Spin and pseudospin symmetries of Dirac equation are solved under scalar and vector generalized isotonic oscillators and Cornell potential as a tensor interaction for arbitrary quantum number via the analytical ansatz approach. The spectrum of the system is numerically reported for typical values of the potential parameters.
1. Introduction
The solution of the fundamental dynamical equations is an interesting phenomenon in many fields of physics. The relativistic Dirac equation which describes the motion of spin 1/2 particle [1]. Within the framework of Dirac equation, pseudospin and spin symmetries are used to study features of deformed nuclei, superdeformation, and effective shell model [2]. The concept of SUSYQM provides theoretical physicists with a powerful tool to deal with nonrelativistic Schrödinger equation. The pseudospin symmetry is referred to as a quasidegeneracy of single nucleon doublets with nonrelativistic quantum number n,l,j=l+1/2 and (n-1, l+2, j=l+3/2), where n, l, and j are single nucleon radial, orbital, and total angular quantum numbers [3] and it was shown that the exact pseudospin symmetry occurs in the Dirac equation when dΣ(r)/dr=0; that is, Σ(r)=V(r)+S(r)=cps=const., where Vr and S(r) are repulsive vector potential and attractive scalar potential, respectively. Details of recent review of spin and pseudospin symmetries are given in [4]. These symmetries, under various phenomenological potentials, have been investigated using various methods such as Nikiforov-Uvarov (NU), supersymmetric quantum mechanics (SUSSYQM) and shape invariance (SI), ansatz approach, and asymptotic iteration (AIM) [5–15]. In recent years, the Dirac equation with different potentials in relativistic quantum mechanics with spin and pseudospin symmetry has been investigated [16–25]. The main aim of the present paper is to obtain approximate solutions of the Dirac equation with scalar and vector generalized isotonic oscillators and Cornell tensor interaction under the above mentioned symmetry limits. The isotonic oscillator potential consists of the harmonic oscillator plus centrifugal barrier which is of great interest in the theory of coherent states [26] and quantum optics [27] and in the analysis of the isochronous oscillator [28]. This potential is important because of its relation with supersymmetric quantum mechanics [29]. The two-dimensional version of the isotonic potential is superintegrable and usually is referred to as the Smorodinsky-Winternitz potential [30]. Here, we make use of the ansatz approach to deal with this complicated equation. A survey on the application of this technique to other wave equations including Dirac, Schrödinger, Klein-Gordon, spinless-Salpeter, and DKP equations can be found in [31–39]. The paper is organized as follows. In Section 2, we give a brief introduction of the supersymmetry quantum mechanics (SUSYQM). In Section 3, the Dirac equation is written for spin and pseudospin including the Cornell tensor interaction term. We next propose a physical ansatz solution to the equation and we solve the Dirac equation under these symmetries in Section 4. Finally, conclusion is presented in Section 5.
2. Dirac Equation including Tensor Coupling
In spherical coordinates, the Dirac equation with both scalar potential S(r) and vector potential V(r) can be expressed as [1, 2]
(1)α→·p→+β(M+S(r))-iβα→·r^U(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; p is the momentum operator, p→=-i∇→. For a particle in a spherical field, the total angular momentum operator J and spin-orbit matrix operator K=(σ→·L→+1) commute with the Dirac Hamiltonian, where σ and L are the Pauli matrix and orbital angular momentum, respectively. The eigenvalues of K are κ=-(j+1/2) for aligned spin (s1/2, p3/2, etc.) and κ=(j+1/2) for unaligned spin (p1/2, d3/2, etc.). The complete set of the conservative quantities can be taken as (H2, K, J2, Jz). As shown in [15], by taking the spherically symmetric Dirac spinor wave functions as
(2)ψnk(r→)=fnk(r→)gnk(r→)=Fnk(r→)rYjml(θ,φ)iGnk(r→)rYjml~(θ,φ),
where Fnk(r→) and Gnk(r→) are the radial wave functions of the upper and lower components, respectively, Yjml(θ,φ) and Yjml~(θ,φ), respectively, stand for spin and pseudospin spherical harmonics that are coupled to the angular momentum j and m is the projection of the angular momentum on the z-axis. The orbital angular momentum quantum numbers l and l~ refer to the upper and lower components, respectively. The quasidegenerate doublet structure can be expressed in terms of pseudospin angular momentum s=1/2 and pseudoorbital angular momentum l~, which is defined as l~=l+1 for aligned spin j=l~-1/2 and l~=l-1 for unaligned spin j=l~+1/2. As shown in [1, 2], substituting (2) into (1) yields two coupled differential equations as follows:
(3)ddr+κr-UrFnk(r)=M+Enk-Vr+SrGnk(r),(4)ddr-κr+UrGnk(r)=M-Enk+Vr+SrFnkr.
We consider the difference potential Δ(r) and sum potential Σ(r) as Δ(r)=V(r)-S(r) and Σ(r)=V(r)+S(r), respectively.
The spin-orbit quantum number κ is related to the orbital angular momentum quantum number l. By eliminating Gnk(r→) in (3) and Fnk(r→) in (4), we obtain the following two second-order differential equations for the upper and lower components:
(5)d2dr2-κ(κ+1)r2+2κrU(r)-dU(r)dr-U2(r)+dΔr/drM+Enκ-Δrddr+κr-UrFnκr=M+Enκ-ΔrM-Enκ+ΣrFnκ(r),(6)d2dr2-κκ-1r2+2κrUr+dUrdr-U2r+dΣr/drM-Enκ+Σrddr-κr+UrGnκr=M+Enκ-ΔrM-Enκ+ΣrGnκr.
We have κ(κ-1)=l~(l~+1) and κ(κ+1)=l(l+1).
3. Pseudospin and Spin Symmetric Solutions3.1. Pseudospin Symmetry Limit
The pseudospin symmetry occurs when dΣr/dr=0 or equivalently Σr=Cps=Const. [1, 2]. Here, we work on
(7)Δr=Ar2-2g2r2-12r2+12.
For the tensor term, we consider the Cornell potential
(8)Ur=apsr+bpsr.
Substitution of these two terms into (6) gives
(9)d2dr2+1r2(-κ(κ-1)+2κbps-bps-bps2)-ApsEnκps-M-Cpsr2-aps2r2-2gps2r2-12r2+12Enκps-M-Cps+Enκps+MEnκps-M-Cps-2κapsd2dr2+aps-2apsbpsGnκps(r)=0,
where κ=-l~ and κ=l~+1 for κ<0 and κ>0, respectively.
3.2. Spin Symmetry Limit
In the spin symmetry limit dΔr/dr=0 or Δr=Cs=Const. [15], we consider(10a)Σ(r)=Ar2-2g2r2-12r2+12,(10b)Ur=asr+bsr.
Substitution of the later into (5) gives
(11)d2dr2+1r2-κκ+1+2κbs+bs-bs2-As(M+Enκs-Cs)r2-as2r2+2gs2r2-12r2+12M+Enκs-Cs+Enκs-MM+Enκs-Cs+2κasd2dr2-as-2asbsFnκs(r)=0,
where κ=l and κ=-l-1 for κ<0 and κ>0, respectively.
4. The Ansatz Solution4.1. Solution of the Pseudospin Symmetry Limit
In the previous section, we obtained a Schrodinger-like equation of the form
(12)d2Gnκps(r)dr2+εps+λpsr2+χpsr2+δps2r2-12r2+12Gnκps(r)=0,
where
(13)εps=(Enκps+M)(Enκps-M-Cps)-2κaps+aps-2apsbps,λps=-κ(κ-1)+2κbps-bps-bps2,χps=-Aps(Enκps-M-Cps)-aps2,δps=-2gpsEnκps-M-Cps.
The latter fails to admit exact analytical solutions. Therefore, we follow the ansatz approach with the starting square
(14)Gnκps(r)=fnpsrexpgκpsr,
where
(15)fnps(r)=1,ifn=0,∏i=1n(r-αin),ifn≥1,(16)gκps(r)=-αpsr2+βpslnr+γpsln(2r2+1),αps>0.
By substitution of fn(r) and gκ(r) into (14) and then taking the second-order derivative of the obtained equation, we can get
(17)Gnκps′′r=gκps′′(r)+gκps′2(r)fnps′′(r)+2gκps′(r)fnps′(r)fnps(r)+fnps′′r+2gκps′rfnps′rfnpsrGnκps(r).
By considering the case n=0, from (14)–(16), we find
(18)G0κps′′r=1r2-βps+βps2+12r2+12×16γps2r2+16αpsγpsr2+16βpsγpsr216γps2r2+16αpsγpsr2+16βpsγpsr2-8γpsr2+4γps+8αpsγ+8βpsγps+4αps2r2-2αps12r2+12-4αpsβps-8αpsγpsG0κps(r).
By comparing the corresponding powers of (12) and (18), we have
(19)-λps=βps(βps-1),-χps=4αps2,-2δps=16γps2+16αpsγps+16βpsγps-8γps,δps=4γps+8αpsγ+8βpsγps,-εps=-2αps-4αpsβps-8αpsγps.
Actually, to have well-behaved solutions of the radial wave function at boundaries, namely, the origin and the infinity, we need to take δps from (19) as
(21)δps=4γps+8αpsγ+8βpsγps.
Form (13), (19), and (20), the ground-state energy satisfies
(22)εps--χps--χps1+1-4λps+4-χps-χps+1+1-4λps=0.
From (14), (16), and (20), we simply have the upper and lower components of the wave function as
(23a)gκpsr=-12-χpsr2+121+1-4λps×lnr-2αps+βpsln(2r2+1),(23b)G0κpsr=N0κr1/21+1-4λps2r2+1-2αps+βps×exp-12-χpsr2,(23c)F0κpsr=1M-E0κps+Cpsddr-κr+UrG0κpsr.
4.2. Solution of the Spin Symmetry Limit
In this case, our ordinary differential equation is
(24)d2Fnκs(r)dr2+εs+λsr2+χsr2+δs2r2-12r2+12Fnκs(r)=0,
where
(25)εs=(Enκs-M)(M+Enκs-Cs)+2κas-as-2asbs,λs=-κ(κ+1)+2κbs+bs-bs2,χs=-As(M+Enκs-Cs)-as2,δs=2gsM+Enκs-Cs.
In this case, we introduce the ansatz
(26)Fnκs(r)=fns(r)exp(gκs(r)),
where
(27)fns(r)=1,ifn=0,∏i=1n(r-αin),ifn≥1,(28)gκsr=-αsr2+βslnr+γsln2r2+1,αs>0,
substitution of the proposed ansatz gives
(29)Fnκs′′(r)=gκs′′r+gκs′2r+fns′′r+2gκs′rfns′rfnsrFnκsr.
For n=0, we have
(30)F0κs′′r=1r2-βs+βs2+12r2+12×16γs2r2+16αsγsr2+16βsγsr216γs2r2+16αsγsr2+16βsγsr2-8γsr2+4γs+8αsγs+8βsγs12r2+12+4αs2r2-2αs-4αsβs-8αsγsF0κs(r).
By comparing the corresponding powers of (24) and (30), we have
(31)βs=121+1-4λs,αs=12-χs,γs=-2αs+βs,εs-2αs-4αsβs-8αsγs=0.
To have physically acceptable solutions, we pick up the value by considering the above equation; the first node eigenvalue satisfies
(32)εs--χs--χs1+1-4λs+4-χs-χs+1+1-4λs=0.
From (26), (28), and (31), the upper component of the wave function is(33a)gκsr=-12-χsr2+121+1-4λslnr-2αs+βsln(2r2+1),(33b)F0κsr=N0κr1/21+1-4λs×2r2+1-2αs+βsexp-12-χsr2,and for the lower component of the wave function, we have
(34)G0κsr=1M+E0κs-Csddr+κr-UrF0κsr.
We have given some numerical values of the energy eigenvalues in Tables 1–6 for various states. We have investigated the dependence of the bound-state energy levels n=0 on potential parameter A. The results in Tables 1 and 2 have found that case A=-0.05 is contrary to case A=-0.5 under the condition of the pseudospin and spin symmetries, respectively. Tables 3 and 4 present the dependence of the bound-state energy levels on parameters Cps and Cs in view of the pseudospin and spin symmetry limits. It is seen in Tables 3 and 4 that although bound states obtained in view of spin symmetry become more bounded with increasing Cs, they become less bounded in the pseudospin symmetry limit with increasing Cps. We show the effects of the M-parameter on the bound states under the conditions of the pseudospin and spin symmetry limits. The results are given in Tables 5 and 6. It is seen that if the M-parameter increases, the bound states become less bounded for both the pseudospin and the spin symmetry limits. In Figure 1, the wave functions are plotted for spin and pseudospin symmetry limits.
Bound state for the pseudospin symmetryaps=-0.02, bps=0.01,M=0.5fm-1, and Cps=-1fm-1.
l~
κ
(l,j)
E0κps(fm-1)
A=-0.05
A=-0.5
1
−1
0S1/2
−0.492577461
−0.499257852
2
−2
0P3/2
−0.492679105
−0.499267983
3
−3
0d5/2
−0.492730875
−0.499273142
4
−4
0f7/2
−0.492762185
−0.499276263
Bound state for the spin symmetryas=0.02, bs=0.01,M=0.5fm-1, and Cs=1fm-1.
l
κ
(l,j)
E0κs(fm-1)
A=-0.05
A=-0.5
1
−2
0p3/2
0.506363937
0.500636263
2
−3
0d5/2
0.506651467
0.500665063
3
−4
0f7/2
0.506779923
0.500677931
4
−5
0g9/2
0.50685245
0.500685197
Energies in the pseudospin symmetry limit forAps=-0.5,aps=-0.02, bps=0.01, and M=0.5fm-1.
Cps
E0κps(fm-1)
0S1/2
0P3/2
0d5/2
0f7/2
−5
−4.499263699
−4.49927201
−4.499276204
−4.49927873
−4
−3.499262219
−3.499270997
−3.499275435
−3.499278111
−3
−2.499260752
−2.499269988
−2.499274669
−2.499277494
−2
−1.499259296
−1.499268983
−1.499273904
−1.499276877
−1
−0.499257852
−0.499267983
−0.499273142
−0.499276263
0
0.50074358
0.500733013
0.500727617
0.500724351
Energies in the spin symmetry limit for As=-0.5,as=0.02, bs=0.01, and M=0.5fm-1.
Cs
E0κs(fm-1)
0p3/2
0d5/2
0f7/2
0g9/2
0
−0.499365805
−0.499336214
−0.499322986
−0.499315517
1
0.500636263
0.500665063
0.500677931
0.500685197
2
1.500638332
1.500666338
1.500678847
1.500685909
3
2.500640401
2.500667612
2.500679761
2.500686621
4
3.50064247
3.500668885
3.500680675
3.500687332
5
4.500644539
4.500670157
4.500681588
4.500688043
Energies in the pseudospin symmetry limit forAps=-0.5,aps=-0.02, bps=0.01, and Cps=-1fm-1.
M(fm-1)
E0κps(fm-1)
0S1/2
0P3/2
0d5/2
0f7/2
0
−0.999259296
−0.999268983
−0.999273904
−0.999276877
0.2
−0.799258717
−0.799268582
−0.799273599
−0.799276631
0.4
−0.59925814
−0.599268183
−0.599273295
−0.599276385
0.6
−0.399257564
−0.399267783
−0.39927299
−0.39927614
0.8
−0.199256991
−0.199267385
−0.199272686
−0.199275894
Energies in the spin symmetry limit for As=-0.5,as=0.02, bs=0.01, and Cs=1fm-1.
M(fm-1)
Enκs+(fm-1)
0p3/2
0d5/2
0f7/2
0g9/2
0
1.000638332
1.000666338
1.000678847
1.000685909
0.2
0.800637505
0.800665828
0.80067848
0.800685624
0.4
0.600636677
0.600665318
0.600678114
0.600685339
0.6
0.40063585
0.400664807
0.400677748
0.400685054
0.8
0.200635022
0.200664297
0.200677381
0.200684769
PSS: wavefunction for pseudospin symmetry limit for Aps=-0.5, aps=-0.02, bps=0.01, M=0.5fm-1, and Cps=-1fm-1. SS: wavefunction for spin symmetry limit for As=-0.5, as=0.02, bs=0.01, M=0.5fm-1, and Cs=1fm-1.
5. Conclusion
In this paper, we have obtained the approximate solutions of the Dirac equation for the isotonic oscillator potential including the Cornell tensor interaction within the framework of pseudospin and spin symmetry limits using the ansatz approach which stands as a strong tool of mathematical physics. We have obtained the energy eigenvalues and corresponding lower and upper wave functions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors wish to give their sincere gratitude to the referees for a technical comment on the paper.
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