application/xmlPseudospin symmetry in the relativistic Manning–Rosen potential including a Pekeris-type approximation to the pseudo-centrifugal termGao-Feng WeiShi-Hai DongPseudospin symmetryDirac equationManning–Rosen potentialPhysics Letters B 686 (2010) 288-292. doi:10.1016/j.physletb.2010.02.070journalPhysics Letters BCopyright © 2010 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26936864-529 March 20102010-03-29288-29228829210.1016/j.physletb.2010.02.070http://dx.doi.org/10.1016/j.physletb.2010.02.070doi:10.1016/j.physletb.2010.02.070http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.2

PLB26585S0370-2693(10)00268-610.1016/j.physletb.2010.02.070Elsevier B.V.Theory

Table 1

The bound state energy levels Enrk are calculated in pseudospin symmetry for different values nr and l˜. For a given Dirac eigenstate there exists a corresponding partner state. The parameters C=−6, A=30.52, α=1.5, β=20, M=1 with atomic units are taken.

l˜nr,k<0l, jEnr,k<0nr−1,k>0l+2,j+1Enr−1,k>011,−12s1/2−4.99866/−1.032380,24d3/2−4.99866/−1.0323821,−23p3/2−4.99772/−1.014940,35f5/2−4.99772/−1.0149431,−34d5/2−4.99656/−1.005610,46g7/2−4.99656/−1.0056141,−45f7/2−4.99517/−1.000800,57h9/2−4.99517/−1.00080Table 2

The bound state energy levels Enrk are calculated numerically in pseudospin symmetry. It is shown that the energy levels approach a constant when potential range parameter β goes to infinity. The parameters C=−6, A=30.52, α=1.5, M=1 with atomic units are taken.

nr−1, kβ=5β=10β=20β=300,1−4.99066/−1.78334−4.99754/−1.22331−4.99961/−1.05774−4.99983/−1.025820,2−4.89661/−0.97286−4.98939/−1.04230−4.99866/−1.03238−4.99978/−1.019300,3−4.74644/−1.01954−4.97825/−0.99836−4.99772/−1.01494−4.99932/−1.012450,4−4.51593/−1.21627−4.96351/−0.99108−4.99656/−1.00561−4.99905/−1.007471,1−4.97889/−1.35296−4.99447/−1.09556−4.99860/−1.02439−4.99955/−1.010881,2−4.85755/−0.974672−4.98322/−1.01478−4.99755/−1.01384−4.99912/−1.008251,3−4.68087/−1.07448−4.96939/−0.99463−4.99629/−1.00576−4.99880/−1.005271,4−4.41409/−1.31375−4.95194/−0.99672−4.99480/−1.00125−4.99843/−1.00295Pseudospin symmetry in the relativistic Manning–Rosen potential including a Pekeris-type approximation to the pseudo-centrifugal term

Gao-Feng

Wei

⁎

fgwei_2000@163.com

Shi-Hai

Dong

dongsh2@yahoo.com

aDepartment of Physics, Xi'an University of Arts and Science, Xi'an 710065, PR ChinabEscuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Edificio 9, Unidad Profesional Adolfo López Mateos, Mexico D.F. 07738, Mexico⁎Corresponding author. Tel.: +86 29 88272280; fax: +86 29 88272280.

Editor: W. Haxton

Abstract

Based on the Sturm–Liouville theorem and shape invariance formalism, we study by applying a Pekeris-type approximation to the pseudo-centrifugal term the pseudospin symmetry of a Dirac nucleon subjected to scalar and vector Manning–Rosen potentials including the spin–orbit coupling term. A quartic energy equation and spinor wave functions with arbitrary spin–orbit coupling quantum number k are presented. The bound states are calculated numerically. The relativistic Manning–Rosen potential could not trap a Dirac nucleon in the limit case β→∞.

Keywords

Pseudospin symmetryDirac equationManning–Rosen potential

Introduction

Pseudospin doublets was suggested forty years ago [1,2] based on the small energy difference between nuclear energy levels with quantum numbers (nr,l,j=l+1/2) and (nr−1,l+2,j=l+3/2), where the nr,l and j are the single-nucleon radial, orbital and total angular momentum quantum numbers, respectively. This doublet structure is expressed in terms of a “pseudo” orbital angular momentum l˜=l+1, the average of the orbital angular momentum of the two states in doublet, and “pseudo” spin s˜=1/2. Since j=l˜±s˜ for the two states in the doublet, the energy of the two states in the doublet are then approximately independent of the orientation of the pseudospin; that is, these doublets are almost degenerate with respect to pseudospin. Six years later, Bell and Ruegg found that pseudospin symmetry is a relativistic symmetry of the Dirac Hamiltonian that occurs when the scalar and vector potentials are equal in magnitude and opposite in sign [3]. This condition approximately holds for the relativistic mean fields of nuclei [4].This symmetry11

Its name arose from the fact that pseudospin was like spin but was not spin.

was considered in the context of deformation [5] and superdeformation [6], magnetic moment interpretation [7–9], identical bands [10,11], and effective shell-model coupling scheme [12]. Although there have been attempts to understand the origin of this “symmetry”[13,14], only recently has it been shown to arise from a relativistic symmetry of the Dirac Hamiltonian [15–17], in which Ginocchio has shown clearly that the quasi-degenerate pseudospin doublets in nuclei arise from the near equality in the magnitudes of an attractive scalar Vs and a repulsive vector Vv, relativistic mean fields Vs∼−Vv, and also revealed that the pseudospin symmetry of nuclear physics was identified as an SU(2) symmetry of the Dirac Hamiltonian discovered in Ref. [3]. Also, Ginocchio and his co-author have shown that the occurrence of approximate pseudospin symmetry in nuclei is connected with certain similarities in the relativistic single-nucleon wave functions of the corresponding pseudospin doublets [18]. Since such connection was made, many predictions follow most of which are summarized by Ginocchio, who recognized that the nuclear energy levels and transition rates are consistent with approximate pseudospin symmetry [17]. Ginocchio [15,19,20] showed that pseudospin symmetry is exact when the sum of the potentials Vs and Vv is equal to zero or a constant. Meng and his co-authors have studied this symmetry [21] again under the condition d[V(r)+S(r)]/dr=0, but they also found that spin symmetry [22] is exact under the condition d[V(r)−S(r)]/dr=0. In addition, Ginocchio studied the U(3) and pseudo-U(3) symmetry for the relativistic harmonic oscillator [23]. Other studies related to this subject have also been carried out [24–31].As an important physical potential, the Manning–Rosen potential [32] can be used to describe molecular vibration with the form(1)V(r)=ℏ22Mβ2[α(α−1)e−2r/β(1−e−r/β)2−Ae−r/β1−e−r/β], where α and A are two dimensionless parameters, and parameter β is related to the range of the potential. This potential keeps invariant by mapping α→1−α. The s-wave bound state energy eigenvalues and the corresponding s-wave scattering solutions are obtained by path integral approach [33] and function analysis method [34], respectively. In this Letter, we shall study the pseudospin symmetry of a Dirac nucleon subjected to scalar and vector Manning–Rosen potentials, which was not considered before to our knowledge.This Letter is organized as follows. In Section 2 using algebraic method we solve approximately the Dirac equation in pseudospin symmetry and obtain a quartic energy equation. In Section 3 the corresponding spinor wave functions are derived by using function analysis method and the bound state energy levels are calculated numerically. We summarize our conclusions in Section 4.2

Algebraic formalism to energy spectra

In the case of spherically symmetric potential, the Dirac equation of a nucleon with mass M moving in scalar and vector potentials can be written as (ℏ=c=1)(2){α⋅p+β[M+S(r)]}ψ(r)=[E−V(r)]ψ(r), where E is the relativistic energy of the system; α and β are the 4×4 Dirac matrices. As shown in Ref. [35], by taking the spherically symmetric Dirac spinor wave functions as(3)ψnk(r)=1r[Fnk(r)Yjml(θ,ϕ)iGnk(r)Yjml˜(θ,ϕ)],k=±(j+1/2), and substituting it into Eq. (2) yields two coupled differential equations as follows,(4)(ddr+kr)Fnk(r)=[M+Enk+S(r)−V(r)]Gnk(r),(5)(ddr−kr)Gnk(r)=[M−Enk+S(r)+V(r)]Fnk(r), from which we obtain the following two second-order differential equations for the upper and lower components,(6){d2dr2−k(k+1)r2−[M+Enk−Δ(r)][M−Enk+Σ(r)]+dΔ(r)dr(ddr+kr)M+Enk−Δ(r)}Fnk(r)=0,(7){d2dr2−k(k−1)r2−[M+Enk−Δ(r)][M−Enk+Σ(r)]−dΣ(r)dr(ddr−kr)M−Enk+Σ(r)}Gnk(r)=0, where Δ(r)=V(r)−S(r) and Σ(r)=V(r)+S(r). The spin–orbit quantum number k is related to the orbital angular momentum quantum number l. For given k=±1,±2,±3,…, we have j=|k|−1/2, l=|k+1/2|−1/2, l˜=|k−1/2|−1/2, and l(l+1)=k(k+1), l˜(l˜+1)=k(k−1).We are now in the position to study the pseudospin symmetry. Here, we take Σ(r)=C=constant, and Δ(r) as the Manning–Rosen potential(8)Δ(r)=12Mβ2[α(α−1)e−2r/β(1−e−r/β)2−Ae−r/β1−e−r/β], and inserting it into Eq. (7) gives us(9){d2dr2−k(k−1)r2+[Enk2−M2−C(M+Enk)]+M−Enk+C2Mβ2[α(α−1)e−2r/β(1−e−r/β)2−Ae−r/β1−e−r/β]}Gnk(r)=0. For this equation, the analytical solutions can be obtained only for k=1. However, if adopting a Pekeris-type approximation [36] to the pseudo-centrifugal term k(k−1)/r2, which can be expanded as the series nearby the minimum value point of the effective potential. For the Manning–Rosen potential, its minimum value is given by V(r0)=−A2/4β2α(α−1)(α>1) at r0=βγ where γ≡log[1+2α(α−1)/A]. Therefore, we may express the centrifugal term as follows:(10)1r2=1r02(1+x)2≈1r02−2xr02+3x2r02+O(x)3, where x=(r−r0)/r0. In addition, we may also approximately express it in the following way(11)1r2≈1r02[c0+c1e−r/β1−e−r/β+c2e−2r/β(1−e−r/β)2]. Comparing Eq. (10) with Eq. (11), we obtain the expansion coefficients c0,c1,c2 as follows:(12)c0=12α2(α−1)2−4α(α−1)[2A+3α(α−1)]γβ2[A+2α(α−1)]2γ4+[A+2α(α−1)]2γ2β2[A+2α(α−1)]2γ4,c1=8α2(α−1)2{−6α(α−1)+[3A+4α(α−1)]γ}Aβ2[A+2α(α−1)]2γ4,c2=−16α3(α−1)3{−3α(α−1)+[A+α(α−1)]γ}A2β2[A+2α(α−1)]2γ4. Obviously, Eq. (11) is a proper approximation to the centrifugal term if r0/β≪1 is satisfied. Inserting Eq. (11) into Eq. (9) allows us to obtain(13)d2dr2Gnk(r)=−{E˜nk−Veff.(r)}Gnk(r), where(14)E˜nk=[Enk2−M2−C(M+Enk)]−c0k(k−1)r02,Veff.(r)=14[−2λ1+2(λ1−λ2)coth(r/2β)+λ2cosh(r/β)csch2(r/2β)],λ1=(M−Enk+C)A2Mβ2+c1k(k−1)r02,λ2=−(M−Enk+C)α(α−1)2Mβ2+c2k(k−1)r02.Now, we apply the Sturm–Liouville theorem22

As Yang said in a talk on monopole: “For the Sturm–Liouville problem, the fundamental trick is the definition of a phase angle, which is monotonic with respect to the energy.”[37]. The phase angle information for a solution to the Schrödinger equation is contained in nothing but the logarithmic derivative G˜nk(r).

and shape invariance formalism to solve Eq. (13) and obtain energy equation. By introducing the logarithmic derivative G˜nk(r)=Gnk(r)−1dGnk(r)/dr, the Schrödinger-like equation (13) is equivalent to following non-linear Riccati equation(15)ddrG˜nk(r)=−[E˜nk−Veff.(r)]−G˜nk2(r). The G˜nk(r) decreases monotonically with respect to r between two turning points, where E⩾Veff.(r). Specifically, as r increases across a node of wave function Gnk(r), G˜nk(r) decreases to −∞, jumps to +∞, and then decreases again. By defining a new variable y=coth(r/2β), one has(16)dydr=−12β(y2−1),Veff.(y)=14[−2λ1+λ2+2(λ1−λ2)y+λ2y2],y∈(1,∞), for r∈(0,∞). For ground state G˜0k(y), the Riccati equation (15) becomes(17)−12β(y2−1)ddyG˜0k(y)=−{E˜0k−14[−2λ1+λ2+2(λ1−λ2)y+λ2y2]}−G˜0k2(y). It is well known that the wave function of ground state has no node and its logarithmic derivative G˜0k(y) has no pole and decreases monotonically as r increases in the region E⩾Veff.(y). For bound state, the possible solution to Riccati equation (17) satisfying above conditions is taken as(18)G˜0k(y)=Q1y+Q2,Q1>0. Substituting this into Eq. (17) yields(19)Q12−Q12β−λ24=0,2Q1Q2−λ12+λ22=0,E˜0k+Q22+Q12β+λ12−λ24=0. Their solutions are given by(20)Q1=14β[1+1+4β2λ2],Q2=λ1−λ24Q1,E˜0k=−(Q1+Q2)2=−(4Q12+(λ1−λ2)4Q1)2,G˜0k(y)=Q1y+Q2. For ground state G0k(r), its logarithmic derivative G˜0k(y) is essentially the same as the superpotential33

The present definition of W(r) is different from the conventional one G0k(r)=exp[−∫W(r)dr]. However, this does not affect the final results.

by the relation G0k(r)=exp[∫W(r)dr]=exp[∫G˜0k(r)dr] corresponding to following two partner Hamiltonians(21)H−=Aˆ†Aˆ=−d2dr2+V−(r),H+=AˆAˆ†=−d2dr2+V+(r), where(22)Aˆ=ddr−W(r),Aˆ†=−ddr−W(r),V±(r)=W2(r)∓W′(r),V−(r)=[Veff.(r)−E˜0k]. From these relations (20)–(22), we can obtain the supersymmetric partner potentials as(23)V±(r)=(λ1−λ2)216Q12+Q12+λ1−λ22coth(r/2β)+Q1(Q1±12β)csch2(r/2β), from which we find that V+(r) and V−(r) are shape invariant(24)V+(r,a0)=V−(r,a1)+R(a1), with(25)a0=Q1,a1=Q1+12β,an=Q1+n2β,R(a1)=(λ1−λ2)216Q12+Q12−[(λ1−λ2)216(Q1+12β)2+(Q1+12β)2],R(an)=[(λ1−λ2)216(Q1+n−12β)2+(Q1+n−12β)2]−[(λ1−λ2)216(Q1+n2β)2+(Q1+n2β)2]. Following the formalism of shape invariance and SUSYQM, we can obtain the energy levels of system as(26)E˜nk=E˜nk(−)+E˜0k=∑i=1nR(ai)+E˜0k=−[λ1−λ24(Q1+n2β)+(Q1+n2β)]2. From Eqs. (14) and (26), we can obtain the energy equation as follows:(27)Enk2−M2−C(M+Enk)=c0k(k−1)r02−[λ1−λ24(Q1+n2β)+(Q1+n2β)]2, where(28)Q1+n2β=1+2n+1+4β2λ24β,λ1−λ2=(M−Enk+C)[A+α(α−1)]2Mβ2+(c1−c2)k(k−1)r02. The energy levels Enk are determined by this quartic energy equation (27) and can be calculated numerically for given parameters.3

Eigenfunctions

The Dirac spinors can be derived principally by recurrence operator methods [38,39]. Nevertheless, we shall employ function analysis approach to obtain them. To this end, by using a new variable z=e−r/β∈(0,1) for r∈(0,∞) and inserting it into Eq. (13), we have(29)z2d2dz2Gnk(z)+zddzGnk(z)+β2[E˜nk+z(−1+z)λ1−z2λ2(−1+z)2]Gnk(z)=0. By taking trial wave function of the form,(30)Gnk(z)=zδ1(1−z)δ2gnk(z), and inserting it into Eq. (29), we have(31)(1−z)zgnk″(z)+[1+2δ1−z(1+2δ1+2δ2)]gnk′(z)−(δ2+2δ1δ2+β2λ1)gnk(z)=0, where(32)δ1=β−E˜nk>0,δ2=12[1+1+4β2λ2]>0. Here, it should be noted for the choices of parameters δ1 and δ2 that the boundary conditions of wave functions should be satisfied, i.e., Gnk(r)/r becomes zero when r is infinity, and Gnk(r)/r is finite when r goes to zero. These regularity conditions require δ1>0,δ2>0 and δ1,δ2∈ℜ. The solutions of Eq. (31) are given by gnk(z)=F12(a;b;c;z), where a=δ1+δ2−δ12−δ2+δ22−β2λ1=−n, b=2(δ1+δ2)+n, c=1+2δ1. According to general quantum condition a=−n, we may obtain the same energy equation as Eq. (27) after some complicated algebraic calculations. The corresponding lower component Gnk(r) can be expressed as(33)Gnk(r)=e−δ1r/β(1−e−r/β)δ2F12(−n;2(δ1+δ2)+n;1+2δ1;e−r/β). By using the recurrence relation of hypergeometric function [40],(34)ddz[F12(a;b;c;z)]=(abc)F12(a+1;b+1;c+1;z) and inserting Eq. (33) into Eq. (5), we obtain the corresponding upper component Fnk(r) as(35)Fnk(r)=Gnk(r)M−Enk+C[δ2e−r/ββ(1−e−r/β)−δ1β−kr]−n[2(δ1+δ2)+n]β(1+2δ1)(M−Enk+C)(1−e−r/β)δ2e(δ1+1)r/βF12(−n+1,2(δ1+δ2)+n+1;2(1+δ1);e−r/β). It is seen from Eqs. (33) and (35) that the spinors Gnk(r) and Fnk(r) satisfy the regularity boundary conditions.It is found from energy equation (27) that we cannot directly find the pseudospin symmetry for a Dirac nucleon subjected to scalar and vector Manning–Rosen potentials except for numerical calculations. For this purpose, we take a set of parameters M=1,A=30.52,C=−6,α=1.5 and β=20 to solve Eq. (27) numerically for the Dirac state 2s1/2 with nr=1 (the principal quantum number n=nr+l+1) and k=−1, from which we obtain two energy values of E1,−1:−4.99866,−1.03238. For the former E1,−1=−4.99866, the parameters are given by δ1=7.18099,δ2=2.47736, which satisfying the regularity boundary restrictions. Similarly, the latter E1,−1=−1.03238 also satisfies the regularity condition. Furthermore, we find that the state 4d3/2 has the same eigenvalue as that of state 2s1/2. This is a common knowledge that the pseudospin symmetry referring to quasi-degeneracy of single-nucleon doublets can be characterized with the non-relativistic quantum numbers (nr,l,j=l+1/2) and (nr−1,l+2,j=l+3/2) as shown in Table 1

Let us consider a special case, i.e., the limit case β→∞. It is found from Eqs. (27), (33) and (35) that, under the condition of pseudospin symmetry, the energy eigenvalue and corresponding upper and lower components become the following forms:(36)limβ→∞Enk=−M,orlimβ→∞Enk=C+M,(37)limβ→∞Fnk(r)=0,andlimβ→∞Gnk(r)=0. Obviously, the Fnk(r) and Gnk(r) become unbound in the limit case β→∞ and energy eigenvalues become a constant. That is to say, under the condition of pseudospin symmetry, there do no exist bound states in this limit. In fact, when β→∞ the Manning–Rosen potential reduces to the following form:(38)limβ→∞VMR(r)=α(α−1)2M1r2. By substituting this into Eq. (9), we are able to obtain the corresponding results as follows:(39)G(r)=rJ(121−4ξ,rE˜), where(40)ξ=(M−E+C)α(α−1)2M−k(k−1),E˜=E2−M2−C(M+E), which implies that this solution diverges for large r. Therefore, we may conclude that the relativistic Manning–Rosen potential could not trap a Dirac nucleon in this limit. The effects of potential parameter β on the energy levels are also shown in Table 2

. Since this quantum system keeps symmetry by interchanging k↔1−k, we calculate energy eigenvalues only for positive k as shown in Table 2. The numerical results show that the energy levels approach the constants Enk=−M or M+C when β→∞.

Concluding remarks

In this Letter we have studied the pseudospin symmetry of a Dirac nucleon subjected to scalar and vector Manning–Rosen potentials. The quartic energy equation and spinor wave functions for bound states have been obtained by algebraic formalism and function analysis method, respectively. It is shown that there exist negative-energy bound states in the case of pseudospin symmetry. It is also shown that energy spectra tend to a constant when potential parameter β goes to infinity, i.e., the relativistic Manning–Rosen potential could not trap a Dirac nucleon in the limit case β→∞. Before ending this Letter, we give two useful remarks. First, in this Letter we have presented a novel algebraic approach to study the pseudospin symmetry in the relativistic Manning–Rosen potential. The key issue is how to find the superpotential W(r), i. e. G˜0k(y) given in Eq. (18), which is found by using the Sturm–Liouville theorem.

Acknowledgement

We would like to thank the kind referee for invaluable and positive suggestions which have improved the present manuscript greatly and also Prof. S.G. Zhou for helpful discussions. This work was partially supported by Project 20100297-SIP-IPN, COFAA-IPN, Mexico.

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