Motivation

Relativistic and NR quantum mechanics have achieved great successes in terms of the convergence of theoretical treatments with experimental measurements. However, some indications show that there are many problems that quantum mechanics known in the literature has not been able to solve, for example, the problem of non-renormalizable electroweak interactions, the problem of quantization of gravity, and addition the problems in string theory. The idea of noncommutativity resulting from properties of deformation of space-space (Heisenberg in 1930 is the first to suggest the idea and then it was developed by Snyder in 1947) was one of the major solutions to these problems. As a result of all these motivational data, it is logical to consider the topographical properties of the noncommutativity space-space and phase-phase have a clear effect on the various physical properties of relativistic and nonrelativistic quantum systems [¹³-¹⁸].

A lot of research has been devoted to study the properties of quarkonium in the noncommutative space phase (NCSP) in the framework of the two cases: the nonrelativistic NR and relativistic, based on the three fundamental equations related to the Yukawa potentials such as modified Klien-Gordon equation (KGE) with modified scalar-vector Yukawa potential [¹⁹] and the relativistic interactions in one-electron atoms with modified Yukawa potential for spin-1/2 particles [²⁰]. Moreover, we have treated the nonrelativistic behavior of hydrogen-like and neutral atoms subjected to the generalized perturbed Yukawa potential with a centrifugal barrier [²¹]. We are studying the modified unequal mixture scalar vector Hulthén-Yukawa potentials model as a quark-antiquark interaction and neutral atoms with relativistic treatment using the approximation of the centrifugal term and Bopp’s Shift method [²²]. We have investigated the approximate solutions of DKG and DSE under the modified more general exponential screened Coulomb potential plus Yukawa potential in NCQM symmetries [²³]. We have constructed a theoretical model of the DKG equation with generalized modified screened Coulomb plus inversely quadratic Yukawa potential in RNCQM symmetries [²⁴]. We have obtained solutions of the KG equation for the modified central complex potential in the symmetries of noncommutative quantum mechanics [²⁵]. We are studying spectra of heavy quarkonia with modified CP in the framework of modified SE [²⁶]. We have obtained the new NR atomic energy spectrum of energy-dependent potential for heavy quarkonium in noncommutative spaces and phases symmetries [²⁷]. In 2019, we constructed a new model for Heavy-Light Mesons (HLM) in the extended NR quark model under a new modified potential containing Cornell, Gaussian, and inverse square terms in the symmetries of NCQM [²⁸]. We investigated a new asymptotic study to the 3D-RSE under modified quark-antiquark interaction potential [²⁹]. We have calculated the new relativistic atomic mass spectra of quarks (u, d and s) for the extended modified CP at nano and Planck scales [³⁰]. In addition, we have built a new model for HLM in the symmetries of the extended NR quark model [³¹]. Furthermore, we have investigated the NR-BS solution at finite temperature using the sum of a modified CP plus inverse quadratic potential in the framework of the DSE [³²]. Motivated by the previous works in ordinary quantum mechanics and NCSP, we hope to investigate the Hulthén plus Hellmann potentials in the 3D-NR quantum mechanics noncommutative phase space 3DNRQM-NCSP symmetries to obtain new applications on the microscopic scale and contribute to the knowledge of elementary particles at the nanoscale. The NR energy levels under the improved Hulthén plus Hellmann potentials model temperature-dependent have not been obtained yet in the 3DNRQM-NCSP symmetries, we propose a new version of the improved Hulthén plus Hellmann potentials model temperature-dependent (IHHPTd model) Vnchhp (r, T) and a corresponding Hamiltonian operator Hnchhp (r, p, T) in the presence of TD confined Coulomb potential in the 3DNRQM-NCSP symmetries as follows:

The two couplings LΘ and Lθ- are equal to L _x Θ₁₂ + L _y Θ₂₃ + L _z Θ₁₃ and L _x θ-₁₂ + L _y θ-₂₃ + L _z θ-₁₃, respectively, and they appear automatically from the influence of noncommutativity NC effect space-space Θ(Θ₁₂ , Θ₂₃ , Θ₁₃) on the potential V _hhp (r, T) and NC effect phase-phase θ- θ-12,θ-23,θ-13 on the kinetic term p22μ with the angular momentum operator L(L _x , L _y , L _z ), and Θ _τυ = θ _{τυ /} /2, (-β ₂, β ₁ ^¡, β ₀ and α ₃) are the new potential parameters defined in Sec. 2, is the distance between the two particles while H _hhp (r, p, T) is the usual Hamiltonian operator of the HHPTd in NRQM symmetries. In the present work, we modify the HHPTd model Vnchhp(r) by adding new terms (l(l + 1)r ^-4 LΘ, β02r ^-3 LΘ, -β12rLΘ and (β ₂ LΘ + Lθ-2μ)) due to the topological properties of the self-quantum influence of space-space and phase-phase including the effect of the centrifugal term l(l + 1)r ^-2 which appears in the first additive part. The study of quarkonia systems with the HHPTd model at finite temperature in 3DNRQM-NCSP symmetries is an essential tool for understanding the status of the matter formed in the heavy-ion collisions as in Ref. [³³], on the other hand, it is a continuation of our previous efforts in this context as in Ref. [³²] and which falls within the context of the investigation of the properties of the quarkonium system under the influence of the sum of modified Cornell plus inverse quadratic potential at finite temperature in 3DNRQM-NCSP symmetries. To the best of our knowledge, this new study of DSE with Vnchhp(r, T) was not done before by any researcher. The structure of 3DNRQM-NCSP symmetries based on NC canonical commutations relations in (Schrödinger, Heisenberg and interactions) pictures (SP, HP and IP), respectively, as follows (throughout this article, the natural units c = ħ = 1 will be applied) (see, e.g., [³⁴-⁴¹]):

where ħ _eff = ħ (1 + Tr (θθ-/4)) and ħ are the effective Planck constant and the usual reduced Planck constant, respectively. However, the unified operators ΥτHnct= xτnc ∨  pτInc (t) and ΥτInct= xτInc ∨  pτInc (t) in (HP and IP, respectively) depend on the corresponding operator ΥτSnct= xτnc ∨  pτnc in SP with the following projection relations:

Here T = t-t ₀ , Υ _τs =x _τ ∨ p _τ , Υ _τH (t) = (x _τ ∨ p _τ ) (t) and ΥτHnct= xτinc ∨  pτinc (t) are the usual three representations of SP, HP and IP in NRQM, while the new dynamics of studied systems dΥτHnc tdt describe from the new following motion equations in 3DNRQM-NCSP symmetries:

Here H _hhp Hohhp and Hnchhp Honchhp denote the ordinary and generalized quantum Hamiltonian (free) operators for the of IHHPTd model in the NRQM and 3DNRQM-NCSP, respectively. The two infinitesimal parameters (θ ^τυ ,θ- ^τυ )=𝜖 ^τυ (θ,θ-) (compared to the energy) are the elements of two antisymmetric real matrices with dimensions of (length)² and (momentum)², respectively while 𝜖 _μv just an antisymmetric number, for example 𝜖 ₁₂ = -𝜖 ₂₁ = 1, 𝜖 ₁₁ = 𝜖 ₂₂ = 𝜖 ₃₃ = 0. Furthermore, the star notation *, denote to the star product, which is generalized between two arbitrary functions (f g) (x, p) of the form (f g) (x ^nc , p ^nc ) ≡ (f * g) (x, p) in 3D-NCSP symmetries (see, e.g., [⁴²-⁵¹]:

This allows the construction of two scales of space and phase cells with elementary volumes τncs3= θ ^3/2 and τncs3= θ ^-3/2 respectively. On the other hand, Eq. (6) allows us to satisfy the postulated algebra in Eq. (3). The second and the third terms in the above equation are the effects of (space-space) and (phase-phase) noncommutativity properties, respectively. This paper aims to present approximate solutions of the deformed Schrödinger equation (DSE) with the improved Hulthén plus Hellmann potentials temperature-dependent (IHHPTd, in short) model in 3DNRQM-NCSP symmetries using the generalized Bopp’s shift method (GBSM), in addition to the standard perturbation theory (SPT).

The organization of the present work is given as follows: in the next section, we briefly review the SE with the HHPTd model. We divided the third section into subsections, the first one reserved to the physical and mathematical model for the HHPTd model in 3D-NCSP by applying the GBSM, in the next subsection, we generate the new spin-orbit Hamiltonian operator for the hydrogenic atoms (HAs) and the HLM under the IHHPTd model and by applying SPT we find the corrected spectrum of n ^th excited levels in the framework of the global group 3D-NCSP and then, we derive the modified magnetic and rotational spectra for the IHHPTd model, which produced with the effect of both perturbed Hamiltonians Hzhhp and Hperthhp-rot due to the topological properties of phase space. In the fourth section, we resume the global spectrum and corresponding NC Hamiltonian operator for the IHHPTd model and corresponding energy levels of the HAs (He⁺, Li⁺² and Be⁺) and the HLM such as charmonium cc and bottomonium bb and we calculate the new mass spectra of (cc-, bb-) in 3D-NCSP symmetries, we will also treat some special cases that demonstrate the correctness of our results. Finally, the paper ends with concluding remarks in Sec. 5.

2.1. Overview of the eigenfunctions and the energy eigenvalues for HHPTd in NRQM

In this section, we shall recall here the SE for the HHPTd model, which is an important short-range potential that behaves like a Coulomb potential for small values and decreases exponentially for large values presented in Eq. (1). For small values of (-r=β ₂) the HHPTd model takes the form [¹¹, ¹²]:

with β ₀ = A ₂ - A ₁ - (A ₀ =m _D (T)), β ₁ = (A ₂ m ² _D (T)/2) - A ₀(m _D (T )/12), β ₂ = (A ₂mD3 (T)/6) and β ₃ = (A ₀ =2) - A ₂ m _D (T ). If we consider a nonrelativistic virtual particle of reduced mass μ in a central potential like V _hhp (r, T), the quantum evolution of this particle governed with the SE in the spherical coordinates (r, θ, φ) as follows:

The operator H _hhp (p, x) is just the ordinary Hamiltonian operator in NRQM, Ψ (r, θ, φ, t) = (R _nl (r)/r)Y _l ^m (θ, φ) e-i/ħEnlhhpt, Ylm(θ,φ) are the spherical harmonic functions, Veffhhp(r, T ) = -(β ₀ /r)+β ₁ r-β ₂ r ² +β ₃ +(l[l + 1]=r ²) is effective potential, Enlhhp are the eigenvalues of the Hulthén plus Hellmann potentials model in the presence of TD while n and l are the radial and orbital angular momentum quantum numbers. The wave function and the energy spectrum of Eq. (8) for HHPTd model (7) are respectively given by [¹¹, ¹²]:

Here α = (6μβ ₁ /δ ₂ ) + 2μβ ₀ - (16μβ ₂ /δ ₃ ), - 𝜖 _nl =2μ (E _nl - β ₃) + (12μβ ₂/δ₂) - (6μβ ₁/δ) and δ =1/r ₀ while r ₀ is a characteristic radius of the meson. The energy Enlhhp of the potential in Eq. (7) is given by [¹¹, ¹²]:

3.1. Review of the concepts of GBSM

In this subsection, we devote this part to studying the nonrelativistic improved Hulthén plus Hellmann potentials model temperature-dependent in the presence of TD confined Coulomb potential Vnchhp (r, T), in 3DNRQM-NCSP symmetries. To perform this task the physical form of the deformed Schrödinger equation DSE, it is necessary to replace the ordinary three-dimensional Hamiltonian operators H _hhp (p, x), ordinary energy Enlhhp and corresponding complex wave function Ψ r→, in the symmetries of NRQM by three-dimensional Hamiltonian operators Hnchhp (p _nc , x _nc ), new unknown values Enchhp of energy and corresponding new complex wave function Ψ rnc→, respectively, in 3DNRQM-NCSP symmetries. In addition, we need to replace the ordinary product with the star product (*), this allows us to construct the DSE in (NC-3D: RSP) symmetries as (see, e.g., [⁵²-⁵⁶]):

allowing us to obtain the modified radial part of the SE as follows:

Among the possible paths to find the solutions to Eqs. (19) and (20), we make use of the Connex method or Seiberg and Witten map. It is known to specialists that the star product can be translated into the ordinary product known in the literature using what is called Bopp’s shift method. Bopp was the first to consider pseudo-differential operators obtained from a symbol by the quantization rules (x, p) → (x^ = x - (i/2)𝜕 _p ,p^ = p + (i/2)𝜕 _x ) instead of the ordinary correspondence (x, p) → (x^ = x,p^ = p + (i/2)𝜕 _x ), respectively. It is known to specialists that BSM has been applied effectively and has succeeded in simplifying the known four fundamental equations: the NRDSE [⁶⁷-⁷⁰], the relativistic deformed Klein-Gordon equation RDKGE [⁶⁰-⁶⁸], the relativistic deformed Dirac equation RDDE [²⁰, ⁴¹, ⁶⁹], and the deformed relativistic Duffin-Kemmer-Petiau equation (DRDKPE) [⁷⁰] with the notion of star product to the NRSE, RKGE, RDE and RDKPE with the notion of ordinary product, respectively. Thus, BSM is based on reducing second-order linear differential equations of DNRSE, DRKGE, DRDE, and DRDKPE with star product to second-order linear differential equations of NRSE, RKGE, RDE, and RDKPE without star products with simultaneous translation in the 3D-NCSP. The CNCCRs with star product in Eqs. (3) become new CNCCRs without the notion of the star product as follows (see, e.g., [⁵⁹-⁶¹]):

The generalized positions and momentum coordinates (xτnc, pτnc) in 3DNRQM-NCSP depend on the corresponding usual generalized positions and momentum coordinates (x _τ , p _τ ) in NRQM by the following, respectively (see, e.g., [⁵⁷-⁵⁹]):

The above equation allows us to obtain the two operators rnc2 and pnc2 in 3DNRQM-NCSP symmetries [⁵²-⁵⁵]:

Thus, the reduced radial part of the SE (without star product) can be written as:

The Hamiltonian operator Hnchhp for the improved Hulthén plus Hellmann potentials model temperature-dependent can be expressed as:

Now, we want to find the new effective potential of IHHPTd Veff-nchhp (r, T ) in 3DNRQM-NCSP symmetries:

The new IHHPTd in the presence of temperature-dependent confined Coulomb potential Vnchhp (r, T ) and the new centrifugal term l(l + 1)rnc-2 in 3DNRQM-NCSP symmetries:

After straightforward calculations, we can obtain the important terms -β ₀ /r _nc , β ₁ r _nc and -β ₂ r ² which will be used to determine the IHHPTd in the presence of TD confined Coulomb potential Vnchhp (r, T ) in 3DNRQM-NCSP symmetries as:

By making the substitution above Eqs. (18) and (17.2) into Eq. (16) and (17.1), we find the global our working Hamiltonian operator Hnchhp and the new effective potential of IHHPTd Veff-nchhp (r) in 3DNRQM-NCSP symmetries:

where the operator H _hhp (p, x) is just the ordinary Hamiltonian operator in usual nonrelativistic quantum mechanics:

while the rest five terms are proportional with two infinitesimal parameters (Θ and θ-) and then we can be considered as perturbations terms Hperthhp and Vperthhp (r, T) in 3DNRQM-NCSP symmetries as:

It is clear that the operator H _hhp (p, x) is just the Hamiltonian operator for HAs such as He⁺, Li⁺² and Be⁺ and HLM in ordinary quantum mechanics while the generated part Hperthhp appears as a result of the deformation of the 3D-NCSP. In the following, we can disregard the second term in Hperthhp because we are interested in the corrections of first-order Θ and θ¯.

3.2. New spin-orbit Hamiltonian operator for HAs and HLM under the IHHPTd model

In this subsection, we want to derive the physical form of the induced perturbed Hamiltonian Hperthhp (p _nc , x _nc ) due to space-phase noncommutativity effects. To achieve this goal, we replace LΘ both and Lθ¯ with the useful physical forms (𝜖ΘLS or g _s ΘLS) and (𝜖θ¯LS or g _s θ¯LS), respectively (see, e.g., [⁵¹-⁵⁵]):

allowing us to construct the induced perturbed spin-orbit Hamiltonian operator as follows Hsohhp :

Here Θ=Θ122+Θ232+Θ1321/2, θ¯=θ¯122+θ¯232+θ¯1321/2, LS     is just the scalar product LxSx+LySy+LzSz, ϵ≈1/137 is the atomic fine structure constant and, gs is the strong coupling constant, and S denotes the spin of the hydrogenic atoms such as (He⁺, Li⁺² and Be⁺) or the heavy-light mesons. Thus, the spin-orbit interactions Hsohhp appear automatically as a result of the deformation of the space phase. Now, physically, we can rewrite the quantum spin-orbit LS coupling as follows:

Here J is the total momentum of the hydrogenic atoms He⁺, Li⁺² and Be⁺ and HLM. Substituting this equation into Eq. (22) yields:

Our recent study can apply in two principal cases: The first case considers A ₁ = Ze ², Z and e are the atomic number and the charge of the electron, the term (-A ₁ /r ) becomes an attractive Coulomb potential, thus, we can consider the Hamiltonian described hydrogenic atoms such as (He⁺, Li⁺² and Be⁺) under the influence of external fields described by other terms (-[A ₀ exp (-m _D (T ) r)/1 - exp (-αr)] + [A ₂ exp (-m _D (T ) r)=r]) in ordinary quantum mechanics and its extension to 3DNRQM-NCSP, which allows us to get the eigenvalues j of the total angular momentum operator J from the interval 〡l - 1/2〡 ≤ j ≤ 〡 l + 1/2〡. Because the operator G² has two eigenvalues, we can obtain two values of energy, as follows

A second way of determining a diagonal matrix Hsohhp of order (3 × 3) with diagonal elements Hsohhp11, Hsohhp22 and Hsohhp33=0 as:

The non-null diagonal elements Hsohhp11 and Hsohhp22 of the perturbed Hamiltonian operator Hperthhp can be influenced by the energy values Enl by creating three new values ΔEn-sou-hhp=ΨHsohhp11Ψ and ΔEn-sod-hhp=ΨHsohhp22Ψ corresponding the polarizations up j=l+1/2 and down l-1/2 that can expressed as:

The second case is for the heavy-light mesons (HLM), for example: scalar, vector, pseudoscalar, and pseudovector for (B, B _s , D and D _s ) mesons, or the heavy quarkonioum systems, such as charmonioum cc¯ and bottomonioum bb¯, which quarks and antiquarks of the same system (QQ¯), the eigenvalues of the spin-orbit coupling operator LS are k (j, l, s) = j(j + 1) - l(l + 1) - s(s + 1) corresponding j = l + 1 (spin great), j = l (spin middle) and j = l - 1 (spin little), respectively. Then, one can form a diagonal matrix for modified nonrelativistic quark-antiquark potential with diagonal element, and in 3DNRQM-NCSP symmetries:

Here 2k1,k2,k3≡l,-2,-2l-2. The non-null diagonal elements Hsohhp11, Hsohhp22 and Hsohhp33 of the perturbed Hamiltonian operator Hperthhp can be influenced to the energy values Enl by creating three new values ΔEn-ghhp=ΨHsohhp11Ψ, ΔEn-mhhp=ΨHsohhp22Ψ and ΔEn-lhhp=ΨHsohhp33Ψ as:

After straightforward calculations, the radial functions Rnlr satisfy the following differential equation in 3D-NCSP for hydrogenic atoms He +, Li +2 and Be  + and HLM systems under the improved Hulthén plus Hellmann potentials model:

With

here Veff-nchhpr,T is the new generalized effective potential in 3D-NCSP symmetries. We have seen previously that the induced spin-orbit Hsohhp is infinitesimal compared to the principal Hamiltonian operator H _hhp (p, x) in NRQM for HAs and HLM, such as charmonioum cc¯ and bottomonioum bb¯ under the improved Hulthén plus Hellmann potentials model, this allows us to apply standard perturbation theory to determine the nonrelativistic energy corrections Esohhp at the first order of two infinitesimal parameters Θ and θ¯ due to noncommutativity space-space and phase-phase properties.

3.3. BS Solution for the spin-orbit operator for HAs and HLM systems under the IHHPTd model

The Hulthén plus Hellmann potentials model is extended by including new radial terms l(l + 1)r ^-4 β ₀ r ^-3 and β ₁ r ^-1 to become an improved Hulthén plus Hellmann potentials model temperature-dependent in 3DNRQM-NCSP symmetries. The additive part Hperthhp (Eq. (19)) of the new Hamiltonian operator Hnchhp is also proportional to the infinitesimal vectors Θ and θ¯. This allows us to consider the additive part Hperthhp as a perturbation potential compared with the main potential H _hhp (p, x) (Eq. (20)) in the symmetries of 3DNRQM-NCSP, that is, the inequality Hperthhp ≪ H _hhp has to become satisfied. That is, all the physical justifications for applying the time-independent perturbation theory become satisfied. This allows us to give a complete prescription for determining the energy level of the generalized n ^th excited states. Now, we use perturbation theory and in the case of relativistic 3DNRQM-NCSP, we find the expectation values of the radial terms 1/r ⁴, 1/r ³ and 1/r taking into account the wave function which we have seen previously in Eq. (9). Thus, we obtain

where we have used the property of the spherical harmonics given by

with dΩ=sinθdθdφ. To ease the notation, we will provide useful abbreviations 〈nlm〡D〡nlm〉 ≡ 〈D〉 _(nlm) . Comparing Eqs. (32) with the integral of the form [⁷¹, ⁷²]:

Where Reν〉0, Rep〉0, m∈N Λ n ∈ N and  3F2-m,ν,ν-β;-n+ν,λ+1;1 is obtained from the generalized hypergeometric function  3F2α1,...,αp;β1,...,βq,z and Γ(v) being the Gamma function. After some manipulation, we can obtain the explicit results:

and

with X ₁ and X ₂ are equal    3F2-n,[α/ϵnl]-1,-1;-n-1,[α/ϵnl]+1;1 and  3F2(-n,[α/ϵnl],0;-n,[α/ϵnl]+1;1), respectively. We have replaced Γ (2) and Γ (1) with a value of 1 in the denominators of both Eqs. (34.1) and (34.2), respectively, and 1/Γ (-1) with zero in Eq. (34.3). Furthermore, we have used the properties Γ (n + 1) = n Γ (n) = n!. The main goal of this subsection is to determine the corrected energy spectrum ΔEn-sou-hhpk+,n,mDT,A1,A2,A0,j,l,s≡ΔEn-sou-hhp and ΔEn-sod-hhpk-,n,mDT,A1,A2,A0,j,l,s≡ΔEn-sod-hhp which come from Hsohhppnc,xnc corresponding to j = l ± 1/2 at the first order of two parameters Θ and θ¯ for hydrogenic atoms He⁺, Li⁺² and Be⁺ for (n, l) states by applying standard perturbation theory and through the structure constants which specified the dimensionality of the improved Hulthén plus Hellmann potentials model:

with

For the HLM, such as charmonioum cc¯ and bottomonioum bb¯, which quarks and antiquarks of the same system (QQ¯), the eigenvalues of the spin-orbit coupling, we obtain the following results, for the excited states ΔEn-ghhp,ΔEn-mhhp,ΔEn-lhhp respectively: