Exact bound state solutions of the radial Schrödinger equation for the Coulomb potential by conformable Nikiforov–Uvarov approach

Emad K. Jaradat; Omar Alomari; Mariam M. Tarawneh; Omar K. Jaradat

doi:10.1515/phys-2025-0211

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Exact bound state solutions of the radial Schrödinger equation for the Coulomb potential by conformable Nikiforov–Uvarov approach

Emad K. Jaradat , Omar Alomari , Mariam M. Tarawneh und Omar K. Jaradat

Veröffentlicht/Copyright: 30. September 2025

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Abstract

In this work, we investigate the use of the conformable Nikiforov–Uvarov (CNU) method to solve the radial Schrödinger equation exactly for the Coulomb potential. Analytical solutions to the radial Schrödinger equation, which describe how a quantum particle behaves under Coulomb potentials, are frequently challenging. The CNU approach allows us to convert the radial Schrödinger problem into a form that can be solved exactly by the Nikiforov–Uvarov technique, which is well known for its capacity to solve a large class of second-order linear differential equations. We apply the method to the Coulomb potential and obtain accurate formulations for the energy eigenvalues and associated wavefunctions by applying suitable boundary conditions. This method offers a strong framework for examining more intricate quantum systems and provides precise solutions for common potential models. The outcomes offer important new information on quantum mechanical systems with central potentials and demonstrate the effectiveness of the CNU approach in solving the radial Schrödinger equation, particularly when considering fractional dynamics.

Keywords: fractional radial Schrödinger equation; Coulomb potential; conformable fractional derivative; conformable Nikiforov–Uvarov method

1 Introduction

Fractional calculus has become a powerful mathematical tool for accurately characterizing physical problems in recent years [1–4]. Although other definitions of the fractional derivative operator have been put forth, the conformable fractional derivative is often considered highly practical due to its properties that closely mirror classical calculus [5,6]. To solve the radial Schrödinger equation for potentials such as the harmonic oscillator, Woods-Saxon, and Hulthén potentials in the fractional domain, the conformable fractional derivative has been used to extend the Nikiforov–Uvarov (NU) technique. Such quantum systems have been efficiently described by the conformable fractional Nikiforov–Uvarov (CF-NU) method [6,7]. The exploration of quantum systems often involves complex interactions and phenomena, as seen in studies of asymmetric two-level atoms [8], long-lived quantum coherence in semiconductor quantum wells [9], geometric phases [10,11], and multi-atom interactions [12,13].

To solve differential equations, usually in the context of quantum mechanics, mathematical physics, and other applied fields, CFNU method is a modified version of the standard NU method that incorporates fractional calculus, specifically conformable fractional derivatives. The technique uses a methodical approach to solving second-order linear differential equations in conjunction with fractional calculus [6,7,14].

By incorporating concepts from conformable fractional calculus, the conformable Nikiforov–Uvarov (CNU) technique expands on the well-established NU method, making it more adaptable and applicable to a greater range of potential forms, including those present in atomic and molecular systems. Notably, the CNU approach can address complicated potentials like the Hulthén, Morse, and Rosen–Morse potentials by offering a methodical framework for determining energy eigenvalues and wave functions [15–18]. Researchers can investigate both bound and scattered states thanks to this feature, which enhances our comprehension of quantum processes in a variety of physical contexts. The general applicability of the NU method and its extensions to various quantum mechanical potentials has been a subject of ongoing research [19–23].

All things considered, CNU approach is a noteworthy development in the search for answers to the radial Schrödinger equation, advancing theoretical knowledge and real-world applications in quantum mechanics, especially in domains like condensed matter physics and quantum chemistry [18,20–23]. Bound state solutions under a range of possible configurations can be obtained by using this method to systematically derive eigenvalues and eigenfunctions. The use of the NU approach to solve the Schrödinger equation in the presence of modified potentials, including the Modified Hylleraas and Hulthén potentials, which involve intricate interactions that call for approximation techniques, has been shown in recent publications [14–17,19].

Advances have been achieved with the development of conformable versions of the conventional NU technique, which are intended to improve the accuracy and computing efficiency of solutions. By refining the conventional method using the ideas of conformable calculus, the conformable NU method provides a more flexible framework for handling different types of potentials, such as those that arise in atomic and molecular systems. This change in approach highlights an important advancement in the search for precise and approximative solutions in the field of quantum mechanics [18,20–23].

In this study, the radial part of the fractional Schrödinger equation for the Coulomb potential is analyzed by CFNU. The wave functions as well as the energy spectrum for this system are obtained. When the fractional order α = 1 , the energy is consistent with that calculated by the integer NU method. The significance of the fractional order α and the practical applications of this method will be further elaborated in subsequent sections.

This article is organized as follows: Section 2 reviews the CF-NU method, including the properties of the conformable fractional derivative. Section 3 details the derivation of exact bound states for the Coulomb system using this CF-NU approach. For comparative purposes and as a baseline, Section 4 presents the solution for the Coulomb system using the standard NU method. Section 5 offers a graphical representation and comparison of these solutions. The physical and mathematical implications of the fractional order parameter α are explored in Section 6. Section 7 discusses practical applications and provides a comparison of the CF-NU method with other techniques. The potential for extending the CF-NU method to other physical potentials is considered in Section 8. Finally, Section 9 summarizes the key findings and concludes the article.

2 CF-NU method

The standard NU method is a technique used to solve second-order linear differential equations, especially of the form [14]:

(1) d 2 y d x 2 + T ( x ) d y d x + V ( x ) y = 0 ,

where T ( x ) and V ( x ) are certain functions, often arising in quantum mechanics. The method transforms this equation into a hypergeometric equation by using an appropriate change of variables and identifying the solution in terms of special functions like Jacobi polynomials.

In the CFNU method, the standard NU method is modified to handle fractional differential equations. This is done by replacing the usual derivatives with conformable fractional derivatives. The conformable fractional derivative was introduced to provide a fractional derivative definition that satisfies familiar properties of classical derivatives, such as the derivative of a constant being zero, and the product and chain rules in a more classical form, making it more accessible for certain applications.

The conformable fractional derivative of order α for a function f ( t ) is defined as follows [5,6]:

(2) D α [ f ( t ) ] = lim ε → 0 f ( t + ε t 1 − α ) − f ( t ) ε , α ∈ ( 0 , 1 ] , t > 0

And the second differential equation in conformable fractional form, analogous to Eq. (1), becomes:

(3) D α D α y ( x ) + T ( x ) D α y ( x ) + V ( x ) y ( x ) = 0 ,

where D α is the conformable fractional derivative of order α . This fractional equation is then treated using the same stepwise process as the standard NU method, except now we solve it in the context of fractional derivatives as follows: transforming the equation into a hypergeometric-type equation, then, identifying possible solutions using special functions, similar to the standard NU method, finally applying the boundary conditions in the fractional setting [7].

This definition (Eq. (2)) satisfies some important properties [6]:

(4) D α [ a f + b g ] = a D α [ f ] + b D α [ g ] (linearity) ,

(5) D α [ f g ] = f D α [ g ] + g D α [ f ] (product rule) ,

(6) D α [ f ( g ) ] = d f d g D α [ g ] (chain rule, for differentiable f ) ,

(7) D α [ f ] = t 1 − α f ′ , where f ′ = d f d t (key property) .

Accordingly, the conformable fractional derivative is considered fruitful compared to other fractional definitions in modeling many physical systems. To solve the time-independent Schrödinger equation using the NU technique, this requires appropriate transformations to convert it to a hypergeometric differential equation as follows [14]:

(8) ψ ″ ( z ) + τ ˜ ( z ) σ ( z ) ψ ′ ( z ) + σ ˜ ( z ) σ 2 ( z ) ψ ( z ) = 0 ,

where σ ( z ) and σ ˜ ( z ) are second-degree polynomials, τ ˜ ( z ) is a first-degree polynomial, and ψ ( z ) is hypergeometric. By means of the special orthogonal functions, Eq. (8) is solved analytically, and then the bound state solutions are calculated. Based on the key property (Eq. (7)), Eq. (8) for the fractional case becomes [7]:

(9) ψ ″ ( z ) + τ ˜ f ( z ) σ f ( z ) ψ ′ ( z ) + σ ˜ ( z ) σ f 2 ( z ) ψ ( z ) = 0 ,

where

(10) τ ˜ f ( z ) = ( 1 − α ) z − α σ ( z ) + τ ˜ ( z )

(11) σ f ( z ) = z 1 − α σ ( z ) ,

and subscript f indicates fractional. The expressions τ ˜ f ( z ) and σ f ( z ) have α th and ( α + 1 ) th degrees, respectively. The parameter σ ˜ ( z ) has a 2 α th degree. Then π f ( z ) is determined through the following:

(12) π f ( z ) = σ f ′ ( z ) − τ ˜ f ( z ) 2 ± σ f ′ ( z ) − τ ˜ f ( z ) 2 2 − σ ˜ ( z ) + k ( z ) σ f ( z ) .

Subsequently, π f ( z ) is a function of α th degree; to execute this, the formula inside the root sign must be equal to the square of a function of degree α . Next, k ( z ) is found. The relevant functions for the energy spectrum are as follows:

(13) τ ( z ) = τ ˜ f ( z ) + 2 π f ( z ) ,

(14) λ ( z ) = k ( z ) + π f ′ ( z ) ,

(15) λ n ( z ) = − n τ ′ ( z ) − n ( n − 1 ) 2 σ f ″ ( z ) , n = 0 , 1 , 2 , …

To obtain the energy spectrum, equate λ ( z ) = λ n ( z ) . The wave functions are given relying on these equations:

(16) ϕ ′ ( z ) ϕ ( z ) = π f ( z ) σ f ( z ) ,

(17) ( σ f ( z ) ρ ( z ) ) ′ = τ ( z ) ρ ( z ) ,

(18) y n ( z ) = B n ρ ( z ) d n d z n [ σ f n ( z ) ρ ( z ) ] ,

where ψ ( z ) = ϕ ( z ) y n ( z ) .

3 Exact bound states of the coulomb system with the CF-NU

The potential of an electron that is moved in the electrostatic field of the nucleus is defined as follows:

(19) V ( r ) = − Z é 2 r ,

where Z is the atomic number and é = e ∕ 4 π ε 0 . To describe the interactions between particles in Coulomb systems, this potential is essential. The total potential energy of a system with numerous charges is equal to the sum of the pairwise Coulomb interactions between each pair of charges. The radial Schrödinger equation for the Coulomb potential after using suitable coordinate transformation is given as follows:

(20) d 2 ψ ( z ) d z 2 + 2 z d ψ ( z ) d z + ( − P z 2 + Q z − R ) z 2 ψ ( z ) = 0 ,

where

P = − 2 E ∕ a é 2 , Q = 2 Z ∕ a , R = l ( l + 1 ) ,

and the constant a is given as a = ℏ 2 ∕ μ é 2 = 4 π ε 0 ℏ 2 ∕ μ e 2 .

Eq. (17) can be rewritten using the conformable fractional derivative as follows:

(21) D α D α ψ ( z ) + 2 z α D α ψ ( z ) + ( − P z 2 α + Q z α − R ) z 2 α ψ ( z ) = 0 .

Therefore, the parameters are obtained by:

(22) τ ˜ ( z ) = 2 , σ ( z ) = z α , σ ˜ ( z ) = − P z 2 α + Q z α − R .

In addition, the fractional form of parameters (referring to τ ˜ f ( z ) , σ f ( z ) ) using the key property is:

(23) τ ˜ f ( z ) = ( 1 − α ) z − α σ ( z ) + τ ˜ ( z ) = 3 − α ,

(24) σ f ( z ) = z 1 − α σ ( z ) = z ,

(25) σ ˜ ( z ) = − P z 2 α + Q z α − R .

After substituting the fractional parameters into Eq. (12), π f ( z ) is expressed as follows:

(26) π f ( z ) = α − 2 2 ± P z 2 α − [ Q z α − 1 − k ] z + ( α − 2 ) 2 4 + R .

To obtain the values of k , the discriminant of the relation inside the root sign in the previous equation must be equal to zero. This leads to:

Δ = [ Q z α − 1 − k ] 2 − 4 P ( α − 2 ) 2 4 + R = 0 .

Thus,

(27) k 2 − 2 Q k z α − 1 + Q 2 z 2 α − 2 − P ( α − 2 ) 2 − 4 P R = 0 ,

and hence,

(28) k ± = ( Q ± P ( α − 2 ) 2 + 4 P R ) z α − 1 .

As a result, the function π f ( z ) takes the forms:

(29) π f ( z ) = α − 2 2 ± P z α + 1 2 ( α − 2 ) 2 + 4 R , for k + = ( Q + P ( α − 2 ) 2 + 4 P R ) z α − 1 P z α − 1 2 ( α − 2 ) 2 + 4 R , for k − = ( Q − P ( α − 2 ) 2 + 4 P R ) z α − 1

The proper form of π f ( z ) to obtain bound state solutions is given as follows:

(30) π f ( z ) = α − 2 2 − P z α + 1 2 ( α − 2 ) 2 + 4 R ,

for k − = ( Q − P ( α − 2 ) 2 + 4 P R ) z α − 1 . Based on Eq. (13), the τ ( z ) can be determined as follows:

(31) τ ( z ) = 1 + ( α − 2 ) 2 + 4 R − 2 P z α .

Using Eqs. (14) and (15):

(32) λ ( z ) = [ Q − P ( α − 2 ) 2 + 4 P R − α P ] z α − 1 .

(33) λ n ( z ) = 2 n α P z α − 1 .

Equate λ ( z ) and λ n ( z ) , and then the energy of this system is expressed as follows:

(34) E n = − 2 Z 2 μ é 4 ∕ ℏ 2 ( ( 2 n + 1 ) α + ( α − 2 ) 2 + 4 l ( l + 1 ) ) 2 .

Besides that the eigenfunctions Φ ( z ) and ρ ( z ) are obtained by means of Eqs. (16) and (17), yielding:

(35) Φ ( z ) = z ( 1 ∕ 2 ) ( α − 2 + ( α − 2 ) 2 + 4 R ) e − P z α ∕ α ,

(36) ρ ( z ) = z ( α − 2 ) 2 + 4 R e − 2 P z α ∕ α .

Then, substituting ρ ( z ) and σ f ( z ) into Eq. (18) yields:

(37) y n ( z ) = B n z − ( α − 2 ) 2 + 4 R e 2 P z α ∕ α d n d z n [ z n + ( α − 2 ) 2 + 4 R e − 2 P z α ∕ α ] .

Hence, the unnormalized wave functions of the Coulomb potential are given as follows ( ψ ( z ) = Φ ( z ) y n ( z ) ):

(38) ψ ( z ) = N n z ( 1 ∕ 2 ) ( α − 2 − ( α − 2 ) 2 + 4 R ) e P z α ∕ α × d n d z n [ z n + ( α − 2 ) 2 + 4 R e − 2 P z α ∕ α ] .

4 Exact bound states of the Coulomb system using the standard NU method

To verify the solution obtained above, particularly for the case α = 1 , we now apply the standard NU method to solve the proposed problem. This method is used to derive the energy eigenvalues and the corresponding radial wavefunction for an electron subject to a Coulomb potential. The resulting solution will then be compared to the fractional solution derived earlier for α = 1 , serving as a consistency check.

As in the previous section, the electrostatic potential experienced by an electron due to a nucleus of atomic number Z is expressed as follows:

(39) V ( r ) = − Z é 2 r ,

where é = e ∕ 4 π ε 0 incorporates the elementary charge e and the vacuum permittivity ε 0 . Upon separation of variables and suitable coordinate and wavefunction transformations, the radial part of the Schrödinger equation can be cast into the following form [24,25]:

(40) d 2 ψ ( z ) d z 2 + 2 z d ψ ( z ) d z + ( − P z 2 + Q z − R ) z 2 ψ ( z ) = 0 ,

where the parameters P , Q , and R are defined as follows:

(41) P = − 2 μ E a 0 2 Z 2 ℏ 2 or equivalently P = − 2 E a N é 2 ,

(42) Q = 2 μ Z é 2 a 0 Z ℏ 2 or equivalently Q = 2 Z a N ,

(43) R = l ( l + 1 ) .

Here, E is the energy eigenvalue, μ is the reduced mass of the system, ℏ is the reduced Planck constant, l is the azimuthal quantum number, and a N = ℏ 2 ∕ ( μ é 2 ) is a characteristic length scale related to the Bohr radius a 0 = 4 π ε 0 ℏ 2 ∕ ( μ e 2 ) = a N ∕ Z (using a N here to match the original problem’s a ). For bound states, E < 0 , and consequently, P > 0 .

The NU method addresses the second-order differential equations that can be written as follows [14]:

(44) ψ ″ ( z ) + τ ˜ ( z ) σ ( z ) ψ ′ ( z ) + σ ˜ ( z ) σ 2 ( z ) ψ ( z ) = 0 ,

where σ ( z ) and σ ˜ ( z ) are polynomials of degree at most two, and τ ˜ ( z ) is a polynomial of degree at most one. The core idea is to find a transformation ψ ( z ) = ϕ ( z ) y ( z ) such that y ( z ) satisfies a simpler, hypergeometric-type equation whose solutions are known classical orthogonal polynomials.

The key components of the method include:

The determination of an auxiliary polynomial π ( z ) , related to σ ( z ) and τ ˜ ( z ) , and a parameter k , through the equation:
(45) π 2 ( z ) = σ ′ ( z ) − τ ˜ ( z ) 2 2 − σ ˜ ( z ) + k σ ( z ) .
π ( z ) must be a polynomial of degree at most one, which imposes conditions on k .
The energy quantization condition arises from the relation λ = λ n r , where
(46) λ = k + π ′ ( z ) ,

(47) λ n r = − n r τ ˜ ′ ( z ) − n r ( n r − 1 ) 2 σ ″ ( z ) , ( n r = 0 , 1 , 2 , … ) .
Here, n r represents the degree of the polynomial solution y n r ( z ) , corresponding to a quantum number (in this case, the radial quantum number).
The first part of the solution, ϕ ( z ) , is found by integrating:
(48) ϕ ′ ( z ) ϕ ( z ) = π ( z ) σ ( z ) .
The second part, y n r ( z ) , comprises the polynomial solutions, typically related to classical orthogonal polynomials by a Rodrigues-type formula involving a weight function ρ ( z ) , which satisfies ( σ ( z ) ρ ( z ) ) ′ = ( τ ˜ ( z ) + 2 π ( z ) ) ρ ( z ) .

By comparing Eq. (40) with the standard NU form Eq. (44), we select σ ( z ) = z . This choice yields: σ ( z ) = z , τ ˜ ( z ) = 2 , σ ˜ ( z ) = − P z 2 + Q z − R .

The necessary derivatives are σ ′ ( z ) = 1 , σ ″ ( z ) = 0 , and τ ˜ ′ ( z ) = 0 .

With τ ˜ ′ ( z ) = 0 and σ ″ ( z ) = 0 , the quantization condition Eq. (47) simplifies to λ n r = 0 . Thus, from Eq. (46), we require k + π ′ ( z ) = 0 .

Substituting the identified polynomials into Eq. (45):

π 2 ( z ) = P z 2 + ( k − Q ) z + R + 1 4 .

For π ( z ) to be a first-degree polynomial, P z 2 + ( k − Q ) z + ( R + 1 ∕ 4 ) must be the square of a first-degree polynomial. Let π ( z ) = c 1 z + c 0 . Then π ′ ( z ) = c 1 . The condition k + c 1 = 0 must hold. For the specific structure of the Coulomb problem, and to ensure physically acceptable (decaying) solutions, the NU method (often by direct comparison with the parameters of the confluent hypergeometric equation to which Eq. (40) can be transformed [25]) consistently yields the quantization condition:

(49) Q 2 P − ( l + 1 ) = n r ,

where n r = 0 , 1 , 2 , … is the radial quantum number. This condition arises fundamentally from the requirement that the polynomial part of the wavefunction, y n r ( z ) , must terminate to ensure normalizability.

Introducing the principal quantum number n = n r + l + 1 (where n = 1 , 2 , 3 , … ), Eq. (49) becomes Q 2 P = n . This leads to P n = Q 2 4 n 2 . Using the definitions of P and Q from Eqs. (41)-(42) (with a N for a ):

− 2 E n a N é 2 = ( 2 Z ∕ a N ) 2 4 n 2 = Z 2 a N 2 n 2 .

Solving for E n and substituting a N = ℏ 2 ∕ ( μ é 2 ) :

(50) E n = − Z 2 é 2 2 a N n 2 = − μ Z 2 é 4 2 ℏ 2 n 2 .

This is the well-established Bohr formula for the energy levels of hydrogen-like atoms.

The wavefunction is ψ n , l ( z ) = ϕ ( z ) y n − l − 1 ( z ) , where n r = n − l − 1 . For ϕ ( z ) , using Eq. (48) with σ ( z ) = z , and selecting π ( z ) to match the known asymptotic behavior and z l dependence near the origin, i.e., π ( z ) = l − P n z :

ϕ ′ ( z ) ϕ ( z ) = l − P n z z = l z − P n .

Integration yields ϕ ( z ) = C z l e − P n z , where C is an integration constant. The polynomial component y n − l − 1 ( z ) is found by substituting ψ n , l ( z ) = ϕ ( z ) y n − l − 1 ( z ) into Eq. (40). This procedure leads to the following differential equation for y n − l − 1 ( z ) , after a change of variable x = 2 P n z :

(51) x d 2 y n − l − 1 d x 2 + ( ( 2 l + 1 ) + 1 − x ) d y n − l − 1 d x + ( n − l − 1 ) y n − l − 1 ( x ) = 0 .

This is the defining equation for the associated Laguerre polynomials L N ( β ) ( x ) , where N = n − l − 1 is the degree (radial quantum number n r ) and β = 2 l + 1 . Thus, y n − l − 1 ( x ) ∝ L n − l − 1 ( 2 l + 1 ) ( x ) . Combining these parts, the radial wavefunction is given as follows:

(52) ψ n , l ( z ) = N n , l z l e − P n z L n − l − 1 ( 2 l + 1 ) ( 2 P n z ) ,

where N n , l is the normalization constant. The argument of the Laguerre polynomial is 2 P n z = 2 Z n a N z . The allowed quantum numbers are n = 1 , 2 , … and l = 0 , 1 , … , n − 1 .

5 Graphical representation of solutions

The obtained wavefunctions are illustrated in Figures 1, 2, 3, 4, 5, 6, which display both the radial wavefunction and the corresponding radial probability density for specific quantum states. Figure 1 presents an overlay comparison of the wavefunctions for the n = 0 , l = 0 state (corresponding to the state) computed using the standard NU method and the CF-NU method, where the fractional order α is set to 1. This comparison visually confirms that the CF-NU solution converges to the classical result when α = 1 . A similar agreement is demonstrated for the radial probability density function in Figure 2. Figures 3–6 illustrate how the wavefunctions and probability density functions obtained using the CF-NU method vary for different values of the fractional parameter α for n = 0 states with l = 0 and l = 1 .

$Figure 1 Combined plot overlaying the radial wavefunction ψ n , l ( z ) {\psi }_{n,l}\left(z) for the state ( n = 0 , l = 0 n=0,l=0 ) of a hydrogen-like atom. The solid line represents the solution from the standard NU method, and the dashed line represents the solution from the CF-NU method with fractional order α = 1 \alpha =1 . Parameters used for this plot: principal quantum number n = 0 n=0 , Azimuthal quantum number l = 0 l=0 , CF-NU radial quantum number n r a d = 0 {n}_{rad}=0 , CF-NU fractional order α = 1 \alpha =1 , atomic number Z = 1 Z=1 , Bohr radius parameter a = 1 a=1 . The perfect overlap demonstrates the consistency of the CF-NU method with the standard approach when α = 1 \alpha =1 .$

Figure 1

Combined plot overlaying the radial wavefunction ψ n , l ( z ) for the state ( n = 0 , l = 0 ) of a hydrogen-like atom. The solid line represents the solution from the standard NU method, and the dashed line represents the solution from the CF-NU method with fractional order α = 1 . Parameters used for this plot: principal quantum number n = 0 , Azimuthal quantum number l = 0 , CF-NU radial quantum number n r a d = 0 , CF-NU fractional order α = 1 , atomic number Z = 1 , Bohr radius parameter a = 1 . The perfect overlap demonstrates the consistency of the CF-NU method with the standard approach when α = 1 .

$Figure 2 Combined plot overlaying the radial probability density z 2 ∣ ψ n , l ( z ) ∣ 2 {z}^{2}| {\psi }_{n,l}\left(z){| }^{2} for the state ( n = 0 , l = 0 n=0,l=0 ) of a hydrogen-like atom. The solid line represents the solution from the standard NU method, and the dashed line represents the solution from the CF-NU method with fractional order α = 1 \alpha =1 . Parameters used for this plot: principal quantum number n = 0 n=0 , Azimuthal quantum number l = 0 l=0 , CF-NU radial quantum number n = 0 n=0 , CF-NU fractional order α = 1 \alpha =1 , atomic number Z = 1 Z=1 , Bohr radius parameter a = 1 a=1 . The perfect overlap demonstrates the consistency of the CF-NU method with the standard approach when α = 1 \alpha =1 .$

Figure 2

Combined plot overlaying the radial probability density z 2 ∣ ψ n , l ( z ) ∣ 2 for the state ( n = 0 , l = 0 ) of a hydrogen-like atom. The solid line represents the solution from the standard NU method, and the dashed line represents the solution from the CF-NU method with fractional order α = 1 . Parameters used for this plot: principal quantum number n = 0 , Azimuthal quantum number l = 0 , CF-NU radial quantum number n = 0 , CF-NU fractional order α = 1 , atomic number Z = 1 , Bohr radius parameter a = 1 . The perfect overlap demonstrates the consistency of the CF-NU method with the standard approach when α = 1 .

$Figure 3 CFNU radial wavefunction ψ n , l ( z ) {\psi }_{n,l}\left(z) for state ( n = 0 , l = 0 n=0,l=0 ) of a hydrogen-like atom. Parameters used for this plot: principal quantum number n = 0 n=0 , Azimuthal quantum number l = 0 l=0 , atomic number Z = 1 Z=1 , Bohr radius parameter a = 1 a=1 .$

Figure 3

CFNU radial wavefunction ψ n , l ( z ) for state ( n = 0 , l = 0 ) of a hydrogen-like atom. Parameters used for this plot: principal quantum number n = 0 , Azimuthal quantum number l = 0 , atomic number Z = 1 , Bohr radius parameter a = 1 .

$Figure 4 CFNU radial probability density z 2 ∣ ψ n , l ( z ) ∣ 2 {z}^{2}| {\psi }_{n,l}\left(z){| }^{2} for the state ( n = 0 , l = 0 n=0,l=0 ) of a hydrogen-like atom. Parameters used for this plot: Principal quantum number n = 0 n=0 , Azimuthal quantum number l = 0 l=0 , atomic number Z = 1 Z=1 , Bohr radius parameter a = 1 a=1 .$

Figure 4

CFNU radial probability density z 2 ∣ ψ n , l ( z ) ∣ 2 for the state ( n = 0 , l = 0 ) of a hydrogen-like atom. Parameters used for this plot: Principal quantum number n = 0 , Azimuthal quantum number l = 0 , atomic number Z = 1 , Bohr radius parameter a = 1 .

$Figure 5 CFNU radial wavefunction ψ n , l ( z ) {\psi }_{n,l}\left(z) for the state ( n = 0 , l = 1 n=0,l=1 ) of a hydrogen-like atom. Parameters used for this plot: principal quantum number n = 0 n=0 , Azimuthal quantum number l = 1 l=1 , atomic number Z = 1 Z=1 , Bohr radius parameter a = 1 a=1 .$

Figure 5

CFNU radial wavefunction ψ n , l ( z ) for the state ( n = 0 , l = 1 ) of a hydrogen-like atom. Parameters used for this plot: principal quantum number n = 0 , Azimuthal quantum number l = 1 , atomic number Z = 1 , Bohr radius parameter a = 1 .

$Figure 6 CFNU radial probability density z 2 ∣ ψ n , l ( z ) ∣ 2 {z}^{2}| {\psi }_{n,l}\left(z){| }^{2} for the state ( n = 0 , l = 1 n=0,l=1 ) of a hydrogen-like atom. Parameters used for this plot: principal quantum number n = 0 n=0 , Azimuthal quantum number l = 1 l=1 , atomic number Z = 1 Z=1 , Bohr radius parameter a = 1 a=1 .$

Figure 6

CFNU radial probability density z 2 ∣ ψ n , l ( z ) ∣ 2 for the state ( n = 0 , l = 1 ) of a hydrogen-like atom. Parameters used for this plot: principal quantum number n = 0 , Azimuthal quantum number l = 1 , atomic number Z = 1 , Bohr radius parameter a = 1 .

6 Significance of the fractional parameter α

The fractional order α within the CFNU framework introduces a degree of freedom that generalizes standard integer-order quantum mechanics. When α = 1 , the CFNU formulation precisely recovers the classical NU results and the well known solutions of the Schrödinger equation. Deviations of α from unity imply a modification to the underlying differential operators governing the system’s dynamics. Physically, such fractional derivatives can model systems exhibiting non-local interactions, memory effects, or dynamics within complex media or fractal structures where standard calculus may be insufficient [2,4]. For the Coulomb potential, varying α alters the effective strength and radial dependence of the interaction as perceived by the fractional derivative, leading to shifts in the quantized energy levels (as shown in Eq. (34)) and modifications to the spatial distribution of the wavefunctions (Eq. (38)). While direct experimental validation of fundamental fractional Coulomb interactions remains an open area, the parameter α provides a valuable theoretical means to explore potential deviations from standard quantum behavior and to model effective interactions in more complex physical scenarios.

7 Practical applications and comparison

The CFNU method extends the utility of the established NU technique to the domain of fractional calculus, offering distinct advantages for specific classes of problems. While fundamental interactions in quantum mechanics are typically described by integer-order derivatives, effective field theories or models of quantum systems in complex environments (e.g., porous media, systems with long-range memory) may naturally lead to Schrödinger-like equations involving fractional derivatives [3]. In such scenarios, the CFNU method provides a robust analytical pathway to exact or approximate solutions that might be intractable with standard methods. Compared with purely numerical approaches for fractional differential equations, which can be computationally intensive and may present challenges regarding stability and convergence, analytical solutions derived via CFNU offer deeper physical insight into the system’s parameter dependencies and overall behavior. The method’s direct incorporation of the fractional order α allows for a continuous bridge between fractional and standard integer-order descriptions, as demonstrated by its convergence to classical NU results when α = 1 . Although applications where α ≠ 1 is a fundamental necessity for describing elementary Coulomb interactions are not yet established, the CFNU framework is valuable for model building in mesoscopic physics, quantum chemistry where effective potentials are used, or any scenario where anomalous quantum dynamics are suspected or observed.

8 Generalization to other potentials

The adaptability of the NU method and its conformable fractional extension (CF-NU) is not limited to the Coulomb potential. These algebraic techniques can be applied to a significant range of other physically relevant potentials, provided the corresponding Schrödinger equation (integer or fractional order) can be transformed into the canonical form required by the NU method. Examples include the harmonic oscillator, Morse potential, Hulthén potential, Rosen-Morse potential, Pöschl-Teller potential, and various empirical potentials used in molecular and nuclear physics [18,20]. The general procedure involves: (i) starting with the radial Schrödinger equation for the target potential, incorporating the conformable fractional derivative D α if a fractional treatment is desired; (ii) performing appropriate coordinate transformations (e.g., z = f ( r ) ) to cast the equation into the NU standard form; (iii) identifying the NU polynomials σ ( z ) , τ ˜ ( z ) , and σ ˜ ( z ) (or their fractional counterparts σ f ( z ) , τ ˜ f ( z ) ); and (iv) systematically applying the NU machinery to derive energy eigenvalues and eigenfunctions. While the core algebraic steps of the NU/CFNU method remain consistent, the specific transformations and the resulting NU polynomials are inherently potential dependent, often requiring ingenuity in the initial setup. Successful applications to potentials like the Hartmann potential [15] and modified Hylleraas-Hulthén potentials [16] further underscore the method’s versatility.

9 Conclusion

In this study, we have investigated the exact bound state solutions of the radial Schrödinger equation for the Coulomb potential using the CFNU approach. The methodology allows for the systematic derivation of energy eigenvalues and unnormalized wavefunctions, where the fractional order α serves as a generalizing parameter. We have explicitly shown that for α = 1 , the CFNU results for the n = 0 state align perfectly with those obtained via the standard NU method, as illustrated by the graphical comparisons (Figures 1 and 2). Furthermore, the influence of varying α on the n = 0 wavefunctions and their corresponding probability densities for different l values was presented (Figures 3–6). The CFNU method extends the powerful algebraic framework of the NU technique into the domain of fractional calculus, providing a valuable tool for solving fractional differential equations that may arise in various physical contexts. This work underscores the consistency of the CFNU approach and its potential for modeling quantum systems where fractional dynamics might offer a more precise description or new theoretical insights.

Funding information: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2502).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2025-03-17

Revised: 2025-06-06

Accepted: 2025-06-12

Published Online: 2025-09-30

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2025-0211

Schlagwörter für diesen Artikel

fractional radial Schrödinger equation; Coulomb potential; conformable fractional derivative; conformable Nikiforov–Uvarov method

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