We solve the D-dimensional Schrödinger equation with hyperbolic Pöschl-Teller potential plus a generalized ring-shaped potential. After the separation of variable in the hyperspherical coordinate, we used Nikiforov-Uvarov (NU) method to solve the resulting radial equation and obtain explicitly the energy level and the corresponding wave function in closed form. The solutions to the energy eigenvalues and the corresponding wave functions are obtained using the NU method as well.

1. Introduction

The noncentral potentials in recent times have been an active field of research in physics and quantum chemistry [1–3]. For instance, the occurrence of accidental degeneracy and hidden symmetry in the noncentral potentials and their application in quantum chemistry and nuclear physics are used to describe ring-shaped molecules like benzene and the interaction between deformed pair of nuclei [4, 5]. It is known that this accidental degeneracy occurring in the ring-shaped potential was explained by constructing an SU (2) algebra [6]. Owing to these applications, many authors have investigated a number of real physical problems on nonspherical oscillator [7], ring-shaped oscillator (RSO) [8], and ring-shaped nonspherical oscillator [9]. Berkdemir [10] had shown that either Coulomb or harmonic oscillator will give a better approximation for understanding the spectroscopy and structure of diatomic molecules in the ground electronic state. Other applications of the ring-shaped potential can be found in ring-shaped organic molecules like cyclic polyenes and benzene [11, 12].

On the other hand, Chen and Dong studied the Schrödinger equation with a new ring-shaped potential [3]. Cheng and Dai investigated modified Kratzer potential plus the new ring-shaped potential using Nikiforov-Uvarov method [13]. Recently, Ikot et al. [14–16] investigated the Schrödinger equation with Hulthen potential plus a new ring-shaped potential [3], nonspherical harmonic and Coulomb potential [15], and pseudo-Coulomb potential in the cosmic string space-time [16]. Many authors have used different methods to obtain exact solutions of the wave equation such as the methods of Supersymmetric Quantum Mechanics (SUSY-QM) [17–19], the Tridiagonal Representation Approach (TRA) [20–23], and Nikiforov-Uvarov (NU) method [24–28].

Motivated by the recent studies of the ring-shaped potential [29–32], we proposed a novel hyperbolical Pöschl-Teller potential plus generalized ring-shaped potential of the form (see Figure 1)(1)Vr,θ=Atanh2λr+Btanh2λr+γcot2θ+ζcotθcscθ+κcsc2θr2,where λ is the screening parameter and A, B, γ, ζ, and κ are real potential parameters. As a special case when λr→0 with A→mω2/2λ2-ħ2α/30mλ2, B→ħ2α/2mλ2, and E→E+2B/3; the potential of (1) turns to nonspherical harmonic oscillator plus generalized ring-shaped potential. (2)Vr,θ=12mω2r2+ħ2α2mr2+γcot2θ+ζcotθcscθ+κcsc2θr2.

The plot of the novel Pöschl-Teller plus ring-shaped potential as a function of r and θ for A=1, B=2, λ=0.1, γ=2, ς=4, and κ=3.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M24"><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula>-Dimensional Schrödinger Equation in Hyperspherical Coordinates

The D-dimensional Schrödinger equation is given as follows [33, 34]:(3)∇D2+2μħ2E-UΨl1,l2,…,lD-2l=lD-1X→=0,where μ is the effective mass of two interacting particles, ħ is Planck’s constant, E is the energy eigenvalue, U is the potential energy function, X→=(r,θ1,θ2,…,θD-1)T is the position vector in D-dimensions, where θ→=θ1,θ2,…,θD-1 is the angular position vector written in terms of hyperspherical coordinates [35, 36], and ∇D2 is the D-dimensional Laplacian operator given in Appendix B.

The solvable potentials that allow separation of variable in (3) must be of the form(4)Ux→=V1r+V2θD-1r2.The separable wave function takes the following form:(5)Ψl1,l2,…,lD-2l=lD-1X→=r-D-1/2grYl1,l2,…,lD-2l=lD-1θ→.Applying (5) to (3) with the use of (4), we obtain the following radial and angular wave equations:(6)d2dr2-D+2l-22-14r2+2μħ2E-V1rgr=0,(7)d2dθj2+j-1cosθjsinθjddθj+Λj-Λj-1sin2θjHθj=0,(8)d2dθD-12+D-2cosθD-1sinθD-1ddθD-1+ll+D-2-ΛD-2sin2θD-1+2μħ2V2θD-1HθD-1=0,where Yl1,l2,…,lD-2l=lD-1θ→=1/2πe±imθ1∏j=2D-1Hθj, (8) holds for j∈2,D-2, with D>3, and Λj=ljlj+j-1. Solutions of (8) will not be affected by the presence of the proposed potential and thus are common to different systems and they were done before using different approaches [37]. Consequently, we will only solve (6) and (8) using the Nikiforov-Uvarov method [24, 25].

3. Nikiforov-Uvarov Method

Many problems in physics lead to the following second-order linear differential equation [24]:(9)d2dx2+τ~xσxddx+σ~xσ2xux=0,where σx and σ~x are polynomials of degree 2 at most and τ~x is at most linear in x. Equation (9) is sometimes called of hypergeometric type. Let us consider ux=ϕxyx; this will transform (9) to the following differential equation foryx:(10)d2dx2+τxσxddx+σ-xσ2xyx=0,where we assumed the following conditions:(11a)ϕ′xϕx=πxσx,(11b)τx=τ~x+2πx,(11c)σ-x=σ~x+π2x+πxτ~x-σ′x+π′xσx,(11d)πx=σ′x-τ~x2±σ′x-τ~x22-σ~x+kσx,(11e)k=η-π′x,where k and η are constants chosen such that πx is polynomial which is at most linear in x and σ-x=ησx. This will transform (10) to the following:(12)σxd2dx2+τxddx+ηyx=0,where σx and τx are polynomials of degrees 2 and 1, respectively. In this case, solutions to (12) are polynomials of degree n; yx=ynx,ηn, where ηn is given as follows:(13)ηn=-nτ′-12nn-1σ″.Equation (13) will be used to obtain the energy spectrum formula of the quantum mechanical system. We should point out here that the polynomial solutions to (12) for τ′<0 and τ=0 on the boundaries of the finite space (the latter case is omitted for infinite space) are the classical orthogonal polynomials. It is well known that each set of polynomials is associated with a weight function ρx. For the polynomial solutions to (12), this function must be bounded on the domain of the system and must satisfy σρ′=τρ. This weight function will be used to construct the Rodrigues formula for these polynomials, which reads(14)ynx=Bnρxdndxnσnxρx,where Bn is just a constant obtained by the normalization conditions and n=0,1,2,….

4. The Solutions of the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M68"><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula>-Dimensional Radial Equation

We use the NU method to solve (6); in the presence of our potential,(15)d2grdr2+2μħ2E-Atanh2λr-Btanh2λr-γDħ22μ1r2gr=0,where γD=D+2l-22-1/4. Equation (15) cannot be solved analytically due to the centrifugal term 1/r2. Different authors used different approximation techniques to allow an approximate analytical solution of (15) and these methods rely on Taylor expansion of the centrifugal potential in terms of the other components of the potential of interest [38]. In this work, we use the following approximation obtained by Taylor expansion [39]:(16)1λ2r2≈-23-13tanh2λr+1tanh2λr.The advantage of this approximation is that it is valid not only for λr≪1 but also for 0≤λr≤2 with high accuracy. Also, it satisfies the limits on 1/r2 at zero and infinity; that is, limr→0RHS=1/λ2r2 and limr→∞RHS=0, where RHS denotes the right-hand side of (16). Now, Using (16) back in (15), we get(17)d2dr2-4λ2A~tanh2λr-4λ2B~tanh2λr+4λ2E~gr=0,where 4λ2E~=2μ/ħ2E+γDħ2λ2/3μ, 4λ2A~=2μ/ħ2A-γDħ2λ2/6μ, and 4λ2B~=2μ/ħ2B+γDħ2λ2/2μ. Making change of variable s=tanh2λr and by writing gs=ϕsys, this transforms (17) to (9) with the polynomials being σ=s-s2, τ~=-3s-1/2, and σ~=E~s-A~s2-B~. We now use (11d) to calculate πs, which reads(18)πs=1-s4±1-s42+A~s2-E~s+B~+ks-s2.The choice of k that makes (18) a polynomial of first degree must satisfy c12=c2c3, where c1=16k-8-16E~, c2=1+16A~-16k, and c3=41+16B~. This gives(19)πs=1-s4±14c2s+c12c2,where we will pick the negative part in (19) which makes τ′<0. The function τs can be easily calculated using (11b):(20)τs=1-2s-12c2s+c12c2.Using (19) and (20) in (13) and (11e), we get(21)4k2n+1-2n+12=1+16A~-16k.Solutions of (21) for k are(22)4k=-2n+12±2n+11+16A~.The next step is to use the value of c2 in (21) with the constraint on k mentioned in (18), which is c12=c2c3; we obtain(23)8+16E~-16k2=41+16B~1+16A~-16k.The conditions for bound states are k≤0 and A~, B~≥-1/16. Using (22) in (23), we write the bound states formula as follows:(24)E~nlD=-142n+12-142n+11+16A~-12±181+16B~1+16A~+42n+12+42n+11+16A~.In terms of the original parameters A, B, and E, the spectrum formula in D-dimensions reads(25)2μħ2EnlD=-2γDλ23-λ22n+12-λ22n+11+8μλ2ħ2A-γDħ2λ26μ-2λ2±λ221+8μλ2ħ2B+γDħ2λ22μ×1+8μλ2ħ2A-γDħ2λ26μ+42n+12+42n+11+8μλ2ħ2A-γDħ2λ26μ,where conditions for bound states become A≥λ2ħ2/2μγD/3-1/4 and B≥-λ2ħ2/2μγD+1/4. We will only take the (−) sign in (25) as explained below. The s-wave spectrum formula in three dimensions is the only exact solution that is obtained by setting γD=0 in (25). However, for other higher states, the above solution is acceptable with high accuracy as far as the condition 0≤λr≤2 is satisfied.

The transformation s=tanh2λx makes the domain of the function be 0,1. This suggests a change of variable z=2s-1 to bring the domain to that of Jacobi polynomials which are well-known classical orthogonal polynomials. By using (18) in (11a), we obtain ϕz=2c4-2c51+zc51-zc5-c4, where 4c4=1+c2 and 8c5=2-c1/c2. The weight function can be easily calculated using (20) in σρ′=τρ, which gives ρ(z)=22c7+c61+z-c7(1-z)-c6-c7, where 2c6=8+c2 and 4c7=c1/c2. The solution of (12) in our case is written in the following Rodrigues formula:(26)ynz=Cn1+zc71-zc6+c7dndzn1+z-c7+n1-zn-c6-c7.By comparison to Jacobi polynomials, we conclude that ynz=Pn-c6-c7,-c7z, where Pn-c6-c7,-c7z is the Jacobi polynomial of order n in z. Thus, the bound state solution of the radial wave equation now reads(27)gnr=Ωntanhλr-c6-c7+1/2sechλr-c7Pn-c6-c7,-c72tanhλr2-1,where Ωn is just a normalization constant. We must clarify here that, for Jacobi polynomials, we have to have -c6-c7,-c7>-1. Thus, the parameters c1 and c2 are chosen to satisfy c1<4c2 and 2c2+c1<-12c2. Moreover, since those parameters depend on the energy as we mentioned previously, this yields us to reject the (+) sign in (24) and (25). Consequently, bound states occur for B~>35/16 (which does not violate the old restriction B~≥-1/16) and E~>k-1/2. The latter condition on E is already satisfied as we can see in (25), so we do not have to worry about it.

To calculate the normalization constant Ωn, we first use the following identity of Jacobi polynomials [40]:(28)Pna,by=12n∑m=0nn+amn+bn-m1-ym-n1+ym.Next, we use the normalization constraint ∫0∞gr2dr=∫-1+1gy2dy/2λ1+y1-y=1, where y=2tanh2λr-1; this gives(29)Ωn2/λ2n+1/2∑m=0nn+amn+bn-m∫-1+11-y2a+m-n-11+yb+mPna,bydy=1.To calculate the integral in (29), we will use the following very useful integral formula [41]:(30)∫-1+11-yc1+ydPna,bydy=2c+d+1Γc+1Γd+1Γn+a+1Γn+1Γc+d+2Γa+1×F23-n,n+a+b+1,c+1;a+1,c+d+2;1,where F23a,b,c;d,e;f is the generalized hypergeometric function [41]. By direct comparison between (29) and (30) we get Ωn=1/Λn, where Λn is given as follows:(31)Λn=1λ2n+1/2∑m=0nn+amn+bn-m22a+2m-n+bΓ2a+m-nΓb+m+1Γn+a+1Γn+1Γ2a+2m-n+b+1Γa+1×F23-n,n+a+b+1,2a+m-n;a+1,2a+2m-n+b+1;1,where a=-c6-c7 and b=-c7.

The only issue that is left for discussion in this section is that the solutions of the radial wave equation gnrn=0N, where N denotes the maximum number in which we get bound states, are not orthogonal! But they are normalized as we discussed above. We know that Hermitian operators with distinct eigenvalues must have orthogonal eigenvectors [42]. To solve this problem, one must use the Gram-Schmidt (GS) method to obtain an orthonormal set ϕnrn=0N by linear combinations [43]. The latter set will be the solutions of the radial wave equation. The process is a bit lengthy and we will not be able to do it here. However, we encourage the interested reader to do these calculations by referring to the process of GS.

5. Solutions of the Angular Equations

It is well known from literature that solutions of (7) are written in terms of Jacobi polynomials as follows [41]:(32)Hy=Nn1-yα1+yβPnc,dy,where Nn is just a constant factor, 2β+j/2=d+1, and 2α+j/2=c+1. Moreover, the latter parameters are written in terms of the quantum numbers as c=d=cj=lj-1+j-2/2, which yields α=β=lj-1/2 and n=lj-lj-1. The above solution was obtained using different methods including the NU technique [24]. Hence, solutions of (7) are written below:(33)Hθj=Nnsinθjlj-1Pncj,cjcosθj.To solve (8), we introduce coordinate transformation as y=cosθD-1, which gives(34)1-y2d2dy2-D-1yddy+ll+D-2-ΛD-21-y2+UyHy=0,where Uy=2μ/ħ2V2y=γ′y2+ζ′y+κ′/1-y2 for real parameters γ′,ζ′,κ′ are related to γ,ζ,κ by a factor of 2μ/ħ2. Equation (34) is of hypergeometric type with the polynomials being σy=1-y2, τ~y=-D-1y, and σ~y=η2y2+η1y+η0, where η2=γ′-ll+D-2, η1=ζ′, and η0=κ′+ll+D-2-ΛD-2. The solutions of (34) are written as Hy=ϕyYy, where ϕy satisfies (11a). Next, we need to find the function πy using (11d); we find that this function takes the following form:(35)πy=D-3-2u0y2+η12u0,where u0=D-3/22-η2-k and the parameter k defined in (11e) must satisfy 4k-4η0=η12/u02. The latter constraint will be used later to obtain the eigenvalues of (34). Now, we use (35) in (11b); we get τy=-21+u0y+η1/u0, which satisfies dτy/dy<0. Using (11a), we can obtain ϕy to be ϕy=1-y-u1+u2/21+yu2-u1/2, where 2u1=(D-3-2u0) and η1=2u0u2. We can also calculate the weight function by solving σρ′=τρ, which gives ρy=1-yu0-u21+yu0+u2. The Rodrigues formula of the polynomials Yy reads(36)Yny=ξn1-yu0-u21+yu0+u2dndyn1-yn+u0-u21+yn+u0+u2,where ξn is just a constant. By direct comparison with the Rodrigues formula of Jacobi polynomials xx, we conclude that Yny=Pnu0-u2,u0+u2y. As required by Jacobi polynomials, we must impose that u0±u2>-1. Now, we use (35) and (13) in (11e) to obtain the following quadratic formula for k:(37)k-n2-n+D-322=2n+12D-322-η2-k.The solutions of (37) are given below:(38)k=122-D-2n-2n2±1+2nD-22-4η2.Moreover, we use 4k-4η0=η12/D-3/22-η2-k to obtain another solution for k:(39)8k=9+D2-6D+4η0-4η2±D-32+4η0-4η22-16η12+η0D-32-4η2.Direct comparison between (38) and (39) gives(40)2n+12=-D-12+4η0-4η2-22,(41)2n+12D-22-4η2=D-32+4η0-4η2216-η12+η0D-32-4η2.In the next section, we will consider different examples and try to obtain the unknown parameters for each case.

6. Results and Discussions

In this section, we will discuss different examples that are considered as special cases of the potential in (1).

As a first example, we consider the case when γ′=ζ′=0 and κ′≠0, which is equivalent to the following noncentral hyperbolic potential:(42)Vr,θ=Atanh2λr+Btanh2λr+κcsc2θr2.In this case, we have u2=0. Thus, solution of (34) reads(43)HnθD-1=NnsinθD-1-u1Pnu0,u0cosθD-1,where 2u1=(D-3-2u0), u0=D-3/22+ll+D-2-k, and k is given below:(44)8k=9+D2-6D+4η0+4ll+D-2±D-32+4η0+4ll+D-22-16η0D-32+4ll+D-2and the corresponding eigenvalue is obtain from (41) as(45)2n+12D-22+4ll+D-2=D-32+4η0+4ll+D-2216-η0D-32+4ll+D-2.(2) The next special case of our potential model is considered when we choose the ring-shaped parameters γ=±ς and κ≠0, which corresponds to the following potential:(46)Vr,θ=Atanh2λr+Btanh2λr+γcot2θ±cotθcscθ+κcsc2θr2.

Under these conditions, we have(47)η1=ς,η2=±ς-ll+D-2,u1=D-3-2u02,u2=ς2u0,u0=D-322∓ς-ll+D-2-k.The k values and the corresponding eigenvalues are obtained as follows:(48)8k=9+D2-6D+4η0∓4ς+4ll+D-2±D-32+4η0∓4ς+4ll+D-22-16ς2+η0D-32∓4ς+4ll+D-2,2n+12D-22∓4ς+4ll+D-2=D-32+4η0∓4ς+4ll+D-2216-ς2+η0D-32∓4ς+4ll+D-2.The associated nonnormalized wave function is obtained as(49)HnθD-1=Nn1-cosθD-1-u1+u2/21+cosθD-1u2-u1/2Pnu0-u2,u0+u2cosθD-1(3) Another special case of our study is when ς=0 and γ, κ≠0, which corresponds to the following potential:(50)Vr,θ=Atanh2λr+Btanh2λr+γ+κcot2θ+κr2.With these assumptions, we have(51)η1=0,η1=κ+ll+D-2-ΛD-2,η2=γ-ll+D-2,u1=D-3-2u02,u2=0,u0=D-322-γ+ll+D-2+k.Under this special case, we obtain the k parameter, the eigenvalues, and the corresponding wave function as follows:(52)8k=9+D2-6D+4κ+ll+D-2-ΛD-2-4γ-ll+D-2±D-32+4κ+ll+D-2-ΛD-2-4γ-ll+D-22-16η0D-32-4η2,2n+12D-22-4γ-ll+D-2=D-32+4κ+ll+D-2-ΛD-2-4γ-ll+D-2216-κ+ll+D-2-ΛD-2D-32-4γ-ll+D-2,HnθD-1=NnsinθD-1-u1Pnu0,u0cosθD-1.However, one needs to be careful here as, for γ=-κ, there will be no ring-shaped term and one ends up with hyperbolic PT potential plus pseudo centrifugal term:(53)Vr,θ=Atanh2λr+Btanh2λr+κr2.(4) We consider the last special case for γ=0 and κ=±ς, which corresponds to the potential of the form(54)Vr,θ=Atanh2λr+Btanh2λr+ςcotθcscθ±csc2θr2.The following parameters are obtained under this case:(55)η1=±ς,η2=-ll+D-2,η0=±ς+ll+D-2-ΛD-2,u1=D-3-2u02,u0=D-322+ll+D-2-k,u2=±ς2u0.Using (55), we obtain the k parameter, the eigenvalues, and the corresponding wave function for this special case as follows:(56)8k=9+D2-6D+4±ς+ll+D-2-ΛD-2+4ll+D-2±D-32+4±ς+ll+D-2-ΛD-2+4ll+D-22-16±ς2+±ς+ll+D-2-ΛD-2D-32+4ll+D-2,2n+12D-22+4ll+D-2=D-32+4±ς+ll+D-2-ΛD-2+4ll+D-2216-±ς2+±ς+ll+D-2-ΛD-2D-32+4ll+D-2,HnθD-1=Nn1-cosθD-1-u1+u2/21+cosθD-1u2-u1/2Pnu0-u2,u0+u2cosθD-1.

7. Conclusions

In this paper, we have obtained analytically the solutions of the D-dimensional Schrödinger potential with hyperbolic Pöschl-Teller potential plus a generalized ring-shaped term. We employed NU and trial function methods to solve the radial and angular parts of the Schrödinger equation, respectively. This result is new and has never been reported in the available literature to the best of our knowledge. Finally, this result can find many applications in atomic and molecular physics and thermodynamic properties [43].

AppendixA. Jacobi Polynomials

Jacobi polynomials Pnμ,νy defined on -1,1 are solutions of the following second-order linear differential equation [8]:(A.1)1-y2d2dy2-μ+ν+2y+μ-νddy+nn+μ+ν+1Pnμ,νy=0.We also mention their orthogonality relation:(A.2)∫-111-yμ1+yνPnμ,νPmμ,νdy=2μ+ν+12n+μ+ν+1Γn+μ+1Γn+ν+1Γn+μ+ν+1n!δn,m.

B. Hyperspherical Coordinates

The D-dimensional position vector x→=r,θ1,…,θD-1 is defined in terms of hyperspherical Cartesian coordinates below [36]:(B.1)x1=rcosθ1sinθ2⋯sinθD-1,x2=rsinθ1sinθ2⋯sinθD-1,xj=rcosθj-1sinθj⋯sinθD-1,where j=3,4,…,D-1, xD=rcosθD-1, and ∑j=1Dxj2=r2. For D=2, this is the case of polar coordinates r,φ with x1=x=rcosφ and x2=y=rsinφ, whereas D=3 represents the spherical coordinates r,φ,θ, where x1=x=rcosφsinθ, x2=y=rcosφsinθ, and x3=z=rcosθ.

The volume element in D-dimension is defined as dV=rD-1dr∏j=1D-1sinθjj-1dθj, where r∈0,∞, θ1∈0,2π, and θj∈0,π for j≥2. The Laplacian operator in D dimensions is defined below:(B.2)∇D2=∂2∂r2+D-1r∂∂r+1r2×1sinD-2θD-1∂∂θD-1sinD-2θD-1∂∂θD-1-LD-22sin2θD-1.Finally, we mention the normalization conditions of the wave function in D-dimensions:(B.3)∫0∞gnr2dr=1,∏j=2D-1∫0πHθj2sinθjj-1dθj=1.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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