BIBECHANA 19(1-2) (2022) 165-169 165 165 Energy Eigenvalue and Thermodynamic Properties of q- deformed Hulthen Potential 1*Bhishma Karki, 2, 3, 4Saddam Husain Dhobi, 2, 4Jeevan Jyoti Nakarmi, 2,4Kishori Yadav 1*Department of Physics, Tri-Chandra Multiple Campus, Tribhuvan University, Kathmandu-44600, Nepal 2Department of Physics, Patan Multiple Campus, Tribhuvan University, Lalitpur-44700, Nepal 3Robotics Academy of Nepal, Lalitpur-44700, Nepal 4Innovative Ghar Nepal, Lalitpur-44700, Nepal *Corresponding Author Email: magnum.photon@gmail.com Article Information: Received: July 28, 2021 Accepted: February 26, 2022 Keywords: Gaussian Hypergeometric Function Thermodynamic Properties Partition Function Nuclear Physics ABSTRACT The objective of this work is to calculate the energy eigenvalue for q-deformed Hulthen potential using a Gaussian hypergeometric function with centrifugal approximation factor and related thermodynamical properties. For this, we develop a mathematical model using the Schrodinger wave equation to find the energy eigenvalue. In addition, the thermodynamic parameters were also calculated for q-deformed Hulthen potential using the partition function. The energy eigenvalue for quantum numbers n=1 to n=5 was observed for screening parameters 0.1, 0.5, and 1. In between, 0.1 to 1 and n=1 to n=2 the energy eigenvalue ranges from -1.048 to -208.572. The energy eigenvalue for considering potential shows that decrease with increasing the quantum number which means electron are loosely bounded nucleus in an atom. Also, the vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity, and vibrational entropy are obtained but due to complexity, the further development of the equation is the limitation of this work. DOI: https://doi.org/10.3126/bibechana.v19i1-2.46416 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.org/ licenses/by-nc/4.0/ BIBECHANA ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Department of Physics, Mahendra Morang A.M. Campus, TU, Biratnagar, Nepal mailto:magnum.photon@gmail.com https://doi.org/10.3126/bibechana.v19i1-2.46416 https://creativecommons.org/%20licenses/by-nc/4.0/ Karki et al. / BIBECHANA 19(1-2) (2022) 165-169 166 1. Introduction Deformation Hulthen Potential (HP) with q-deformation is well-defined as 𝑉(π‘Ÿ) = βˆ’ 𝑉0𝑒 βˆ’2π›Όπ‘Ÿ 1 βˆ’ π‘’βˆ’2π›Όπ‘Ÿ Here 𝑉0 = 𝑍𝑒 2𝛼 is known as coupling strength, and 𝛼 is the screening parameter [1-5]. HP has introduced to studies the detail of deuteron and has extensive applications in physics like Nuclear physics, particle physics, Atomic physics, Condensed Matter Physics, etc.). HP is one of the short-range potentials like Yukawa potential, which behaves as a Coulomb potential at a small distance and exponentially decreases with increasing the distance from the nucleus. For spin-zero particles, Klein-Gordon uses HP for their equation to solve as 𝑉(π‘Ÿ) = βˆ’π‘π›Ό exp(βˆ’ π‘Ÿ π‘Ž ) 1βˆ’exp(βˆ’ π‘Ÿ π‘Ž ) Or 𝑉(π‘Ÿ) = βˆ’π‘ 𝑒2𝛿 π‘’βˆ’π›Ώπ‘Ÿ 1βˆ’π‘’βˆ’π›Ώπ‘Ÿ Different methods like the asymptotic iteration method, supersymmetry method, shift N1 expression, factorization method, Nikiforov-Uvarove, etc., are used to calculate energy eigenvalue for further potential. The q-deformed HP and modified inversely quadratic Yukawa (qDHMIQY) potential can be represented as, 𝑉(π‘Ÿ) = βˆ’ 𝑉0𝑒 βˆ’2π›Όπ‘Ÿ 1 βˆ’ π‘žπ‘’βˆ’2π›Όπ‘Ÿ βˆ’ 𝑉1𝑒 βˆ’2π›Όπ‘Ÿ π‘Ÿ2 Here 𝑉0, 𝑉1 , π‘Žπ‘›π‘‘ π‘ž are coupling strength and deformation parameters, respectively. Such potential is used to describe different interactions such as nucleon-nucleon interactions, meson-meson interactions, the various field of nuclear physics, and quantum chemistry (Hulthen, 1942 and Tencan & Sever, 2009). 1.1. Differntent method to calcaulte energy eigenvalue 1.1.1. Asymptotic iteration method This method is used to solve homogenous linear second-order differential equations defined as, 𝑦" = πœ†0(π‘₯)𝑦′ + 𝑠0(π‘₯)𝑦 , here πœ†0(π‘₯) and 𝑠0(π‘₯) function in 𝐢∞(π‘Ž, 𝑏). On taking (n+1)th and (n+2)the derivative, we get, 𝑦(𝑛+1) = πœ†π‘›βˆ’1(π‘₯)𝑦′ + π‘ π‘›βˆ’1(π‘₯)𝑦 and 𝑦(𝑛+2) = πœ†π‘›(π‘₯)𝑦′ + 𝑠𝑛(π‘₯)𝑦 . Here πœ†π‘› = πœ†π‘›βˆ’1 β€² + π‘ π‘›βˆ’1 + πœ†0πœ†π‘›βˆ’1 and 𝑠𝑛 = π‘ π‘›βˆ’1 β€² + 𝑠0πœ†π‘›βˆ’1. On taking the ratio of (n+2)th and (n+1)th and applying the asymptotic condition 𝑠𝑛 πœ†π‘› = π‘ π‘›βˆ’1 πœ†π‘›βˆ’1 ≔ 𝛼, the condition gives the eigenvalue of the considered equation, and we obtained 𝑦(𝑛+1)(π‘₯) = 𝐢1πœ†π‘›βˆ’1 exp (∫ (𝛼 + πœ†0)𝑑𝑑 π‘₯ ) This equation yields the solution of the asymptotic consider equation as 𝑦(π‘₯) = [𝐢2 + 𝐢1∫ exp (∫ (πœ†0(𝜏) + 2𝛼(𝜏))π‘‘πœ 𝑑 )𝑑𝑑 π‘₯ ] 1.1.2. Nikiforov-Uvarov This method is used to solve the solution of an equation (e.g., Schrodinger) by transformation as πœ“β€²β€²(𝑠) + οΏ½ΜƒοΏ½(𝑠) 𝜎(𝑠) πœ“β€²(𝑠) + οΏ½ΜƒοΏ½(𝑠) 𝜎2(𝑠) πœ“(𝑠) = 0, Here οΏ½ΜƒοΏ½(𝑠) is polynomial of degree at most one, 𝜎(𝑠) and οΏ½ΜƒοΏ½(𝑠) are second-degree polynomials. On applying the condition of πœ† = πœ†π‘› one can obtain the energy eigenvalue. Here πœ† = π‘˜βˆ’ + πœ‹β€²(𝑠) and πœ†π‘› = βˆ’π‘›πœ β€²(𝑠) βˆ’ 𝑛(π‘›βˆ’1)πœŽβ€²β€²(𝑠) 2 , πœ‹(𝑠) is also polynomial with four values obtained by comparing the standard equation. 1.1.3. Supersymmetry method To calculate the energy eigenvalue, this method assumes a particular type of wave function πœ“π‘  = exp[βˆ«π‘Š0(π‘Ÿ)π‘‘π‘Ÿ + 𝛽(π‘Ÿ)]πœ™π‘ (π‘Ÿ), here π‘Š0(π‘Ÿ) is supersymmetry in quantum mechanics assumed as Witten superpotential, πœ™π‘ (π‘Ÿ) is a new function, and 𝛽(π‘Ÿ) is also a wave function that leads to correct asymptotic value. As we knew Schrodinger equation is used to obtain the energy Eigenvalue by replacing the general wave function by πœ“π‘  and solving similar to the conventional method, we get energy eigenvalue for any potential. 1.1.4. Factorization method In this method, the wave function for a particular coordinate system (example, Schrodinger equation spherical coordinate system) contains angular and coordinate system, and this method separates the angular and radial part by factorization method known as separation of variable in general like Ξ¨(π‘Ÿ, πœƒ, πœ™) = 𝑅(π‘Ÿ)Θ(πœƒ)Ξ¦(πœ™) after separating the angular and radial parts. The radial part is defined as 𝑅(π‘Ÿ) = π‘ˆ(π‘Ÿ)𝐿(π‘Ÿ), Here U(r) is related to potential and L(r) is associated with Laguerre differential equation. The angular part is defined as Θ(π‘₯) = π‘ˆ(π‘₯)𝑃(π‘₯), here π‘₯ = π‘π‘œπ‘ πœƒ, P(x) related to the Jacobi function. 2. Theoretical Formulation We have radial form part of SE is πœ“β€²β€²(π‘Ÿ) + 2π‘š ℏ2 [𝐸𝑛𝑙 βˆ’ 𝑉(π‘Ÿ) βˆ’ 𝑙(𝑙 + 1)ℏ 2π‘šπ‘Ÿ2 ]πœ“(π‘Ÿ) = 0 Using centrifugal approximation 1 π‘Ÿ2 = 4𝛼2π‘’βˆ’2π›Όπ‘Ÿ (1βˆ’π‘’βˆ’2π›Όπ‘Ÿ)2 , a radical form of SE become πœ“β€²β€²(π‘Ÿ) + 2π‘š ℏ2 [𝐸𝑛𝑙 + 𝑉0𝑒 βˆ’2π›Όπ‘Ÿ 1 βˆ’ π‘žπ‘’βˆ’2π›Όπ‘Ÿ βˆ’ 𝑙(𝑙 + 1)ℏ2 2π‘š 4𝛼2π‘’βˆ’2π›Όπ‘Ÿ (1 βˆ’ π‘’βˆ’2π›Όπ‘Ÿ)2 ] πœ“(π‘Ÿ) = 0 Let us consider 𝑍 = 1 1βˆ’π‘’βˆ’2π›Όπ‘Ÿ then the radial form of the equation become πœ“β€²β€²(π‘Ÿ) + 2π‘š ℏ2 [𝐸𝑛𝑙 + 𝑉0 (𝑍 βˆ’ 1) βˆ’ 𝑙(𝑙 + 1)ℏ2 2π‘š 𝑍24𝛼2 (𝑍 βˆ’ 1) 𝑍 ]πœ“(π‘Ÿ) = 0 (1) On solving equation (1), we get πœ“β€²β€²(𝑍) 𝑍(𝑍 βˆ’ 1) + πœ“β€²(𝑍)(2𝑍 βˆ’ 1) + [ π‘šπΈπ‘›π‘™ 2ℏ2𝛼2 1 𝑍(𝑍 βˆ’ 1) + π‘šπ‘‰0 2ℏ2𝛼2 1 𝑍 βˆ’ 𝑙(𝑙 + 1)]πœ“(𝑍) = 0 (2) Again supposed π‘šπΈπ‘›π‘™ 2ℏ2𝛼2 = βˆ’πœ–, 𝛿 = π‘šπ‘‰0 2ℏ2𝛼2 , 𝜁 = 𝑙(𝑙 + 1), therefore equation (2) becomes πœ“β€²β€²(𝑍) 𝑍(1 βˆ’ 𝑍) + πœ“β€²(𝑍)(1 βˆ’ 2𝑍) + [ βˆ’πœ– 𝑍(𝑍 βˆ’ 1) βˆ’ 𝛿 𝑍 + 𝜁]πœ“(𝑍) = 0 (3) Now asymptotic behavior of equation (3), at π‘Ÿ β†’ 0 (𝑍 β†’ 1) and π‘Ÿ β†’ ∞ (𝑍 β†’ 0), Let us introduce a new function 𝑓(𝑍) as πœ“(𝑍) = π‘πœ‡(1 βˆ’ 𝑍)πœ™π‘“(𝑍) (4) Combining equations (4) and (3), we get a new equation of the form 𝑍(1 βˆ’ 𝑍)𝑓′′(𝑍) + [1 + 2πœ‡ βˆ’ (2πœ‡ βˆ’ 2πœ™ βˆ’ 2)𝑍]𝑓′(𝑍) βˆ’ [(πœ‡ + πœ™)2 + (πœ‡ + πœ™) + 𝜁)]𝑓(𝑍) + [ βˆ’πœ– βˆ’ 𝛿 + ΞΌ2 𝑍(1 βˆ’ 𝑍) + πœ™2 βˆ’ πœ‡2 + 𝛿 (1 βˆ’ 𝑍) ]𝑓(𝑍) = 0 (5) Now equation (5) becomes the Gauss hypergeometric equation when the square bracket term equal to zero βˆ’πœ– βˆ’ 𝛿 + ΞΌ2 = 0, πœ™2 βˆ’ πœ‡2 + 𝛿 = 0 Karki et al. / BIBECHANA 19(1-2) (2022) 165-169 167 Therefore, equation (5) is written as 𝑍(1 βˆ’ 𝑍)𝑓′′(𝑍) + [1 + 2πœ‡ βˆ’ (2πœ‡ βˆ’ 2πœ™ βˆ’ 2)𝑍]𝑓′(𝑍) βˆ’ [(πœ‡ +πœ™ + 1 2 + √ 1 4 βˆ’ 𝜁) Γ— (πœ‡ + πœ™ + 1 2 βˆ’ √ 1 4 βˆ’ 𝜁)]𝑓(𝑍) = 0 (6) The Gaussian hypergeometric function (GHF), 2F1(π‘Ž, 𝑏; 𝑐; 𝑧) can be expressed as infinite series for |𝑧| < 1 as, 2F1(π‘Ž, 𝑏; 𝑐; 𝑧) = βˆ‘ (π‘Ž)𝑛(𝑏)𝑛 (𝑐)𝑛 ∞ 𝑛=0 𝑧𝑛 𝑛! = 1 + π‘Žπ‘ 𝑐 𝑧 1! + π‘Ž(π‘Ž+1)𝑏(𝑏+1) 𝑐(𝑐+1) 𝑧2 2! +⋯……. Here (π‘Ž)𝑛 ≔ Ξ“(π‘Ž+𝑛) Ξ“(π‘Ž) , (𝑏)𝑛 ≔ Ξ“(𝑏+𝑛) Ξ“(𝑏) , (𝑐)𝑛 ≔ Ξ“(𝑐+𝑛) Ξ“(𝑐) putting these in the above equation and simplifying we get, 𝑓(𝑍) = 𝐹1(π‘Ž1, 𝑏2; 𝑐1; 𝑍)2 = Ξ“(𝑐1) Ξ“(π‘Ž1)Ξ“(𝑏1) Here we consider π‘Ž1, , 𝑏1 and 𝑐1 are unknown parameters whose values are expressed as π‘Ž1 = (πœ‡ + πœ™ + 1 2 + √ 1 4 βˆ’ 𝜁) , 𝑏1 = (πœ‡ + πœ™ + 1 2 βˆ’ √ 1 4 βˆ’ 𝜁) , 𝑐1 = 1 + 2πœ‡ . Now substituting the value of 𝑓(𝑍) with these parameters in πœ“(𝑍) = π‘πœ‡(1 βˆ’ 𝑍)πœ™π‘“(𝑍) in this, we get πœ“(𝑍) = π‘πœ‡(1 βˆ’ 𝑍)πœ™ 𝐹1(πœ‡ + πœ™ + 1 2 + √ 1 4 βˆ’ 𝜁, πœ‡ + πœ™ + 1 2 βˆ’ √ 1 4 βˆ’ 𝜁; 1 + 2πœ‡; 𝑍)2 If π‘Ž1, 𝑏1, and 𝑐1 is equal to the negative of integer (𝑛) then hypergeometric function 𝑓(𝑍) will become a polynomial with 𝑛 = 0, 1, 2, 3, . . . , π‘›π‘šπ‘Žπ‘₯ integer, applying quantum condition we have, π‘Ž1 = βˆ’π‘› and 𝑏1 = βˆ’π‘›. Now using πœ™2βˆ’ πœ‡2 + 𝛿 = 0 to calculate the value of πœ™ and πœ‡ from the above suitable value we get πœ™ = ( πœ”βˆ’π‘› 2 + 𝛿 2(πœ”βˆ’π‘›) ) and πœ‡ = ( πœ”βˆ’π‘› 2 βˆ’ 𝛿 2(πœ”βˆ’π‘›) ) where πœ” = √ 1 4 βˆ’ 𝜁 βˆ’ 1 2 = πœ” Now to calculate the energy we have from βˆ’πœ– βˆ’ 𝛿 + ΞΌ2 = 0 and πœ™2βˆ’ πœ‡2 + 𝛿 = 0 is πœ– = πœ™2. Since we have π‘šπΈπ‘›π‘™ 2ℏ2𝛼2 = βˆ’πœ– and πœ™ = ( πœ”βˆ’π‘› 2 + 𝛿 2(πœ”βˆ’π‘›) ), therefore we have, 𝐸𝑛𝑙 = βˆ’ 2ℏ2𝛼2 π‘š ( πœ” βˆ’ 𝑛 2 + 𝛿 2(πœ” βˆ’ 𝑛) ) 2 (7) Substituting the value of 𝛿,πœ”, 𝜁 in equation (7) we get, 𝐸𝑛𝑙 = βˆ’ 2ℏ2𝛼2 π‘š ( √ 1 4 βˆ’ 𝑙(𝑙 + 1) βˆ’ 1 2 βˆ’ 𝑛 2 + π‘šπ‘‰0 2ℏ2𝛼2 2(√ 1 4 βˆ’ 𝑙(𝑙 + 1) βˆ’ 1 2 βˆ’ 𝑛)) 2 𝐸𝑛𝑙 = βˆ’ ℏ2𝛼2 2π‘š ( (√ 1 4 βˆ’ 𝑙(𝑙 + 1) βˆ’ 1 2 βˆ’ 𝑛) + π‘šπ‘‰0 2ℏ2𝛼2 (√ 1 4βˆ’ 𝑙 (𝑙 + 1) βˆ’ 1 2 βˆ’ 𝑛) ) 2 (8) This equation is dependent upon the quantum number and screening parameters. It is a non-relativistic energy spectrum that was calculated using HP, which is short-range potential. Table1: Energy eigenvalue of q-deformed HP Quantum Number Energy eigenvalue (𝑬𝒏𝒍) 𝑛 𝑙 𝑉0 = 1, 𝛼 = 1 𝑉0 = 1, 𝛼 = 0.5 𝑉0 = 0.5, 𝛼 = 1 𝑉0 = 0.1, 𝛼 = 1 𝑉0 = 0.1, 𝛼 = 0.1 1 0 - 2.20977 531 -2.2942 - 1.511428 202 - 1.047957 128 -0.37392 1 -3.909 -2.0596 - 3.342473 165 - 2.921198 927 -0.21685 2 0 - 3.90895 5159 - 2.059648 762 - 3.342473 165 - 2.921198 927 -0.21685 1 - 6.73920 8248 - 2.599411 115 - 6.206286 437 - 5.795751 457 -0.20085 2 - 15.4626 1915 - 4.680554 734 - 14.94963 916 - 14.54548 557 -0.26176 3 0 - 6.73920 8248 - 2.599411 115 - 6.206286 437 - 5.795751 457 -0.20085 1 - 10.5964 3495 - 3.497067 921 - 10.07684 311 - 9.670573 724 -0.22183 2 - 21.3326 1546 - 6.129289 967 - 20.82338 824 - 20.42043 553 -0.31551 3 - 44.9510 0139 - 12.00908 368 - 44.44673 472 - 44.04536 939 -0.54514 4 0 - 15.4626 1915 - 3.497067 921 - 10.07684 311 - 9.670573 724 -0.22183 1 - 21.3326 1546 - 4.680554 734 - 14.94963 916 - 14.54548 557 -0.26176 2 - 36.0774 4104 - 7.835599 168 - 27.69761 069 - 27.29540 443 -0.38115 3 - 65.6991 7677 - 14.47365 847 - 54.32147 377 - 53.92035 895 -0.64285 4 - 119.197 4276 - 26.83975 676 - 103.8209 174 - 103.4203 367 -1.13562 5 0 - 15.4626 1915 - 4.680554 734 - 14.94963 916 - 14.54548 557 -0.26176 Karki et al. / BIBECHANA 19(1-2) (2022) 165-169 168 1 - 21.3326 1546 - 6.129289 967 - 20.82338 824 - 20.42043 553 -0.31551 2 - 36.0774 4104 - 9.796092 271 - 35.57209 463 - 35.17038 379 -0.45783 3 - 65.6991 7677 - 17.18928 52 - 65.19627 857 - 64.79535 114 -0.75082 4 - 119.197 4276 - 30.55728 87 - 118.6958 413 - 118.2953 337 -1.28407 5 - 208.571 5169 - 52.89739 577 - 208.0706 136 - 207.6703 245 -2.17691 2.1. Thermodynamic Properties HP is also used to explain the electronic properties of some alkali halides, study the bound state and scattering properties, etc. Moreover, statistical physics and quantum statistical mechanics help predict, interpret, analyze, etc., different thermodynamic properties such as vibrational and rotational energy levels of various molecules [6]. Thermodynamics properties of HP are also studied for statistical quantum chromodynamics (QCD), nuclear matter, a color deconfined partonic phase, and the quark-gluon plasma (QGP) at sufficiently high temperature/density [7]. Now to calculate the thermodynamic properties for HP, now we develop (8) equation to study the thermodynamic properties; we use the vibrational partition function for this we summarized 𝐸𝑛𝑙 as, 𝐸𝑛𝑙 = βˆ’ ℏ2𝛼2 2π‘š ( 𝑃 (𝜎 + 𝑛) βˆ’ (𝜎 + 𝑛)) 2 (9) Here, βˆ’πœŽ = βˆ’ 1 2 + √ 1 4 βˆ’ 𝑙(𝑙 + 1), 𝑃 = βˆ’ π‘šπ‘‰0 2ℏ2𝛼2 , since vibrational partition function is defined as 𝑍𝑣𝑖𝑏(𝛽) = βˆ‘ π‘’βˆ’π›½πΈπ‘›π‘™ πœ‚ 𝑛=0 , 𝛽 = 1 π‘˜π‘‡ Therefore, substituting the value of 𝐸𝑛𝑙 in 𝑍𝑣𝑖𝑏 HP with partition function modified as, 𝑍𝑣𝑖𝑏(𝛽) = βˆ‘ 𝑒 βˆ’π›½[βˆ’ ℏ2𝛼2 2π‘š ( 𝑃 (𝜎+𝑛) βˆ’(𝜎+𝑛)) 2 ] πœ‚ 𝑛=0 (10) Replacing the sum by an integral in the classical limit of equation (10), we have 𝑍𝑣𝑖𝑏(𝛽) = ∫ 𝑒 βˆ’π›½[βˆ’ ℏ2𝛼2 2π‘š ( 𝑃 (𝜎+𝑛) βˆ’(𝜎+𝑛)) 2 ]πœ‚ 0 𝑑𝑛 (11) Supposing 𝜎 + 𝑛 = 𝜌, equation (becomescome [8] 𝑍𝑣𝑖𝑏(𝛽) = ∫ 𝑒 ( 𝑏𝛽 𝜌2 +π›½π‘ŽπœŒ2+𝑐𝛽) π‘‘πœŒ πœ‚+𝜎 𝜎 (12) Here π‘Ž = ℏ2𝛼2 2π‘š , 𝑏 = ℏ2𝛼2𝑃2 2π‘š , 𝑐 = βˆ’ ℏ2𝛼2𝑃 π‘š , now equation (12) becomes 𝑍𝑣𝑖𝑏(𝛽) = π‘’π‘π›½βˆ’2βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½βˆšπœ‹ (βˆ’ erf[Ξ“1 βˆ’ Ξ“2] + 𝑒 4βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½(erf[Ξ“1 + Ξ“2] βˆ’ erf[Ξ©1 + Ξ“2 + Ξ©2]) βˆ’ Ξ) 4βˆšβˆ’π‘Žπ›½ (13) Where Ξ“1 = βˆšβˆ’π‘π›½ 𝜎 , Ξ“2 = βˆšβˆ’π‘Žπ›½πœŽ,Ξ©1 = βˆšβˆ’π‘Žπ›½πœ‚ , Ξ©2 = βˆšβˆ’π‘π›½ πœ‚+𝜎 π‘Žπ‘›π‘‘ Ξ = erf [Ξ©1 + Ξ“2 βˆ’ Ξ©2] and the error function well-defined as erf(𝑍) = 2 βˆšπœ‹ ∫ 𝑒𝑑 2 𝑑𝑑 𝑧 0 Equation (13) plays an integral equation used to describe the thermodynamic properties of q-deformed HP. This equation described vibrational mean and free energy, vibrational entropy, and specific heat capacity. The Integrals error function erf (𝑍) has essential applications in atomic physics, astrophysics, statistical analysis, etc. [9] 3. Results and discussion 2.2. Energy eigenvalue of q-deformed HP The energy eigenvalue is represented in atom unit with π‘š = ℏ = 1, at 𝑉0 = 0.5, 1 and 𝛼 = 0.5, 1. The energy eigenvalue is developed of q-deformed HP. It is beneficial to calculate the thermodynamic properties of physics fields like quantum chromodynamics, meson-meson interaction, nuclear matter color chromodynamics, etc. The nature of energy eigenvalue with quantum number is shown in figure 1, which shows that with an increased quantum number the electron bounded energy decreases which means the electron goes loosely bounded with the nucleus in an atom. Figure 1: Energy eigenvalue of q-deformed Hulthen Potential 2.3. Thermodynamic properties of q-deformed HP 2.3.1. Vibrational mean energy of q-deformed HP (VMEqHP) The mean vibrational energy for the HP model is obtained as, π‘ˆ(𝛽) = πœ• πœ•π›½ (𝑙𝑛𝑍𝑣𝑖𝑏(𝛽)) π‘ˆ(𝛽) = πœ• πœ•π›½ 𝑙𝑛 ( π‘’π‘π›½βˆ’2βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½βˆšπœ‹ (βˆ’erf[Ξ“1βˆ’Ξ“2]+𝑒 4βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½(erf[Ξ“1+Ξ“2]βˆ’erf[Ξ©1+Ξ“2+Ξ©2])βˆ’Ξž) 4βˆšβˆ’π‘Žπ›½ ) By solving this equation, we get, π‘ˆ(𝛽) = 𝑑(ln{ (π‘’π‘Ÿπ‘“[𝛀2βˆ’π›€2])}) 𝑑𝛽 + 𝑑 ln{erf[Ξ“1+Ξ“2]} 𝑑𝛽 βˆ’ 𝑑 ln{erf[Ξ©1+Ξ“2+Ξ©2]} 𝑑𝛽 (14) 2.3.2. Vibrational mean free energy of q-deformed HP (VMEFqHP) The vibrational mean free energy is obtained as 𝐹(𝛽) = βˆ’π‘˜π‘‡π‘™π‘›(𝑍𝑣𝑖𝑏(𝛽)) 𝐹(𝛽) = βˆ’π‘˜π‘‡π‘™π‘›( π‘’π‘π›½βˆ’2βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½βˆšπœ‹ (βˆ’ erf[Ξ“1 βˆ’ Ξ“2] + 𝑒 4βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½(erf[Ξ“1 + Ξ“2] βˆ’ erf[Ξ©1 + Ξ“2 + Ξ©2]) βˆ’ Ξ) 4βˆšβˆ’π‘Žπ›½ ) 𝐹(𝛽) = βˆ’π‘˜π‘‡[ln{ (π‘’π‘Ÿπ‘“[𝛀2 βˆ’ 𝛀2])} + ln{erf[Ξ“1 + Ξ“2]} βˆ’ ln{erf[Ξ©1 + Ξ“2 + Ξ©2]} + ln{Ξ}] (15) Karki et al. / BIBECHANA 19(1-2) (2022) 165-169 169 2.3.3. Vibrational specific heat capacity of q-deformed HP (VSHCqHP) The vibrational specific heat capacity (Cs) is given as 𝐢𝑠(𝛽) = π‘˜π›½ 2 πœ•2 πœ•π›½2 (𝑙𝑛(𝑍𝑣𝑖𝑏(𝛽))) 𝐢𝑠(𝛽) = π‘˜π›½2 πœ•2 πœ•π›½2 𝑙𝑛( π‘’π‘π›½βˆ’2βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½βˆšπœ‹ (βˆ’erf[Ξ“1 βˆ’ Ξ“2] + 𝑒 4βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½(erf[Ξ“1 + Ξ“2] βˆ’ erf[Ξ©1 + Ξ“2 +Ξ©2]) βˆ’ Ξ) 4βˆšβˆ’π‘Žπ›½ ) 𝐢𝑠(𝛽) = π‘˜π›½ 2 πœ•2 πœ•π›½2 ([ln{ (π‘’π‘Ÿπ‘“[𝛀2 βˆ’ 𝛀2])} + ln{erf[Ξ“1 + Ξ“2]} βˆ’ ln{erf[Ξ©1 + Ξ“2 + Ξ©2]} + ln{Ξ}]) (16) (16) 2.3.4. Vibrational entropy of q-deformed HP (VEqHP) The vibrational entropy is obtained as 𝑆(𝛽) = π‘˜π‘™π‘›(𝑍𝑣𝑖𝑏(𝛽)) + π‘˜π‘‡ πœ• πœ•π›½ (𝑙𝑛𝑍𝑣𝑖𝑏(𝛽)) 𝑆(𝛽) = π‘˜π‘™π‘›( π‘’π‘π›½βˆ’2βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½βˆšπœ‹ (βˆ’ erf[Ξ“1 βˆ’ Ξ“2] + 𝑒 4βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½(erf[Ξ“1 + Ξ“2] βˆ’ erf[Ξ©1 + Ξ“2 +Ξ©2]) βˆ’ Ξ) 4βˆšβˆ’π‘Žπ›½ ) + π‘˜π‘‡ πœ• πœ•π›½ ( π‘’π‘π›½βˆ’2βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½βˆšπœ‹ (βˆ’ erf[Ξ“1 βˆ’ Ξ“2] + 𝑒 4βˆšβˆ’π‘Žπ›½βˆšβˆ’π‘π›½(erf[Ξ“1 + Ξ“2] βˆ’ erf[Ξ©1 + Ξ“2 + Ξ©2]) βˆ’ Ξ) 4βˆšβˆ’π‘Žπ›½ ) 𝑆(𝛽) = π‘˜[ln{ (π‘’π‘Ÿπ‘“[𝛀2 βˆ’ 𝛀2])} + ln{erf[Ξ“1 + Ξ“2]} βˆ’ ln{erf[Ξ©1 + Ξ“2 + Ξ©2]} + ln{Ξ}] + π‘˜π‘‡ 𝑑(ln{ (π‘’π‘Ÿπ‘“[𝛀2βˆ’π›€2])}) 𝑑𝛽 + 𝑑 ln{erf[Ξ“1+Ξ“2]} 𝑑𝛽 βˆ’ 𝑑 ln{erf[Ξ©1+Ξ“2+Ξ©2]} 𝑑𝛽 (17) Thermodynamical properties have error functions with different parameters and in literature [9] authors discuss its multiuse use. Therefore, studying the thermodynamic properties for consideration in this paper has equal importance as discussed in the literature [8, 9]. The thermodynamic properties of the electron in considering potential is obtained in equation (14), (15), (16), and (17) showing that the thermodynamic properties depend upon quantum number also. Therefore, the thermodynamic properties for considering potential depend upon quantum number as well as temperature. 3. Conclusion The development of energy eigenvalue for q-deformation HP potential develops in equation (8) with the help of this energy vibrational thermodynamic properties are calculated. The development of a mathematical model for thermodynamic properties for the electron in q-deformation potential is completely new. In addition, the thermodynamic properties also depend upon the quantum number which means vary with the quantum number. For example, the thermodynamic properties of the electron in the n=1 orbit are different from the thermodynamic properties of the electron in the n=2 orbit and so on. This is because the thermodynamic properties depend upon the energy eigenvalue of the electron and the energy eigenvalue of the electron depends upon the quantum number. The developed mathematical model is based on a Gaussian hypergeometric function with a partition function. To study the more detailed thermodynamic properties of an electron in considering potential one may extend the development theory because due to the complexity of mathematical derivation authors consider the limitations of their work. Acknowledgment The authors would like to thank the Department of Physics, Tri-Chandra Multiple Campus, Tribhuvan University and Patan Multiple Campus, Tribhuvan University, Nepal; Innovative Ghar Nepal, and National Research Council Nepal for supporting this work. References [1] C.O. Edet and P.O. 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Chukwuocha, Thermodynamic Properties of Improved Deformed Exponentialtype Potential (Idea) For Some Diatomic Molecules, accessed at 1st June 2021 https://arxiv.org/ftp/arxiv/papers/2001/2001.04799.pdf [7] M. A. Shady, T. A. A. Karim and SY E. Alarab, Masses and thermodynamic properties of heavy mesons in the non- relativistic quark model using the Nikiforov–Uvarov method, Journal of the Egyptian Mathematical Society, 27(14)(2019) 1-15, https://doi.org/10.1186/s42787-019-0014-0 [8] U. S. Okorie, E. E. Ibekwe, A. N. Ikot, M. C. Onyeaju, and E. O. Chukwuocha, Thermodynamic Properties of the Modified Yukawa Potential, Journal of the Korean Physical Society, 73(9)(2018)1211-1218. [9] E. W. Ng and M. Geller, A Table of Integrals of the Error Functions, Journal of Research of the National Bureau of Standards B. 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