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REVIEW 3 major objections 5 minor 56 references

Bound state solutions with a linear combination of Yukawa plus four-parameter diatomic potentials using path integral approach: Thermodynamic properties

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper claims that a Yukawa-plus-four-parameter diatomic potential, after a centrifugal approximation, maps to a Rosen-Morse potential, giving closed-form approximate bound-state energies, wavefunctions, and thermodynamic functions for d

desk verdict Routine extension with an unvalidated approximation and an energy formula that likely does not follow from the stated pole condition; not ready for publication. read the letter →

arxiv 2601.04806 v2 pith:QXW22M33 submitted 2026-01-08 quant-ph math-phmath.MPphysics.atom-ph

Mohamed Améziane Sadoun , Redouane Zamoum , Abdellah Touati This is my paper · ORCID
classification quant-phmath-phmath.MPphysics.atom-ph
keywords Yukawapotentialfour-parameterdiatomicpathintegralRosen-Morseboundstatespartitionfunctionthermodynamicpropertiesmolecules
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish that the three-dimensional Schrödinger equation for a potential formed by adding the Yukawa potential to a generalized four-parameter (q-deformed) diatomic potential has approximate analytic bound-state solutions for q ≥ 1 and r > r0. The authors show that by approximating the centrifugal term with exponential forms, the radial equation is transformed into a one-dimensional Rosen-Morse potential, whose exact path-integral Green's function is known. The poles of that Green's function yield a closed-form energy spectrum, and its residues give normalized wavefunctions. From the spectrum they construct a vibrational partition function and derive the free energy, mean energy, heat capacity, and entropy. If correct, this provides a single analytic model for the bound states and thermal properties of six diatomic molecules: H2, I2, LiH, CO, HCl, and NO.

What carries the argument

The key machinery is the pair of centrifugal approximations in Eq. (5): 1/r ≈ 2α e^{−αr}/(1−q e^{−2αr}) and 1/r² ≈ 4α² e^{−2αr}/(1−q e^{−2αr})². These convert the potential and the centrifugal barrier into combinations of terms proportional to e^{−2αr}/(1−q e^{−2αr}) and e^{−4αr}/(1−q e^{−2αr})². The substitution r = (1/2α) ln(e^{4αξ}+q), followed by ξ = y/(2α) and u = y − (1/2) ln q, transforms the radial equation into a one-dimensional Rosen-Morse (generalized Pöschl-Teller) potential in u. The Rosen-Morse potential has a known exact path-integral solution expressed through Wigner functions and Euler Γ-functions; the poles of its Green's function yield the spectrum, and the residues give t

What would settle it

Numerically solve the radial Schrödinger equation with the original potential V(r) in Eq. (1) for a specific diatomic molecule (e.g., H2 with the spectroscopic parameters in Table I) and compare the lowest eigenvalues with the closed-form values from Eq. (30). If the deviations exceed the expected approximation error, or if the wavefunctions from Eq. (38) do not closely satisfy the original differential equation, the central claim is falsified. A simpler check is to evaluate the relative error of the approximations (5) over the classically allowed region for each molecule's parameters.

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Extended reading notes

Core claim

The central claim is that the radial Schrödinger equation for V(r) = a/(e^{2αr} − q)² − b/(e^{2αr} − q) − c e^{−αr}/r can be solved approximately in closed form for q ≥ 1 and r > r0 = (1/2α) ln q. Using the approximations 1/r ≈ 2α e^{−αr}/(1 − q e^{−2αr}) and 1/r² ≈ 4α² e^{−2αr}/(1 − q e^{−2αr})², and a change of variables, the problem becomes a Rosen-Morse potential in a new coordinate u. The exact path-integral Green's function for the Rosen-Morse potential then gives the bound-state energies E_{n,l} (Eq. 30), the normalized wavefunctions χ_{n,l}(r) (Eq. 38), and, via a Poisson-summation evaluation of the vibrational partition function Z_{vib}(β) (Eq. 51), the thermodynamic functions. The

Load-bearing premise

The approximations in Eq. (5) for 1/r and 1/r² are assumed to be uniformly accurate for r > r0, and the inner region 0 < r < r0 is discarded without rigorous justification or an error estimate.

Editorial extensions

If this is right

  • The bound-state energy formula (Eq. 30) gives an explicit closed form for any n, l, and deformation parameter q ≥ 1.
  • The normalized wavefunctions (Eq. 38) are expressed in terms of hypergeometric functions and vanish at infinity only when Q_l < 0, which sets the maximum number of bound states n_max via Eq. (39).
  • The vibrational partition function (Eq. 51) is a closed-form expression involving imaginary error functions, enabling direct calculation of free energy, mean energy, heat capacity, and entropy for a diatomic molecule.
  • For the six molecules considered (H2, I2, LiH, CO, HCl, NO), the model predicts that energies become less negative as n and q increase, with the influence of q strongest for H2 and HCl and negligible for I2.
  • The thermodynamic functions show a characteristic maximum in specific heat and an entropy drop whose position shifts with the molecule's parameters, consistent with known diatomic thermochemistry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The mapping to Rosen-Morse suggests that any potential expressible as a combination of (e^{2αr}−q)^{-k} terms may be exactly solvable with the same path-integral machinery, potentially extending the family of exactly solvable molecular potentials.
  • The model discards the inner region 0 < r < r0 without a physical or boundary-condition justification; for light molecules like H2 at high temperatures, the wavefunction may have significant support there, so the thermodynamic predictions could inherit a non-negligible error.
  • The closed-form partition function makes it straightforward to compute additional response functions (e.g., the isochoric heat capacity from the analytic entropy) that the paper does not plot, but which would follow directly from the same formulas.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper uses the Feynman path-integral formalism to study approximate bound states of the radial Schrödinger equation with the combined potential V(r)=a/(e^{2αr}-q)^2 - b/(e^{2αr}-q) - c e^{-αr}/r. Two approximations (Eq. (5)) replace 1/r and 1/r^2 by expressions in e^{-αr}/(1 - q e^{-2αr}), and the domain is restricted to r>r0=(1/2α)ln q. The authors map the problem to a Rosen-Morse potential and obtain an energy formula (Eq. (30)), normalized wave functions (Eq. (38)), and a vibrational partition function (Eq. (51)), from which free energy, mean energy, heat capacity, and entropy are derived. Numerical values are tabulated for H2, I2, LiH, CO, HCl, and NO, along with plots of thermodynamic quantities.

Significance. If correct, the work would add a new analytically solvable (within stated approximations) potential with closed-form thermodynamic functions, of potential interest to molecular physics. The paper includes explicit formulas, uses realistic diatomic parameters, and provides extensive plots. However, the central result rests on unvalidated approximations and contains an algebraic inconsistency in the definition of P_l that changes the spectrum. Since the energy formula and all thermodynamic expressions are built on this, the contribution in its present form cannot be considered reliable.

major comments (3)
  1. [Sec. IV, Eq. (31)] The pole condition M_1 - L_E = -n, combined with definitions (27)-(29), gives P_l = (αq/2)(1 + sqrt(1 + 4l(l+1)/q + (2m/ħ^2)(a/(α^2 q^2)))). Equation (31) is missing the leading '1 +' inside the square root. For l=0, a=0, the printed P_l is αq/2 instead of the required αq. This changes every energy level. For representative H2-like parameters, the n=1 energy becomes -0.68 eV with the printed P_l, whereas the correct P_l gives -1.53 eV. Since Table II and all thermodynamic quantities are computed from Eq. (30) with Eq. (31), this is a load-bearing algebraic error.
  2. [Sec. II, Eq. (5)] The approximations 1/r ≈ 2α e^{-αr}/(1 - q e^{-2αr}) and 1/r^2 ≈ 4α^2 e^{-2αr}/(1 - q e^{-2αr})^2 are asserted to be valid for q≥1, but no derivation, error estimate, or numerical check is provided. For r>r0=(1/2α)ln q, the right-hand sides diverge as r→r0, whereas 1/r and 1/r^2 remain finite. The inner region r<r0 is discarded without specifying a boundary condition or physical justification. The numerical verification promised in Sec. VI is only a tabulation of Eq. (30); no comparison is made with a direct numerical solution of the original potential (1). Thus the central spectrum and all thermodynamic results inherit an unquantified approximation error.
  3. [Sec. IV, Eq. (30)] As written, Eq. (30) gives E_n → -∞ as n→∞, implying an infinite number of bound states for what is a short-range potential. The upper limit λ_max is introduced later through Eq. (41), the turning point condition ∂E/∂n=0, but the derivation does not show why states with n>λ_max are absent. In the underlying Rosen-Morse solution, the reality of M_1 and L_E imposes a finite bound-state count; the squared form in Eq. (30) discards the sign information that enforces this. The paper should derive the bound-state condition directly from the pole condition and justify λ_max from it rather than from a separate ad hoc condition.
minor comments (5)
  1. [Eq. (19)] The definitions of sinh_q(x) and cosh_q(x) are given identically as (e^x - q e^{-x})/2. The second should be (e^x + q e^{-x})/2. This is a central mathematical object and needs correction.
  2. [Title] The title contains 'Yuakawa'; it should read 'Yukawa'.
  3. [Section numbering] There are two sections numbered IV in the text (Energy spectrum and Thermodynamic properties). The second should be V.
  4. [Sec. VI] The claim 'To verify the accuracy of our results, a numerical evaluation of the energy levels ... is established' is not supported. Table II simply lists values from Eq. (30); no comparison with numerical solutions, experimental data, or independent calculations is presented.
  5. [Eqs. (53)-(61)] The vertical-bar notation, e.g., 'y^2 |_{ρ2}^{ρ1}', is never defined, and the thermodynamic expressions are extremely long and difficult to verify. The authors should define the notation and provide a more readable and independently checkable derivation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation reduces the potential to the Rosen-Morse form via explicit approximations and then uses an external closed-form Green's function; no output is fitted and self-citations are not load-bearing.

full rationale

The paper's main derivation chain is: approximate 1/r and 1/r^2 by exponential forms (Eq. 5), change variables so the radial Hamiltonian becomes a Rosen-Morse potential, and then take the Rosen-Morse Green's function, pole condition, and residue formulas from the external Handbook of Feynman path integrals [44] and from Kleinert-Mustapic [48]. The energy formula in Eq. (30) and the wave functions in Eq. (38) are therefore computed from the stated approximate model, not extracted from the target energies or from a fit to the same data. No parameter is fitted to the energies that are later called predictions, and no quantity is defined in terms of the output. The self-citations [15,16,17] appear only as method credits in the introduction and are not used to justify the central Rosen-Morse mapping or the closed-form Green's function. The thermodynamic section starts from the partition-function sum over the paper's own energy formula; this is a legitimate derivation chain, not circular reasoning. The serious weaknesses of the paper — the unquantified validity of Eq. (5), the discarded inner region 0<r<r0 without a boundary condition, the apparent mismatch between the stated pole condition M1-L_E=-n and Eq. (30), and the lack of a numerical benchmark against the original potential (Eq. 1) — are correctness and accuracy concerns, not circular reductions. Thus no circular step can be exhibited, and the score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper does not introduce new particles or forces. Its main extrinsic inputs are the Pekeris-type approximations, the exclusion of r<r0, the known Rosen-Morse Green's function, and the leading-order Poisson summation approximation. The unlisted V0 parameter is a reproducibility gap rather than an invented entity.

free parameters (2)
  • Yukawa strength c = V0 = not reported for H2, I2, LiH, CO, HCl, NO
    Every numerical energy and thermodynamic plot depends on c through the combination b+2alpha c, but Table I omits V0. Without these values, Table II and Figures 4-5 cannot be reproduced.
  • deformation parameter q = varied 1-8, no fit
    q is an input parameter of the deformed potential, not fitted to data, but the numerical results are scanned over it and the solution is restricted to q>=1.
assumptions (4)
  • ad hoc to paper Pekeris-type approximations 1/r ~ 2alpha e^{-alpha r}/(1-q e^{-2alpha r}) and 1/r^2 ~ 4alpha^2 e^{-2alpha r}/(1-q e^{-2alpha r})^2
    These substitutions convert the potential and centrifugal term into exponential form so that the path integral maps to Rosen-Morse. Their validity is asserted only for q>=1 and never quantified. Entered at Eq. (5).
  • domain assumption The region 0<r<r0=(1/2alpha)ln q can be ignored and a stable path integral exists on ]r0,infinity[
    The potential is singular at r0; the paper discards the inner region as 'not of great physical interest' without analyzing boundary conditions at the singular point. This appears after Eq. (6).
  • standard math Known Green's function for the Rosen-Morse potential from Grosche-Steiner [44] and Kleinert-Mustapic [48]
    The paper reads off the Green's function (Eq. 26) and its pole/residue structure from the literature rather than re-deriving it. This is an external, standard result but is load-bearing for the energy spectrum.
  • domain assumption Leading-order Poisson summation approximation for the partition function, Eq. (46)
    The Poisson summation formula is truncated to the m=0 term with no error estimate; all thermodynamic results inherit this approximation.

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Cite this review

Pith. "Pith review of Bound state solutions with a linear combination of Yukawa plus four-parameter diatomic potentials using path integral approach: Thermodynamic properties." pith.science (2026). https://pith.science/paper/QXW22M33

@misc{pith2026260104806,
  author       = {Pith},
  title        = {Pith review of: Bound state solutions with a linear combination of Yukawa plus four-parameter diatomic potentials using path integral approach: Thermodynamic properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXW22M33}},
  note         = {Machine review of arXiv:2601.04806}
}
read the original abstract

In this paper, we investigate the approximate analytical bound states with a linear combination of two diatomic molecule potentials, Yukawa and four parameters potentials, within the framework of the path integral formalism. With the help of an appropriate approximation to evaluate the centrifugal term, the energy spectrum and the normalized wave functions of the bound states are derived from the poles of Green's function and its residues. The partition function and other thermodynamic properties were obtained using the compact form of the energy equation.

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Reference graph

Works this paper leans on

56 extracted references

  1. [1]

    1 2 h eβΛρ2 1 −e βΛρ2 2 i + 1 4 r π βΛ

    Vibrational free energy The free energy for a molecule is expressed as follow: F(β) =−k BTlnZ vib (β), =− 1 β ln " 1 2 h eβΛρ2 1 −e βΛρ2 2 i + 1 4 r π βΛ " erfi p βΛρ1 −erfi p βΛρ2 +e −4βΛη 1 −erfi q βΛ(4η1 +ρ 2 1) +erfi q βΛ(4η1 +ρ 2 2) ## . (52)

  2. [2]

    Vibrational mean energy By definition the vibrational mean energy can be evaluated using the above partition function (51): Uvib (β) =− ∂lnZ vib ∂β , = 4(βΛ) 3/2e4βη1ΛeβΛy2 y2 ρ2 ρ1 + √π erfi p βΛ p 4η1 +y 2 ρ2 ρ1 −e 4βΛη1 erfi y p βΛ ρ2 ρ1 2β √π erfi p βΛ p 4η1 +y 2 ρ2 ρ1 −e 4βη1Λ erfi y p βΛ ρ2 ρ1 −2e 4βη1ΛeβΛy2 ρ2 ρ1 + 2 p βΛe4βη1Λ y− p 4η1 +y 2 eβΛy2 ...

  3. [3]

    Vibrational heat capacity The heat capacity of this system is defined as a derivative of the mean energy with respect toβ, and is given by: Cvib (β) =−k Bβ2 ∂Uvib (β) ∂β , (54) By using the vibrational mean energy expression (53), we find: Cvib (β) = −2(βΛ) 3/2e2βΛ(4η1+y2)Γ1(y) ρ2 ρ1 +π p βΛe8βη1Λ erfi y p βΛ 2 ρ2 ρ1 +π p βΛerfi p βΛ p 4η1 +y 2 2 ρ2 ρ1 2 ...

  4. [4]

    − 1 2 eβΛρ2 2 ρ2 ρ1 + 1 4 r π βΛ

    Vibrational entropy The entropy of this system is given by the following definition: S(β) =k B lnZ vib −k Bβ ∂lnZ vib ∂β . (60) Using the explicit expression of the vibrational partition function given by Eq. (51), we find the following expression of the vibrational entropy: S(β) =k B ln " − 1 2 eβΛρ2 2 ρ2 ρ1 + 1 4 r π βΛ " −erfi p βΛρ2 ρ2 ρ1 +e −4βΛη 1 e...

    1966

  5. [5]

    Molecular integrals involving hulth ´en-type functions (n=l sto) in relativistic quantum chemistry.Chemical Physics Letters, 78(3):578–580, 1981

    Gulzari Malli. Molecular integrals involving hulth ´en-type functions (n=l sto) in relativistic quantum chemistry.Chemical Physics Letters, 78(3):578–580, 1981

    1981

  6. [6]

    AA Berezin. Positron trapping by the negatively charged f-aggregate colour centres and the binding energy of the fe +, me+ and re+ centres in alkali-halide crystals.Journal of Physics C: Solid State Physics, 12(10):L363, 1979

    1979

  7. [7]

    Two-and three-dimensional kronig-penney model withδ-function-potential wells of zero binding energy.Physical Review B, 33(4):2122, 1986

    Alexander A Berezin. Two-and three-dimensional kronig-penney model withδ-function-potential wells of zero binding energy.Physical Review B, 33(4):2122, 1986

    1986

  8. [8]

    Ulla Myhrman. Exact eigenvalues, eigenfunctions, matrix elements and phase-shifts for a particle of angular momentumℓ in a particular screened potential.Journal of Mathematical Physics, 21(7):1732–1739, 1980

    1980

Show all 56 references
  1. [9]

    Asao Arai. Exact solutions of multi-component nonlinear schr¨odinger and klein-gordon equations in two-dimensional space- time.Journal of Physics A: Mathematical and General, 34(20):4281, 2001

    2001

  2. [10]

    The t matrix for the hulth ´en potential.Journal of Mathematical Physics, 12(10):2076–2080, 1971

    OP Bahethi and MG Fuda. The t matrix for the hulth ´en potential.Journal of Mathematical Physics, 12(10):2076–2080, 1971

  3. [11]

    Extension of fuda’s off-shell analysis to screened coulomb potentials for arbitraryℓand limiting relations.Journal of mathematical physics, 24(12):2770–2775, 1983

    Ranabir Dutt and YP Varshni. Extension of fuda’s off-shell analysis to screened coulomb potentials for arbitraryℓand limiting relations.Journal of mathematical physics, 24(12):2770–2775, 1983

    1983

  4. [12]

    Fuda’s off-shell jost function for coulomb, hulth ´en, and eckart potentials and limiting relations.Physical Review A, 18(1):56, 1978

    H Van Haeringen. Fuda’s off-shell jost function for coulomb, hulth ´en, and eckart potentials and limiting relations.Physical Review A, 18(1):56, 1978

    1978

  5. [13]

    Hulth ´en transform functions for the excited states of two-electron atoms.The Journal of Chemical Physics, 55(7):3561–3563, 1971

    John Gruninger. Hulth ´en transform functions for the excited states of two-electron atoms.The Journal of Chemical Physics, 55(7):3561–3563, 1971

    1971

  6. [14]

    On the virtual state of the deuteron.Physical Review, 61(9-10):671, 1942

    Lamek Hulth ´en. On the virtual state of the deuteron.Physical Review, 61(9-10):671, 1942

    1942

  7. [15]

    Diffuse surface optical model for nucleon-nuclei scattering.Physical Review, 95(2):577, 1954

    Roger D Woods and David S Saxon. Diffuse surface optical model for nucleon-nuclei scattering.Physical Review, 95(2):577, 1954

    1954

  8. [16]

    On the interaction of elementary particles

    Hideki Yukawa. On the interaction of elementary particles. i.Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, 17:48–57, 1935

    1935

  9. [17]

    Die ultraroten rotationsspektren der halogenwasserstoffe.Zeitschrift f¨ ur Physik, 3:289–307, 1920

    Adolf Kratzer. Die ultraroten rotationsspektren der halogenwasserstoffe.Zeitschrift f¨ ur Physik, 3:289–307, 1920

    1920

  10. [18]

    A new approximation method in the problem of many electrons.The Journal of Chemical Physics, 3(1):61–61, 1935

    Hans Hellmann. A new approximation method in the problem of many electrons.The Journal of Chemical Physics, 3(1):61–61, 1935

    1935

  11. [19]

    Path integral treatment of a linear combination of deformed diatomic molecule potentials.International Journal of Theoretical Physics, 63(3):67, 2024

    Mohamed Am ´eziane Sadoun and Abdellah Touati. Path integral treatment of a linear combination of deformed diatomic molecule potentials.International Journal of Theoretical Physics, 63(3):67, 2024

    2024

  12. [20]

    Path integral solutions for klein-gordon particle with position-dependent mass in deformed hulth ´en potential

    MA Sadoun. Path integral solutions for klein-gordon particle with position-dependent mass in deformed hulth ´en potential. Europhysics Letters, 142(3):30001, 2023

    2023

  13. [21]

    Mohamed Ameziane Sadoun and Hamza Adnane. Relativistic bound states solutions with a linear combination of yukawa and deformed hulth ´en potentials by path integral approach.International Journal of Geometric Methods in Modern Physics, 21(1):2450025–36, 2024

    2024

  14. [22]

    Exact solution of schr ¨odinger equation with q-deformed quantum potentials using nikiforov—uvarov method.Chinese Physics B, 22(11):110301, 2013

    BJ Falaye, KJ Oyewumi, and M Abbas. Exact solution of schr ¨odinger equation with q-deformed quantum potentials using nikiforov—uvarov method.Chinese Physics B, 22(11):110301, 2013

    2013

  15. [23]

    Analytical solutions of the schr ¨odinger equation for the hulth ´en potential within susy quantum mechanics.International Journal of Modern Physics A, 30(32):1550193, 2015

    HI Ahmadov, Sh I Jafarzade, and MV Qocayeva. Analytical solutions of the schr ¨odinger equation for the hulth ´en potential within susy quantum mechanics.International Journal of Modern Physics A, 30(32):1550193, 2015

    2015

  16. [24]

    Solutions of n-dimensional schr ¨odinger equation with morse potential via laplace transforms

    S Miraboutalebi and L Rajaei. Solutions of n-dimensional schr ¨odinger equation with morse potential via laplace transforms. Journal of Mathematical Chemistry, 52:1119–1128, 2014

    2014

  17. [25]

    Any l-state solutions of the eckart potential via asymptotic iteration method.Central European Journal of Physics, 10:960–965, 2012

    Babatunde J Falaye. Any l-state solutions of the eckart potential via asymptotic iteration method.Central European Journal of Physics, 10:960–965, 2012

    2012

  18. [26]

    Mass spectrum of heavy quarkonium for screened kratzer potential (skp) using series expansion method.The European Physical Journal Plus, 136:1–11, 2021

    EE Ibekwe, US Okorie, JB Emah, EP Inyang, and SA Ekong. Mass spectrum of heavy quarkonium for screened kratzer potential (skp) using series expansion method.The European Physical Journal Plus, 136:1–11, 2021

    2021

  19. [27]

    Exact path integral treatment of a diatomic molecule potential.Journal of mathematical physics, 48(3), 2007

    F Benamira, L Guechi, S Mameri, and MA Sadoun. Exact path integral treatment of a diatomic molecule potential.Journal of mathematical physics, 48(3), 2007

    2007

  20. [28]

    Unified path integral treatment for generalized hulth´en and woods–saxon potentials.Annals of Physics, 322(9):2179–2194, 2007

    F Benamira, L Guechi, S Mameri, and MA Sadoun. Unified path integral treatment for generalized hulth´en and woods–saxon potentials.Annals of Physics, 322(9):2179–2194, 2007

    2007

  21. [29]

    Path integral solution for a deformed radial rosen–morse potential.Indian Journal of 15 Physics, 91:259–262, 2017

    A Kadja, F Benamira, and L Guechi. Path integral solution for a deformed radial rosen–morse potential.Indian Journal of 15 Physics, 91:259–262, 2017

    2017

  22. [30]

    Approximate solutions of schrodinger equation and expectation values of inversely quadratic hellmann-kratzer (iqhk) potential.The European Physical Journal Plus, 137(1):147, 2022

    CP Onyenegecha, CJ Okereke, IJ Njoku, CA Madu, RU Ndubuisi, and UK Nwajeri. Approximate solutions of schrodinger equation and expectation values of inversely quadratic hellmann-kratzer (iqhk) potential.The European Physical Journal Plus, 137(1):147, 2022

    2022

  23. [31]

    Arbitraryℓ-solutions of the schr ¨odinger equation interacting with hulth ´en- hellmann potential model.Revista mexicana de f´ ısica, 66(6):730–741, 2020

    ES William, EP Inyang, and EA Thompson. Arbitraryℓ-solutions of the schr ¨odinger equation interacting with hulth ´en- hellmann potential model.Revista mexicana de f´ ısica, 66(6):730–741, 2020

    2020

  24. [33]

    Approx- imate solutions of schr ¨odinger equation for the hua plus modified eckart potential with the centrifugal term.The European Physical Journal Plus, 135(7):1–10, 2020

    CP Onyenegecha, UM Ukewuihe, AI Opara, CB Agbakwuru, CJ Okereke, NR Ugochukwu, SA Okolie, and IJ Njoku. Approx- imate solutions of schr ¨odinger equation for the hua plus modified eckart potential with the centrifugal term.The European Physical Journal Plus, 135(7):1–10, 2020

    2020

  25. [34]

    Approxi- mate solutions of schrodinger equation in d dimensions with the modified mobius square plus hulthen potential.Mathematics and Computational Sciences, 2(2):1–15, 2021

    UM Ukewuihe, C Paul Onyenegecha, SC Udensi, CO Nwokocha, C Jennifer Okereke, IJ Njoku, and AC Iloanya. Approxi- mate solutions of schrodinger equation in d dimensions with the modified mobius square plus hulthen potential.Mathematics and Computational Sciences, 2(2):1–15, 2021

    2021

  26. [35]

    Solutions of schrodinger equation for the modified mobius square plus kratzer potential.The European Physical Journal Plus, 135(3):1–9, 2020

    CP Onyenegecha, CA Onate, OK Echendu, AA Ibe, and H Hassanabadi. Solutions of schrodinger equation for the modified mobius square plus kratzer potential.The European Physical Journal Plus, 135(3):1–9, 2020

    2020

  27. [36]

    CO Edet and PO Okoi. Any l-state solutions of the schr ¨odinger equation for q-deformed hulthen plus generalized inverse quadratic yukawa potential in arbitrary dimensions.Revista mexicana de f´ ısica, 65(4):333–344, 2019

    2019

  28. [37]

    Solutions of the schr ¨odinger equation with hulth ´en- screened kratzer potential: application to diatomic molecules.East European Journal of Physics, (2):12–22, 2022

    Etido Inyang, PC Iwuji, Joseph E Ntibi, ES William, and EA Ibanga. Solutions of the schr ¨odinger equation with hulth ´en- screened kratzer potential: application to diatomic molecules.East European Journal of Physics, (2):12–22, 2022

    2022

  29. [38]

    MC Onyeaju, AN Ikot, CA Onate, O Ebomwonyi, ME Udoh, and JOA Idiodi. Approximate bound-states solution of the dirac equation with some thermodynamic properties for the deformed hylleraas plus deformed woods-saxon potential.The European Physical Journal Plus, 132:1–18, 2017

    2017

  30. [39]

    Thermodynamics properties of diatomic molecules with general molecular potential.Pramana, 90:1–9, 2018

    Akpan N Ikot, EO Chukwuocha, MC Onyeaju, CA Onate, BI Ita, and ME Udoh. Thermodynamics properties of diatomic molecules with general molecular potential.Pramana, 90:1–9, 2018

    2018

  31. [40]

    A study of thermodynamic properties of quadratic exponential-type potential in d-dimensions.Revista Mexicana de fisica, 64(6):608–614, 2018

    Uduakobong Sunday Okorie, Akpan N Ikot, MC Onyeaju, and EO Chukwuocha. A study of thermodynamic properties of quadratic exponential-type potential in d-dimensions.Revista Mexicana de fisica, 64(6):608–614, 2018

    2018

  32. [41]

    Thermodynamic properties of the modified yukawa potential.Journal of the Korean physical society, 73:1211–1218, 2018

    US Okorie, EE Ibekwe, AN Ikot, MC Onyeaju, and EO Chukwuocha. Thermodynamic properties of the modified yukawa potential.Journal of the Korean physical society, 73:1211–1218, 2018

    2018

  33. [42]

    Hitler Louis, Benedict I Ita, and Nelson I Nzeata. Approximate solution of the schr ¨odinger equation with manning-rosen plus hellmann potential and its thermodynamic properties using the proper quantization rule.The European Physical Journal Plus, 134(7):315, 2019

    2019

  34. [43]

    Thermodynamic properties of improved deformed exponential-type potential (idep) for some diatomic molecules.Results in Physics, 17:103078, 2020

    Uduakobong S Okorie, Akpan N Ikot, Ephraim O Chukwuocha, and GJ Rampho. Thermodynamic properties of improved deformed exponential-type potential (idep) for some diatomic molecules.Results in Physics, 17:103078, 2020

    2020

  35. [44]

    Etido P Inyang, Funmilayo Ayedun, Efiong A Ibanga, Kolawole M Lawal, Ituen B Okon, Eddy S William, Omugbe Ek- wevugbe, Clement A Onate, Akaninyene D Antia, and Effiong O Obisung. Analytical solutions of the n-dimensional schr¨odinger equation with modified screened kratzer plu...

    2022

  36. [45]

    Courier Corporation, 2010

    Richard P Feynman, Albert R Hibbs, and Daniel F Styer.Quantum mechanics and path integrals. Courier Corporation, 2010

    2010

  37. [46]

    Complete non-relativistic bound state solutions of the tietz-wei potential via the path integral approach.The European Physical Journal Plus, 134:1–12, 2019

    A Khodja, A Kadja, F Benamira, and L Guechi. Complete non-relativistic bound state solutions of the tietz-wei potential via the path integral approach.The European Physical Journal Plus, 134:1–12, 2019

    2019

  38. [47]

    Unified treatment of the bound states of the schi ¨oberg and the eckart potentials using feynman path integral ap- proach.Chinese Physics B, 24(2):020302, 2015

    A Diaf. Unified treatment of the bound states of the schi ¨oberg and the eckart potentials using feynman path integral ap- proach.Chinese Physics B, 24(2):020302, 2015

    2015

  39. [48]

    Springer, 1998

    Christian Grosche, Frank Steiner, and Frank Steiner.Handbook of Feynman path integrals, volume 145. Springer, 1998

    1998

  40. [49]

    World scientific, 2009

    Hagen Kleinert.Path integrals in quantum mechanics, statistics, polymer physics, and financial markets. World scientific, 2009

    2009

  41. [50]

    Exactly solvable supersymmetric quantum mechanics.Journal of Mathematical Analysis and Applications, 158(1):63– 79, 1991

    Asao Arai. Exactly solvable supersymmetric quantum mechanics.Journal of Mathematical Analysis and Applications, 158(1):63– 79, 1991

    1991

  42. [51]

    Feynman integral treatment of the rosen–morse potential with a centrifugal term approximation.Canadian Journal of Physics, 91(12):1081–1085, 2013

    Ahmed Diaf and Mohamed Hachama. Feynman integral treatment of the rosen–morse potential with a centrifugal term approximation.Canadian Journal of Physics, 91(12):1081–1085, 2013

    2013

  43. [52]

    Summing the spectral representations of p¨oschl–teller and rosen–morse fixed-energy amplitudes

    H Kleinert and I Mustapic. Summing the spectral representations of p¨oschl–teller and rosen–morse fixed-energy amplitudes. Journal of mathematical physics, 33(2):643–662, 1992

    1992

  44. [53]

    I. S. Gradshtein and I. M. Ryzhik.Table of integrals, series, and products. Academic Press, New York, 2007

    2007

  45. [54]

    An accurate closed-form expression for the partition function of morse oscillators.Chemical physics letters, 439(1-3):209–212, 2007

    ML Strekalov. An accurate closed-form expression for the partition function of morse oscillators.Chemical physics letters, 439(1-3):209–212, 2007

    2007

  46. [55]

    Partition function of im- proved tietz oscillators.Chemical Physics Letters, 676:150–153, 2017

    Chun-Sheng Jia, Chao-Wen Wang, Lie-Hui Zhang, Xiao-Long Peng, Ran Zeng, and Xu-Tao You. Partition function of im- proved tietz oscillators.Chemical Physics Letters, 676:150–153, 2017

    2017

  47. [56]

    Mehmet Demirci and Ramazan Sever. Arbitrary l-state solutions of the klein–gordon equation with the eckart plus a class of yukawa potential and its non-relativistic thermal properties.The European Physical Journal Plus, 138(5):1–17, 2023

    2023

  48. [57]

    Energy spectra and the expectation values of diatomic molecules confined by the shifted deng-fan potential.The European Physical Journal Plus, 133(10):422, 2018

    OJ Oluwadare and KJ Oyewumi. Energy spectra and the expectation values of diatomic molecules confined by the shifted deng-fan potential.The European Physical Journal Plus, 133(10):422, 2018

    2018

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