# Spinors and Rodrigues Representations Associated with Orthogonal Polynomials

Authors: Zahra Bakhshi
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

An effective approach is presented to produce Schrödinger-like equation for the spinor components from Dirac equation. Considering electrostatic potential as a constant value yields a second-order differential equation that is comparable with the well-known solvable models in the nonrelativistic quantum mechanics for the certain bound state energy spectrum and the well-known potentials. By this comparison, the gauge field potential and the relativistic energy can be written by the nonrelativistic models and the spinors will be related to the orthogonal polynomials. It has also shown that the upper spinors wave functions based on the orthogonal polynomials can be given in terms of the Rodrigues representations. Association with the Rodrigues representations of orthogonal polynomials has also been investigated in the lower spinor components, since they are related to the upper spinor components according to first-order differential equation that is attained from Dirac equation.


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## Body

## 1. Introduction

In recent years, there has been a developing interest in search for exactly solvable systems in nonrelativistic and relativistic quantum mechanics. The expression exactly solvable means the eigenvalues and the eigenfunctions of the Hamiltonian operator of the physical system can be derived analytically in closed form. Solvable models are noteworthy because understanding of physics can only be brought with such solutions. Moreover, exact solutions are valuable tools for testing and improving numerical methods introduced to solve problems physically more interesting [1]. Since relativistic extensions of the exact solvable potentials are very useful to study the relativistic effects, various methods were employed to obtain the exact solution of the problem. Point canonical transformation [2–4], dynamical group [5, 6], factorization method [7], supersymmetric quantum mechanics, and shape invariance [8–10] are methods among many which were used in the search for exact solutions of wave function. Also, there are a lot of investigations that show how methods used to obtain analytical solutions of the Schrödinger equation can be extended to Dirac case [11–15].Alhaidari [11–13] applies a unitary transformation to Dirac equation such that the resulting second-order differential equation becomes Schrödinger-like equation so that comparison with the well-known nonrelativistic problems is transparent. If the electrostatic potential is assumed as a constant value, the second-order differential equation can be constituted for upper component by eliminating lower component, without applying a general local unitary transformation that eliminates the first-order derivative such as what Alhaidari has considered.In this method, by assuming electrostatic potential as a constant value, the second-order differential equation can be compared with the well-known solvable Schrödinger equation in the nonrelativistic quantum models. The wave functions in Schrödinger equation for the well-known potentials have been obtained on the orthogonal polynomials, such as Jacobi, generalized Laguerre, and Hermite polynomials and the energy eigenvalues spectrum can be accessible for each case. By comparing the second-order differential equation that has been obtained from Dirac equation with Schrödinger equation for the well-known potential such as Scarff-II, Pöshel-Teller, Mörse, 3D-oscillator, and shift-oscillator potentials, the gauge field potential can be written based on the well-known superpotentials that are related to the mentioned potentials. Therefore, the second-order differential is transformed to the solvable models with the exact solutions; it means that the relativistic energy eigenvalues can be gotten based on the nonrelativistic models, and also the spinors will be related to the orthogonal polynomials according to the nonrelativistic models. Then, Rodrigues representations and the differential equation of them are calculated for orthogonal polynomials. Moreover, the second-order differential equation also can be considered as a product of two first-order differential operators and the spinor wave function related to the differential equation that is expressed in terms of Rodrigues representations related to the orthogonal polynomials. Therefore, the solution of second-order differential equation can be considered with the determined relativistic energy and association with Rodrigues representations can be gotten for each orthogonal polynomial.This paper is organized as followed: In Section2, by using the point canonical transformations, the second-order differential equation is constituted with the gauge field potential and the energy spectrum that will be introduced based on the nonrelativistic models. Then, the association of Rodrigues representation with orthogonal polynomials is shown in the Sections 3, 4, and 5 for Jacobi, generalized Laguerre, and Hermite polynomials, respectively. In each section, all of the gauge field potentials are considered to have the ability to constitute the solvable models with the certain energy eigenvalues, for each orthogonal polynomial. Therefore, in the each section, Rodrigues representations of the orthogonal polynomials have been calculated for some gauge field potentials. In Section 6, the paper ends with a brief conclusion.


## 2. The Three-Dimensional Dirac Equation for a Free Structure

Particle of spin1/2 reads iħγμ∂μ-mcΦ=0, where m is the rest mass of the particle, c is the speed of light, and Φ is a four-component wave function. The four matrices γμμ=03 are given the following standard representation [16]:(1)γ0=I00-I,γ→=0σ→-σ→0,where I is the 2×2 unit matrix and σ are the usual 2×2 Pauli spin matrices. In atomic units m=e=ħ=1, Dirac equation reads iγμ∂μ-α-1Φ=0, where α=ħ/mc=1/c is the Compton wavelength of the particle. In the presence of the electromagnetic potential, Aμ=A0,cA→, gauge invariant coupling to the charged spinor is accomplished by the minimal substitution ∂μ→∂μ+iαAμ, which transforms free Dirac equation into(2)iγμ∂μ+iαAμ-α-1Φ=0.For time independent potential, (2) gives the following matrix representation of Dirac Hamiltonian (in units of mc2=α2) [14]:(3)H=α2A0+1-iασ→·∇→+ασ→·A→-iασ→·∇→+ασ→·A→α2A0-1.Taking into consideration gauge invariance, the form of electromagnetic potential for static charge distribution with spherical symmetry is(4)A0,A→=υr,r^ωr,where r^ is radial unit vector; υr and ωr are electrostatic potential and gauge field potential, respectively. By substituting the two off-diagonal terms ασ→·A→ by ±iασ→·A→ in (3), the Hamiltonian leads to the following two-component radial Dirac equation [17]:(5)α2υr+1αkr+ωr-ddrαkr+ωr+ddrα2υr-1φrθr=εφrθr,where ε are the relativistic energy eigenvalues and k is the spin-orbit coupling parameter defined as k=±j+1/2 for l=j±1/2. Equation (5) gives two coupled first-order differential equations for the radial spinor components. By eliminating lower spinor component and by assuming the electrostatic potential υr to be a constant value η, the second-order differential equation can be gotten for upper spinor wave function as(6)-d2φdr2+ωr+kr2-dωdr-kr2-α2η-ε2-1α2φr=0.Equation (5) also gives the lower spinor component in terms of the upper component as follows:(7)θr=αη-ε+1α-1ωr+krφr+dφdr.By comparing (6) with the solvable Schrödinger equation in the nonrelativistic models, the relation can be considered between the well-known potential in the nonrelativistic quantum models and the gauge field potential in the relativistic system as Vmr=ωr+k/r2-dω/dr-k/r2. Also, nonrelativistic energy eigenvalues can be related to the relativistic energy eigenvalues as E=(α2η-ε2-1)/α2. So, the gauge field potential and the relativistic energy due to solvability of Dirac equation based on the nonrelativistic quantum mechanics are easily available.


## 3. Association of Rodrigues Representation with Jacobi Polynomials

Let us consider the gauge field potentials where their wave functions are related to Jacobi polynomials such as Pöschl-Teller potentialω1r=-Acoth⁡r+B/sinh⁡r-k/r and Scarf-II potential ω2r=-Atanh⁡r-B/cosh⁡r-k/r, where A and B are real parameters. For each potential, respectively, (6) gives the following second-order differential equations for upper spinor component:(8)-d2φn,m1rdr2+A2+B2-A2+Asinh2⁡r+B-2ABcosh⁡rsinh2⁡rφn,m1r=α2η-ε2-1α2φn,m1r,(9)-d2φn,m2rdr2+A2+B2-A2+Acosh2⁡r+2AB-Bsinh⁡rcosh2⁡rφn,m2r=α2η-ε2-1α2φn,m2r,where A=(λ+γ+2m-1)/2 and B=(γ-λ)/2 such that λ,γ>-1 in (8) and A=m+λ-1/2 and B=γ/2 such that λ>-1 and -∞>γ>+∞ in (9). There will be the well-known nonrelativistic energy spectrum as En,m1=λ+γ+n+mm-n-1 and En,m2=2λ+n+mn-m+1 for Pöschl-Teller potential and Scarf-II potential, respectively. As mentioned before, they can be used to calculate relativistic energy spectrum of Dirac equation as the following forms:(10)εn1=α2η∓-α2λ+γ+n+mn-m+1+11/2,εn2=α2η∓-α22λ+n+mn-m+1+11/2.The bound states wave functions of the nonrelativistic problem [6] are mapped into the following upper spinor components wave functions:(11)φn,m1x∝x-12λ+2m-1/4×x+12γ+2m-1/4Pnλ+m-1,γ+m-1x,(12)φn,m2x∝1+x2-1/2×exp⁡γ2tanh-1⁡xPniγ/2+m+λ-1/4,-iγ/2+m+λ-1/4x,where Pnμ,νx is Jacobi polynomial with μ,ν>-1, and x=cosh⁡r, μ=λ+m-1, and ν=γ+m-1 in (11) and x=sinh⁡r, μ+i(γ/2)+m+λ-1/4, and ν=-i(γ/2)+m+λ-1/4 in (12). By substituting upper spinor components (11) and (12) into (7) and using recursion properties of Jacobi polynomials, lower spinor components are given as(13)θn,m1x∝1α∓1α2-n2-nλ+γ+2m-11/2-1×nx-λ-γ2n+2m+λ+γ-2x2-1-1/2φn,m1x-2n+m+λ-1n+m+γ-12n+2m+λ+γ-2x2-1-1/2φn-1,m1x,θn,m2x∝1α∓1α2+n-m+1m-n+2λ-21/2-1×-n+λ+34x+γ22n-3m-λ+3-n+2m-2x2+1-1/2+γ21-x-1x2+11/2φn,m2x-i-n+2m+λ-22+γ2/8-n+2m+λ-2x2+1-1/2φn-1,m2x.Raising and lowering operators B±m=±d/dr+Wmxr, where the superpotential Wmxr satisfies in the Riccati equation Vm=Wm2±Wm′, can be written as the following forms according to Pöschl-Teller and Scarf-II potentials, respectively:(14)B+1=ddr+-Acoth⁡r+Bsinh⁡r,B-1=-ddr+-Acoth⁡r+Bsinh⁡r,(15)B+2=ddr+-Atanh⁡r-Bcosh⁡r,B-2=-ddr+-Atanh⁡r+Bcosh⁡r.It is obvious that the second-order differential equations can always be considered in a factorization form as a product of a pair of linear differential operators (14) and (15). Therefore,(16)B+1,2mB-1,2mφn,m1,2r=E1,2n,mφn,m1,2r,B-1,2mB+1,2mφn,m-11,2r=E1,2n,mφn,m-11,2r.In the above equations, for a given n, the operator B+m raises the index m while the operator B-m lowers it. We can also obtain the highest state φn,n by solving the first-order differential equation B+n+1φn,nr=0 because the nonrelativistic energy spectrum En,m vanishes for m=n+1 [10], since by introducing a new function as φn,mx=A1/4xW1/2xψn,mx and changing the variable dx/dr=Ax Schrödinger equation (6) has been obtained from the general form associated with second-order differential equation in terms of master function Ax and the wave function Wx as follows [18–20]: (17)Axψ¨n,mx+AxWx′Wxψ˙n,mx+-12n2+n-m2A¨x+m-nAxW˙xWx′-m24A˙x2Ax-m2A˙xW˙xWxψn,mx=0,where ψn,mx is Rodrigues representation of the orthogonal polynomials in (17). For a positive integer n, ψn,mx as Rodrigues representation is given by(18)ψn,mx=-1mAm/2xddxmψnx,m=0,1,2,…,n,where(19)ψnx=NWxddxnAnxWx,with N which is normalization constant. So, for each case of the gauge field potentials, the Rodrigues representations of upper and lower spinors and differential equations associated with them are available. Furthermore, it can be shown that they are clear examples of connection between Jacobi polynomials and Rodrigues representations in Dirac equation. Thus, the wave functions ψn,mx are related to the upper spinors as the following forms:(20)ψn,m1x=x-1-λ/2+1/4x+1-γ/2+1/4φn,m1x,for A1x=x2-1 and W1x=x-1λx+1γ in Pöschl-Teller potential and(21)ψn,m2x=x2+1-λ/2+1/4exp⁡-γ2tan-1⁡xφn,m2x,when A2x=x2+1 and W2x=x2+1λexp⁡γtan-1⁡x in Scarf-II potential. Since the lower spinors can be connected to the upper spinors according to (13), therefore, Rodrigues representations of upper spinors also can be associated with lower spinors. If the wave function Θn,mx is introduced for lower spinor θn,mx, it can be written based on Rodrigues representations ψn,mx. The wave function Θn,mx that is connected to the lower spinors θn,mx can be written as follows for Pöschl-Teller and Scarf-II potentials, proportionately:(22)Θn,m1x∝1α∓1α2-n2-nλ+γ+2m-11/2-1×nx-λ-γ2n+2m+λ+γ-2x2-1-1/2ψn,m1x-2n+m+λ-1n+m+γ-12n+2m+λ+γ-2x2-1-1/2ψn-1,m1x,Θn,m2x∝1α∓1α2+n-m+1m-n+2λ-21/2-1×-n+λ+34x+γ22n-3m-λ+3-n+2m-2x2+1-1/2+γ21-x-1x2+11/2ψn,m2x-i-n+2m+λ-22+γ2/8-n+2m+λ-2x2+1-1/2ψn-1,m2x.The wave functions ψn,mx can be also satisfied in the second-order differential equations for each potential, correlatively:(23)x2-1ψ¨n,m1x+121-γ-λx3+12γ-λx2-1ψ˙n,m1x+1-mλ+γ+mx2+m-2γ-λx+λ+γ+n+mm-n-1x2-1+λ+γ+2m-122-m+1ψn,m1x=0,(24)x2+1ψ¨n,m2x+2λ+1x+γψ˙n,m2x+m2-n2-n+2λm-n-m2+2mλx2-γmxx2+1ψn,m2x=0.Rodrigues representations of the associated polynomialsψn,mx are given by(25)ψn,m1x=-1mx2-1m/2ddxmψn1x,m=0,1,2,…,n,ψn,m2x=-1mx2+1m/2ddxmψn2x,m=0,1,2,…,n,where ψnx satisfies in Jacobi differential equation whose Rodrigues representations, respectively, are(26)ψn1x=Nx-1-λx+1-γddxnx-1n+λx+1n+γ,(27)ψn2x=Nx2+1-λexp⁡-γtan-1⁡xddxnx2+1n+λexp⁡-γtan-1⁡x,where N is a normalization constant. Since the wave functions Θn,mx connected to the lower spinor θn,mx that have been calculated based on upper spinors φn,mx and Rodrigues representations of upper spinors can be generalized to the wave function Θn,mx as ψn,mx and ψn-1,mx, therefore, the above Rodrigues representation can also be related to the lower spinor components.


## 4. Association of Rodrigues Representation with Generalized Laguerre Polynomials

When Mörse potentialω1r=-(γ/2)e-r-m-λ/2+1/2-k/r and 3-dimensional oscillator potential ω2r=(γ/4)r-λ+m-1/2(2/r)-k/r are considered as the gauge field potentials, the upper spinor components are associated with generalized Laguerre polynomials. So, the second-order differential equations for the upper spinor components are written according to (6):(28)-d2φn,m1rdr2+γ24e-2r+γm+λ2-1e-rφn,m1r=α2η-ε2-1α2φn,m1r,-d2φn,m2rdr2+γ216r2+λ+m-12λ+m-321r2γ2λ+mφn,m2r=α2η-ε2-1α2φn,m2r.According to the nonrelativistic energy spectrum En,m1=-n-m+1λ+n+m and En,m2=γn-m+1, the relativistic energy spectrums are obtained as(29)εn1=α2η∓-α2n-m+1λ+n+m+11/2,εn2=α2η∓-α2γn-m+1+11/2.Second-order differential equations (28) are due to the solutions based on the generalized Laguerre polynomials as upper spinor wave functions(30)φn,m1x∝γx-n+λ/2+1/2×exp⁡-γ2xLn-2n-λ-1γx,(31)φn,m2x∝γxλ+m-1/2/2×exp⁡-γx2Lnλ+m-1/4γx,where Lnαx is generalized Laguerre polynomial with α>-1. In the upper spinor (30), x=er and α=-2n-λ-1 and, in the other upper spinor (31), x=r2/4 and α=λ+m-1/4. Lower spinor components can be attained by (7) for each potential(32)θn,m1x∝1α∓1α2+n-m+1-m-n-λ1/2-1×-2m-λ+1-n-m+1γxφn-1,m1x,θn,m2x∝1α∓1α2+γn-m+11/2-1×x-1/2n-m+1φn,m2γx-n+λ+34φn-1,m2γx.As mentioned in pervious section, the raising and loweringB+ and B- operators based on the superpotentials are given as(33)B+1=ddr-γ2e-r-m-λ2+12,B-1=-ddr-γ2e-r-m-λ2+12,B+2=ddr+γ4r-λ+m-122r,B-2=-ddr+γ4r-λ+m-122r.The pair of linear differential operators can factorize Schrödinger equation for each potential. Similar to the pervious section, in Mörse potential, if A1x=x2 and W1x=xλe-γ/x, the wave function ψn,m1x is written based on upper spinor as(34)ψn,m1x=exp⁡γ/2xxφn,m1x,and, in 3-dimensional oscillator potential, when A2x=x and W2x=xλe-γx, the wave function ψn,m2x is obtained as(35)ψn,m2x=x-λ/2+1/4exp⁡γx2φn,m2x.Therefore, they are also related to the generalized Laguerre polynomials. It is clear that both of them are examples of associating generalized Laguerre polynomial with Rodrigues representation in Dirac equation. Also, Rodrigues representation of lower spinors θn,mx that are called Θn,mx will be in the following forms based on ψn,mx and ψn-1,mx, for Mörse and 3-dimensional oscillator potentials:(36)Θn,m1x∝1α∓1α2+n-m+1-m-n-λ1/2-1×x-m+n+λγψn-1,m1x,Θn,m2x∝1α∓1α2+γn-m+11/2-1×xλ+1/2n-m+1ψn,m2x+n+λ+34ψn-1,m2x.The wave functions ψn,mx can be also satisfied in second-order differential equations for each potential, proportionately:(37)x2ψ¨n,m1x+λx+γ+2ψ˙n,m1x+-mγx+m2-m-2n2-nλ+1ψn,m1x=0,(38)xψ¨n,m2x+2λ+12-γxψ˙n,m2x+γn-m+12-γλ2-m-12λ+14x-1/2ψn,m2x=0.Rodrigues representations of associated polynomialsψn,mx are given as(39)ψn,m1x=-1mxmddxmψn1x,m=0,1,2,…,n,ψn,m2x=-1mxm/2ddxmψn2x,m=0,1,2,…,n,where ψnx satisfies in Laguerre differential equation whose Rodrigues representations are, respectively,(40)ψn1x=Nx-λexp⁡γxddxnx2n+λexp⁡-γx,(41)ψn2x=Nx-λexp⁡γxddxnxλ+nexp⁡-γx.The above Rodrigues representations can be also related to lower spinor components, because there are the wave function Θn,mx based on the wave function ψn,mx and ψn-1,mx according to the lower spinor components.


## 5. Association of Rodrigues Representation with Hermite Polynomials

The upper spinor component will be considered as Hermite polynomials, if the gauge field potentialωr=(γ/2)r-λ-k/r is written based on shift-oscillator potential. This upper spinor component satisfies in (6) as(42)-d2φn,mrdr2+γ2r-λ2-γ2φn,mr=α2η-ε2-1α2φn,mr.For this potential, the nonrelativistic energyEn,m=γn-m+1 can be used in the following relativistic energy spectrum:(43)εn=α2η∓α2γn-m+1+11/2.The upper spinor wave function based on the Hermite polynomials can be obtained from (42):(44)φn,mx∝exp⁡-γ4x2×Hnγ21/2x,where Hnx is Hermite polynomial. In the wave function (44) x=r-2λ/γ and -∞<x<+∞. According to (7), the lower spinor wave function is calculated as(45)θn,mx∝1α∓1α2+γn-m+11/2-1×2γ1/2n-m+1φn-1,mx.To factorize Schrödinger equation, there are pairs of linear differential operator as(46)B+=ddr+γ2r-λ,B-=-ddr+γ2r-λ.As mentioned before, the wave function ψn,mx that is related to upper spinor is gotten by(47)ψn,mx=exp⁡γ4x2φn,mx,for Ax=1 and Wx=exp⁡-(γ/2)x2. In this potential, Hermite polynomial can be associated with Rodrigues representation in Dirac equation. Also, similar to pervious section, the wave function Θn,mx that is connected to lower spinor θn.mx can be written based on Rodrigues representation:(48)Θn,mx∝1α∓1α2+γn-m+11/2-1×2γ1/2n-m+1ψn-1,mx.The second-order differential equation for shift-oscillator potential will be(49)ψ¨n,mx+-γxψ˙n,mx-γm-nψn,mx=0,where Rodrigues representation of the associated polynomial ψn,mx is considered by(50)ψn,mx=-1mddxmψnx,m=0,1,2,…,nand Rodrigues representation of ψnx, will be in the following form:(51)ψnx=Nexp⁡γ2x2ddxnexp⁡-γ2x2.Since Θn,mx has been used for lower spinor Θn,mx and it has been associated with ψn-1,mx according to (48), therefore, (49), (50), and (51) can be considered for the lower spinor component in shift-oscillator potential.


## 6. Conclusion

A procedure for connecting the methods used in the analysis of exactly solvable potentials in the nonrelativistic quantum mechanics with the solution of Dirac equation has presented. A gauge field potential and the bound states energy spectrum have been defined for the Dirac equation with a constant electrostatic potential that can be constituted by a Schrödinger-like equation. Since orthogonal polynomials are considered as the solution of Schrödinger-like equation that have been obtained from Dirac equation, Rodrigues representations of the orthogonal polynomials can be associated with upper and lower spinor components.



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