Partition function of improved Tietz oscillators

Jia, Chun‐Sheng; Wang, Chaowen; Zhang, Liehui; Peng, Xiaolong; Zeng, Ran; You, Xu-Tao

doi:10.1016/j.cplett.2017.03.068

Cited by 153 publications

(64 citation statements)

References 26 publications

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“…After identifying the corresponding parameters, Equation matches with results given in Tang et al On the other hand, for s ‐states (ℓ = 0) we have

\begin{matrix} lefttrue \\ E_{n}^{ITP} = D_{e} - \frac{{normalℏ}^{2} α^{2}}{2 m} \\ \times {([], \frac{2 n + 1 + frakturh ((), q)}{4} - \frac{\frac{2 {mD}_{e}}{ℏ^{2} α^{2} Q^{2}} ((), e^{2 α r_{e}} - Q^{2})}{2 n + 1 + frakturh ((), q)})}^{2} . \end{matrix}

with

frakturh ((), q) = \sqrt{1 + \frac{8 {mD}_{e}}{α^{2} ℏ^{2} Q^{2}} {(e^{α r_{e}} + Q)}^{2}},

in accordance with Jia et al…”

Section: Some Useful Applicationssupporting

confidence: 89%

“…Before considering the Tiez model, it is important to mention that the improved version of this potential, given in terms of dissociation energy and bond length for diatomic molecules, has been successfully used by professor Jia's research group to obtain important predictions on thermodynamical properties of pure substances . So, returning to the useful applications of this work, in this case we select the parameters

A ((), q) = D_{e}, B ((), q) = D_{e} ((), \frac{q - 2}{q}), C ((), q) = \frac{D_{e}}{q^{2}}, k = \frac{1}{α}

for having

V ((), r) = \frac{D_{e} {qe}^{- α false(r - r_{e false)}}}{1 - {qe}^{- α false(r - r_{e false)}}} + \frac{D_{e} ((), q - 2) e^{- α false(r - r_{e false)}}}{{((), 1 - {qe}^{- α false(r - r_{e false)}})}^{2}} + \frac{D_{e} e^{- 2 α false(r - r_{e false)}}}{{((), 1 - {qe}^{- α false(r - r_{e false)}})}^{2}}

such that, after use

Q = - {qe}^{α r_{e}}

the improved q ‐deformed Tietz potential is obtained,

V^{ITP} ((), r) = D_{e} + V ((), r) = D_{e} {(1 - \frac{e^{- α r_{e}} + Q}{e^{- αr} + Q})}^{2} .

…”

Section: Some Useful Applicationsmentioning

confidence: 99%

“…To deal with this potential, we propose the choice

A ((), q) = 4 U_{2} - 2 U_{1}, B ((), q) = 0, C ((), q) = 4 U_{2}, q = \overset{q}{true} e^{- 2 α r_{e}}

and

k = \frac{1}{2 α}

in the Equation such that

V ((), r) = \frac{((), 4 U_{2} - 2 U_{1}) \overset{q}{true} e^{- α r_{e}} e^{- α ((), r - r_{e})}}{1 - \overset{q}{true} e^{- α r_{e}} e^{- α ((), r - r_{e})}} + \frac{4 U_{2} \overset{q}{true} e^{- α r_{e}} e^{- 2 α ((), r - r_{e})}}{{((), 1 - \overset{q}{true} e^{- α r_{e}} e^{- α ((), r - r_{e})})}^{2}},

leading to

\begin{matrix} lefttrue \\ V^{italicmMR} (r) = U_{0} - U_{1} + U_{2} + V (r) \\ = U_{0} - U_{1} \cot h_{\overset{q}{true}} (italicαr) + U_{2} \cot h_{\overset{q}{true}}^{2} (italicαr) \end{matrix}

which is identified as the q ‐deformed modified Manning‐Rosen potential ( mMR ) as given in Jia et al where cot h q ( αr ), cot h q ( αr ) are the q ‐deformed tri...…”

Section: Some Useful Applicationsmentioning

confidence: 99%

See 2 more Smart Citations

Approximate ℓ‐bound state solutions of q‐deformed exponential‐type potentials

Peña

García-Ravelo

Ovando

et al. 2020

Int J of Quantum Chemistry

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In this work, the exactly solvable Schrödinger equation for s-states of a class of multiparameter exponential-type potential (E-tP) is used to obtain the approximate ℓ 6 ¼ 0 bound state solutions for the corresponding q-deformed radial potentials. To deal with the effective potential, the Pekeris approximation for the centrifugal term is applied. The proposal has the advantage that, depending on the choice of the parameters involved in the E-tP, several q-deformed exponential potentials are obtained here as particular cases. That is, our proposition indicates that it is not necessary to use specialized methods to solve the Schrödinger equation for specific q-deformed exponential potentials. At this regard, in order to show the usefulness of the proposed method, we consider the ℓ 6 ¼ 0 bound state solutions of the q-deformed Tietz, Hulthén, Manning-Rosen, Shioberg, quadratic exponential, Wei and Hua among other potential models. Furthermore, with the example of the Hua potential we are contributing to solve the recent controversy on the energy spectra of this q-deformed model. Moreover, in some applications considered in this work the results can be used to correct similar findings already published. Besides, the proposal is useful when considering new q-deformed potentials with hypergeometric wavefunctions to be used in quantum chemical applications of diatomic molecules. K E Y W O R D Sℓ 6 ¼ 0 bound-state solutions, exponential-type models, Pekeris approximation, q-deformed potentials, Schrödinger equation

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“…After identifying the corresponding parameters, Equation matches with results given in Tang et al On the other hand, for s ‐states (ℓ = 0) we have

\begin{matrix} lefttrue \\ E_{n}^{ITP} = D_{e} - \frac{{normalℏ}^{2} α^{2}}{2 m} \\ \times {([], \frac{2 n + 1 + frakturh ((), q)}{4} - \frac{\frac{2 {mD}_{e}}{ℏ^{2} α^{2} Q^{2}} ((), e^{2 α r_{e}} - Q^{2})}{2 n + 1 + frakturh ((), q)})}^{2} . \end{matrix}

with

frakturh ((), q) = \sqrt{1 + \frac{8 {mD}_{e}}{α^{2} ℏ^{2} Q^{2}} {(e^{α r_{e}} + Q)}^{2}},

in accordance with Jia et al…”

Section: Some Useful Applicationssupporting

confidence: 89%

A ((), q) = D_{e}, B ((), q) = D_{e} ((), \frac{q - 2}{q}), C ((), q) = \frac{D_{e}}{q^{2}}, k = \frac{1}{α}

for having

V ((), r) = \frac{D_{e} {qe}^{- α false(r - r_{e false)}}}{1 - {qe}^{- α false(r - r_{e false)}}} + \frac{D_{e} ((), q - 2) e^{- α false(r - r_{e false)}}}{{((), 1 - {qe}^{- α false(r - r_{e false)}})}^{2}} + \frac{D_{e} e^{- 2 α false(r - r_{e false)}}}{{((), 1 - {qe}^{- α false(r - r_{e false)}})}^{2}}

such that, after use

Q = - {qe}^{α r_{e}}

the improved q ‐deformed Tietz potential is obtained,

V^{ITP} ((), r) = D_{e} + V ((), r) = D_{e} {(1 - \frac{e^{- α r_{e}} + Q}{e^{- αr} + Q})}^{2} .

…”

Section: Some Useful Applicationsmentioning

confidence: 99%

“…To deal with this potential, we propose the choice

A ((), q) = 4 U_{2} - 2 U_{1}, B ((), q) = 0, C ((), q) = 4 U_{2}, q = \overset{q}{true} e^{- 2 α r_{e}}

and

k = \frac{1}{2 α}

in the Equation such that

V ((), r) = \frac{((), 4 U_{2} - 2 U_{1}) \overset{q}{true} e^{- α r_{e}} e^{- α ((), r - r_{e})}}{1 - \overset{q}{true} e^{- α r_{e}} e^{- α ((), r - r_{e})}} + \frac{4 U_{2} \overset{q}{true} e^{- α r_{e}} e^{- 2 α ((), r - r_{e})}}{{((), 1 - \overset{q}{true} e^{- α r_{e}} e^{- α ((), r - r_{e})})}^{2}},

leading to

\begin{matrix} lefttrue \\ V^{italicmMR} (r) = U_{0} - U_{1} + U_{2} + V (r) \\ = U_{0} - U_{1} \cot h_{\overset{q}{true}} (italicαr) + U_{2} \cot h_{\overset{q}{true}}^{2} (italicαr) \end{matrix}

which is identified as the q ‐deformed modified Manning‐Rosen potential ( mMR ) as given in Jia et al where cot h q ( αr ), cot h q ( αr ) are the q ‐deformed tri...…”

Section: Some Useful Applicationsmentioning

confidence: 99%

See 1 more Smart Citation

Approximate ℓ‐bound state solutions of q‐deformed exponential‐type potentials

Peña

García-Ravelo

Ovando

et al. 2020

Int J of Quantum Chemistry

View full text Add to dashboard Cite

show abstract

“…Not long ago, Jia et al used the Poisson summation formula to calculate the vibrational partition function of the ITP [78]. Very recently, ITP energy is employed to describe the internal vibration of a molecule via ITP energy to investigate the entropy and specific heat for some gaseous substances [79,80].…”

Section: Introductionmentioning

confidence: 99%

A comparative interpretation of the thermodynamic functions of a relativistic bound state problem proposed with an attractive or a repulsive surface effect

Lütfüoğlu

Křı́ž

2019

Eur. Phys. J. Plus

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Generalized symmetric Woods-Saxon potential (GSWSP) energy well has a significant importance not only in nuclear physics, but also in atomic and molecular physics. A GSWSP energy well takes the surface effects (describing e.g. repulsive interaction at the nucleus edge in nuclear physics) into account in addition to the volume effect. These effects can be, in general, both repulsive or attractive. In this paper, the recently obtained bound state solution of the Klein-Gordon equation with vector and scalar GSWSP energy in the spin symmetry limit is used to calculate a neutral pion's energy spectra in attractive and repulsive cases via various potential parameters.Then, the spectra are employed to find the thermodynamic functions such as Helmholtz free energy, entropy, internal energy, and specific heat. These functions in the attractive and repulsive cases are compared comprehensively. Finally, the role of the shape parameters on the thermodynamic functions in repulsive and attractive cases, respectively, is analyzed.

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“…In the nonrelativistic case, the investigation of thermodynamic functions of some types of potentials, such as Morse and improved Manning-Rosen potentials and improved Rosen-Morse and Tiez oscillators, through a partition function and its derivatives with respect to temperature, was an important field of research in the literature (see [36][37][38][39]). The extension of the nonrelativistic -harmonic oscillator to the relativistic case, to the best of our knowledge, is not available in the literature.…”

Section: Advances In High Energy Physicsmentioning

confidence: 99%

The Statistical Properties of theq-Deformed Dirac Oscillator in One and Two Dimensions

Boumali

Hassanabadi

2017

Advances in High Energy Physics

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We study the behavior of the eigenvalues of the one and two dimensions of -deformed Dirac oscillator. The eigensolutions have been obtained by using a method based on the -deformed creation and annihilation operators in both dimensions. For a twodimensional case, we have used the complex formalism which reduced the problem to a problem of one-dimensional case. The influence of the -numbers on the eigenvalues has been well analyzed. Also, the connection between the -oscillator and a quantum optics is well established. Finally, for very small deformation , we (i) showed the existence of well-known -deformed version of Zitterbewegung in relativistic quantum dynamics and (ii) calculated the partition function and all thermal quantities such as the free energy, total energy, entropy, and specific heat. The extension to the case of Graphene has been discussed only in the case of a pure phase ( = 푖휂 ).

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Partition function of improved Tietz oscillators

Cited by 153 publications

References 26 publications

Approximate ℓ‐bound state solutions of q‐deformed exponential‐type potentials

Approximate ℓ‐bound state solutions of q‐deformed exponential‐type potentials

A comparative interpretation of the thermodynamic functions of a relativistic bound state problem proposed with an attractive or a repulsive surface effect

The Statistical Properties of theq-Deformed Dirac Oscillator in One and Two Dimensions

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