2 + 4m20 − δ(2n+1)]2 +(4m20√δ2+4m20−δ(2n+1))2.The energy expression in Eq. (43) gives zero energy in the limit of δ → 0 as in the first massdistribution case. Both of the eigenfunctions are dependent on the rest mass of the particle.In summary, we have investigated the position-dependent mass energy spectra, and thecorresponding wave functions of the Dirac equation in two-component approach. We havetransformed it into a Schrödinger-like equation with the help of a transformation applied onone of the Dirac wave functions by using two different mass distributions. Thus two newnon-PT -symmetric and non-Hermitian complex potentials are produced. We have foundthat the Dirac equation can be turned into a Schrödinger-like equation by using a suitabletransformation depending on the mass function m(x) in two-component approach. Thisformalism gives us two new non-PT -symmetric and non-Hermitian exponential and inverse-exponential potentials. We have obtained a complex energy spectrum for the exponentiallypotential given by Eq. (22), and the potential given by Eq. 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Exact Solutions of Effective Mass Dirac Equation with non-PT-Symmetric and non-Hermitian Exponential-type Potentials

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