Exact Solutions of the Razavy Cosine Type Potential

Dong, Shishan; Qian, Dong; Sun, Guo-Hua; Femmam, Smain; Dong, Shi‐Hai

doi:10.1155/2018/5824271

Cited by 18 publications

(22 citation statements)

References 16 publications

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“…The one-dimensional time-independent Schrödinger equation (SE) for a quantum particle moving in some double-well potential (DWP) is ubiquitous in physics and chemistry (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and refs. therein).…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand the exact analytic solutions [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [16] remain an important tool for understanding the reality. Such solutions are expressed via confluent Heun's function (CHF) [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [14] (implemented in Maple), spheroidal function (SF) [7], [8], [9] or derived via the functional Bethe ansatz [16]. Unfortunately these solutions can be obtained only for some particular DWPs and lack the universality of numerical methods.…”

Section: Introductionmentioning

confidence: 99%

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Analytic treatment of IR-spectroscopy data for double well potential

Sitnitsky

2019

Computational and Theoretical Chemistry

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A theoretical scheme for the analysis of experimental data on IR spectroscopy for a quantum particle in a double well potential (DWP) is suggested. The analysis is based on the trigonometric DWP for which the exact analytic solution of the Schrödinger equation is available. The corresponding energy levels along with their wave functions are expressed via special functions implemented in Mathematica (spheroidal function and its spectrum of eigenvalues). As a result trigonometric DWP makes the calculation of the energy levels an extremely easy procedure. It contains three parameters allowing one to model the most important characteristics of DWP (barrier height and the distance between the minima of the potential) along with the required asymmetry. Our approach provides an accurate calculation of the energy spectrum for hydrogen bonds in chromous acid (CrOOH) and potassium dihydrogen phosphate (KH 2 PO 4 ) along with their polarizability in agreement with available experimental data.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analytic treatment of IR-spectroscopy data for double well potential

Sitnitsky

2019

Computational and Theoretical Chemistry

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“…Polynomial solutions for the differential Equations (27) and (32) can be easily obtained using suitable values of the equation parameters to terminate the recurrence relations (28) and (33).…”

Section: Remarkmentioning

confidence: 99%

“…Remark 4. For the admissible values of the equation parameters, the recurrence relations (28) and (33) are generic formulas for the two-term recurrence relations for the series solutions of the differential Equations (27) and (32) respectively. That is to say, by assigning admissible values of the parameters, (27) and (32) will generate solvable differential equations.…”

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confidence: 99%

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

et al. 2020

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The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, n−1, and n−2 respectively. We show that for n≥3 the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of n=4. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.

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“…During last years a number of DWPs for SE with constant mass suitable for problems of chemistry (infinite at the boundaries of the interval for the spatial variable) was suggested. For them analytic solutions were obtained via the confluent Heun's function (implemented in Maple) [24], [25], [26], [27], [28], [29], [30], [31], [32], [33] and the spheroidal function (implemented in Mathematica) [25], [23]. Unfortunately they are not suitable for the problem of phosphine because in this case we deal with a position dependent mass [34], [35] (see below).…”

Section: Introductionmentioning

confidence: 99%

Exactly solvable double-well potential in Schrödinger equation for inversion mode of phosphine molecule

Sitnitsky

2021

Preprint

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The reduced mass of the effective quantum particle for the inversion mode ν 2 of phosphine molecule PH 3 is known to be a position dependent one. In the present article the inversion spectrum of PH 3 is considered with the help of the Schrödinger equation (SE) with position dependent mass and corresponding modified doublewell potential. The SE is shown to be exactly solvable. The results are used for the analysis of the pertinent experimental data available in literature. We are based on the reliable value ν3895.9 cm −1 ) obtained by Špirko et al. Also we use the value for the barrier height E b = 12300 cm −1 that seems to be commonly accepted at present and the hypothetical value for the energy splitting of the 11-th doublet of the 10ν 2 band s 10 = E 21 − E 20 ≈ 7.2 cm −1 suggested in the literature. Definite predictions are derived for the energy splitting of the 4-th doublet of the 3ν 2 band s 3 = E 7 − E 6 that is a test one for the observation in the experiment of Okuda et al. SE with position dependent mass provides selfconsistently the required values of {ν 2 ; E b ; s 10 } yielding s 3 = 6.21 • 10 −12 cm −1 .

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Exact Solutions of the Razavy Cosine Type Potential

Cited by 18 publications

References 16 publications

Analytic treatment of IR-spectroscopy data for double well potential

Analytic treatment of IR-spectroscopy data for double well potential

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

Exactly solvable double-well potential in Schrödinger equation for inversion mode of phosphine molecule

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