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PERIÓDICO TCHÊ QUÍMICA

ANALYTICAL SOLUTION OF SCHRÖDINGER EQUATION IN BISPHERICAL COORDINATE FOR MOBIUS SQUARE PLUS MODIFIED YUKAWA POTENTIAL USING NIKIFOROV UVAROV FUNCTIONAL ANALYSIS (NUFA) METHOD WITH ITS THERMODYNAMICS PROPERTIES FOR DIATOMIC MOLECULES

Briant Sabathino Harya Wibawa, Suparmi Suparmi, C. Cari

Published Mar 20, 2021• 0 citations
Publisher

Abstract

Background: The analytical solution of the Schrödinger equation in bispherical coordinates has attracted a great deal of interest for theoretical physics researchers in the branch of quantum physics. The energy and wave function are solutions of the Schrödinger equation which are very important because it contains all necessary information regarding the behavior of quantum systems. Aim: This study aimed to obtain energy, radial wave functions and thermodynamic properties for diatomic molecules from the radial part of the Schrödinger equation in bispherical coordinates for the modified Mobius square plus Yukawa potential using the Nikiforov Uvarov Functional Analysis (NUFA) method. Methods: The variable separation method was applied to reduce the Schrodinger equation in bispherical coordinates to the radial part and angular part Schrodinger equation. The Schrodinger equation of the radial part in bispherical coordinates was solved using the Nikiforov Uvarov Functional Analysis (NUFA) method to obtain the energy equation and radial wave function. Furthermore, the vibrational partition function 𝑍 was obtained from the energy equation. The vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆 were obtained from the vibrational partition function 𝑍. Results and Discussion: The results showed that the increase of parameters of 𝑛 and 𝛼 caused the decrease of energy, but the increase of parameters of 𝐿 and 𝑚0 caused the increase of energy. The radial quantum number 𝑛 and the potential range 𝛼 had the most effect to the wave functions. The parameters 𝑛𝑚𝑎𝑥, 𝑇, and 𝛼 had effect to the vibrational partition function 𝑍, vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆. Conclusions: From the results of this study, it can be concluded that the energy, radial wave function, and thermodynamic properties for diatomic molecules have been obtained using the Nikiforov Uvarov Functional Analysis (NUFA) method.

DOI
10.52571/ptq.v18.n37.2021.08_wibawa_pgs_111_134.pdf

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Background: The analytical solution of the Schrödinger equation in bispherical coordinates has attracted a great deal of interest for theoretical physics researchers in the branch of quantum physics. The energy and wave function are solutions of the Schrödinger equation which are very important because it contains all necessary information regarding the behavior of quantum systems. Aim: This study aimed to obtain energy, radial wave functions and thermodynamic properties for diatomic molecules from the radial part of the Schrödinger equation in bispherical coordinates for the modified Mobius square plus Yukawa potential using the Nikiforov Uvarov Functional Analysis (NUFA) method. Methods: The variable separation method was applied to reduce the Schrodinger equation in bispherical coordinates to the radial part and angular part Schrodinger equation. The Schrodinger equation of the radial part in bispherical coordinates was solved using the Nikiforov Uvarov Functional Analysis (NUFA) method to obtain the energy equation and radial wave function. Furthermore, the vibrational partition function 𝑍 was obtained from the energy equation. The vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆 were obtained from the vibrational partition function 𝑍. Results and Discussion: The results showed that the increase of parameters of 𝑛 and 𝛼 caused the decrease of energy, but the increase of parameters of 𝐿 and 𝑚0 caused the increase of energy. The radial quantum number 𝑛 and the potential range 𝛼 had the most effect to the wave functions. The parameters 𝑛𝑚𝑎𝑥, 𝑇, and 𝛼 had effect to the vibrational partition function 𝑍, vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆. Conclusions: From the results of this study, it can be concluded that the energy, radial wave function, and thermodynamic properties for diatomic molecules have been obtained using the Nikiforov Uvarov Functional Analysis (NUFA) method.
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In this work, we proposed a screened modified Krazer potential and use the newly proposed Nikiforov-Uvarov-Functional Analysis (NUFA) method to obtain the energy spectrum and the corresponding wave function. With the obtained energy spectrum, we studied the numerical results for some selected diatomic molecules and our results are in good agreement with other analytical method. We also evaluated the vibrational partition function for , and diatomic molecules via the Euler–Maclaurin approach and other thermodynamic functions such as free energy, entropy, mean energy and specific heat Capacity. The expectations values of and are also calculated numerically for different diatomic molecules using the normalized wave function for the two low lying states corresponding to the ground and first excited states. Our numerical results for the selected diatomic molecules validate the Heisenberg uncertainty relation, .
In this paper, we solved the Schrodinger equation with Hellmann-modified Kratzer potential using Nikiforov-Uvarov-Functional Analysis (NUFA) method. The obtained energy is used to study the numerical results of the ro-vibrational energy spectra for some selected diatomic molecules and their thermodynamic properties. In addition, we also investigated the Fisher information for three diatomic molecules and they all satisfied the Stam-Cramer-Rao inequalities uncertainty relations. Special cases of the potential are discussed and we compute the numerical eigenvalue of the modified Kratzer, Kratzer-Feus and Hellmann potentials for comparison with other analytical methods. The results of the present study agree with the results obtained with other known methods.
The Schrödinger equation was solved for the Hulthén plus screened Kratzer Potential (HSKP) via the Nikiforov–Uvarov – functional analysis (NUFA) method. The bound-state energy and wave function for the HSKP were obtained in a closed form by applying the Greene–Aldrich approximation scheme to the inverse-square term. Using the resulting energy equation, we computed the energy spectra for 12 diatomic molecules (CuLi, TiH, VH, TiC, HCl, LiH, H2, ScH, CO, I2, N2, and NO) for various quantum states. To show the accuracy of our procedure, four special cases of the collective potential were obtained, and the results are in excellent agreement with the existing literature.
In this work, we study an approximate bound state solution of a 2D Schrodinger equation under the influence of external magnetic and Aharonov–Bohm (AB) fields with Hellmann plus screened Kratzer potential (HPSKP) using the Nikiforov–Uvarov-functional- analysis method (NUFA). We obtained the normalised wave function expressed in terms of the Jacobi polynomial. The energy spectra were also obtained in closed form and used to study the energy levels of hydrogen and chromium hydride molecules. The partition function was evaluated using Poisson’s summation approach while other thermomagnetic properties such as free energy, mean energy, entropy, heat capacity, magnetisation, magnetic susceptibility, persistent current at zero and finite temperatures were evaluated analytically.
The aims of the research are to determine energy spectra and radial wave function of Quantum system in D-Dimensions for Eckart plus Manning Rosen potential with Nikiforov-Uvarov method. NikiforovUvarov method is method for solving a second-order of differential equations by transforming the equation into the hypergeometric equation by the method of separation of intermediate variables by appropriate variable substitution. The analytical results is energy spectra and radial wave function visualization by Microsoft Excel 2010 and Mathematical Laboratory 2013 Programing. The visualization result can be showed that the orbital quantum number increase and the dimension increase causing the bounding energy decrease in the –n spectra energy. The probability visualization for ground state condition result can be showed that the orbital increase, so probability decrease, while the dimension increase for ground state condition is decreasing probability.