Exact Quantization Rule Research Articles

Theoretical Investigation of Meson Spectrum via Exact Quantization Rule Technique

The energy eigenvalues with the Extended Cornell potential were obtained by analytically solving the radial Schrödinger equation using the Exact Quantization Rule technique (ECP). It was then used for computing the mass spectra of the heavy mesons like charmonium (cc-) and bottomonium (bb-) as well as heavy-light mesons such as bottom-charm bc- and charm-Strange cs- for various quantum states. Two exceptional cases such as the Coulomb and Cornell potentials, were taken into consideration when some of the potential parameters were set to zero. The current potential offers good outcomes when compared to experimental data and work of other researchers with a maximum error of 0.0065 GeV.

Determination of thermodynamic properties of [formula omitted] and [formula omitted] diatomic molecules with the linear combination of Hulthen-type potential plus Yukawa potential

In this work, we investigate the thermodynamic properties of CrH,NiC and CuLi diatomic molecules with a linear combination Hulthen and Yukawa potentials in the presence and absence of magnetic and Aharanov-Bohm (AB) fields. The Schrödinger equation in 3D and 2D were solved using the Nikiforov-Uvarov (NU) method and the exact quantization rule (EQR) respectively. To determine the thermodynamic properties for the selected diatomic molecules, we first used the energy spectrum to evaluate the partition function and other thermodynamic functions such as entropy, Helmholtz free energy, internal energy and specific heat capacity. In addition, it is found that in the classical limit, the specific heat capacity saturates for large values of the principal quantum number, nmax for the selected diatomic molecules at a fixed temperature except CrH. Our results can be applied to molecular physics and chemical physics.

Eigensolutions and expectation values of shifted-rotating Möbius squared oscillator

In this paper we have employed exact quantization rule and the Hellman–Feynman theorem to obtain bound states eigensolutions and expectation values of the shifted-rotating Mobius squared oscillator, and we have used the Pekeris-like approximation recipe to effect solution for bound states energy eigenspectra, and in solving the Riccati equation for the eigenfunctions, cases of Q ≠ 0 and Q = 0 were considered in our solutions. We have also applied our derived formulas to compute numerical values to the spectroscopic parameters of four diatomic molecules: CO, LiH, H2 and ICl. Our analysis showed that numerical results we obtained for potential parameters, bound states energy eigenspectra and expectation values are consistent with available data in the literature and that molecular interactions in CO, LiH, H2 and ICl are best described by improved Wei oscillator than by Morse oscillator.

Energy Eigenvalues and Eigenfunctions of a Diatomic Molecule in Quadratic Exponential-type Potential

We have employed the exact quantization rule to obtain closed form expression for the bound state energy eigenvalues of a molecule in quadratic exponential-type potential. To deal with the spin-orbit centrifugal term of the effective potential energy function, we have used a Pekeris-type approximation scheme, we have also obtained closed form expression for the normalized radial wave functions by solving the Riccati equation with quadratic exponential-type potential. Using our derived energy eigenvalue formula, we have deduced expressions for the bound state energy eigenvalues of the Hulthén, Eckart and Deng-Fan potentials, considered as special cases of the quadratic exponential-type potential. Our deduced energy eigenvalues are in excellent agreement with those in the literature. We have computed bound states energy eigenvalues for six diatomic molecules viz: HCl, LiH, H2, SeH, VH and TiH. Our results are in total agreement with existing results in the literature for the s-wave and in good agreement for higher quantum states. By solving the Riccati equation, we have obtained normalized radial wave functions of the quadratic exponential-type potential, our results show higher probabilities of finding the molecule in the region 0.1 ≤ y ≤ 0.2

ROTATIONAL-VIBRATIONAL EIGEN SOLUTIONS OF THE D-DIMENSIONAL SCHRÖDINGER EQUATION FOR THE IMPROVED WEI POTENTIAL

In this present study, we have employed the techniques of exact quantization rule and ansatz solution method to obtain closed form expressions for the rotational-vibrational eigensolutions of the D-dimensional Schrödinger equation for the improved Wei potential, for cases of h′ ≠ 0 and h′ = 0. By using our derived energy equation and choosing arbitrary values of n and ℓ, we have computed the bound state rotational-vibrational energies of CO, H2 and LiH for various quantum states. The mean absolute percentage deviation (MAPD) and the Lippincott criterion ware used as a goodness-of-fit indices to compare our result with the Rydberg-Klein-Rees (RKR) and improved Tietz potential data in the literature. MAPD of 0.2862%, 0.2896% and 0.0662% relative to the RKR data for CO ware obtained. For the improved Wei and Morse potential, our computed energy eigenvalues for CO, H2 and LiH are in excellent agreement with existing results in the literature

SWKB and proper quantization conditions for translationally shape-invariant potentials

Using a recently proposed classification for the primary translationally shape-invariant potentials, we show that the exact quantization rule formulated by Ma and Xu is equivalent to the supersymmetric JWKB quantization condition. The energy levels for the two considered categories of shape-invariant potentials are also derived.

Shifted Tietz–Wei oscillator for simulating the atomic interaction in diatomic molecules

The shifted Tietz-Wei (sTW) oscillator is as good as traditional Morse potential in simulating the atomic interaction in diatomic molecules. By using the Pekeris-type approximation to deal with the centrifugal term, we obtain the bound-state solutions of the radial Schr\"odinger equation with this typical molecular model via the exact quantization rule (EQR). The energy spectrum for a set of diatomic molecules ($NO \left(a^4\Pi_i\right)$, $NO \left(B^2\Pi_r\right)$, $NO \left(L'^2\phi\right)$, $NO \left(b^4\Sigma^{-}\right)$, $ICl\left(X^1\Sigma_g^{+}\right)$, $ICl\left(A^3\Pi_1\right)$ and $ICl\left(A'^3\Pi_2\right)$ for arbitrary values of $n$ and $\ell$ quantum numbers are obtained. For the sake of completeness, we study the corresponding wavefunctions using the formula method.

Quantization Rule for Relativistic Klein-Gordon Equation

Based on the exact quantization rule for the nonrelativistic Schrodinger equation, the exact quantization rule for the relativistic one-dimensional Klein-Gordon equation is suggested. Using the new quantization rule, the exact relativistic energies for exactly solvable potentials, e.g. harmonic oscillator, Morse, and Rosen-Morse II type potentials, are obtained. Consequently the new quantization rule is found to be exact for one-dimensional spinless relativistic quantum systems. Though the physical meanings of the new quantization rule have not been fully understood yet, the present formal derivation scheme may shed light on understanding relativistic quantum systems more deeply.

Energy spectrum of the Manning-Rosen potential including centrifugal term solved by exact and proper quantization rules

The energy spectrum of the Manning-Rosen potential including centrifugal term in higher dimensions is presented by exact quantization rule approach. The result is compared with that by proper quantization rule method. It is found that the latter is better than that of the exact quantization rule. We find that the interdimensional degeneracy exists for the states in different dimensions. For the special case D = 3, the results agree well with those obtained by other methods.

Proper quantization rule as a good candidate to semiclassical quantization rules

In this article, we present proper quantization rule, ∫k(x) dx - ∫k0(x) dx = nπ, where and study solvable potentials. We find that the energy spectra of solvable systems can be calculated only from its ground state obtained by the Sturm-Liouville theorem. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. The beauty and simplicity of proper quantization rule come from its meaning – whenever the number of the nodes of the logarithmic derivative ϕ(x) = ψ(x)-1dψ(x) /dx or the number of the nodes of the wave function ψ(x) increases by one, the momentum integral will increase by π. We apply two different quantization rules to carry out a few typically solvable quantum systems such as the one-dimensional harmonic oscillator, the Morse potential and its generalization as well as the asymmetrical trigonometric Scarf potential and show a great advantage of the proper quantization rule over the original exact quantization rule.

Approximate analytical solutions for arbitrary l-state of the Hulthén potential with an improved approximation of the centrifugal term

Abstract An approximate analytical solution of the radial Schrödinger equation for the generalized Hulthén potential is obtained by applying an improved approximation of the centrifugal term. The bound state energy eigenvalues and the normalized eigenfunctions are given in terms of hypergeometric polynomials. The results for arbitrary quantum numbers n r and l with different values of the screening parameter δ are compared with those obtained by the numerical method, asymptotic iteration, the Nikiforov-Uvarov method, the exact quantization rule, and variational methods. The results obtained by the method proposed in this work are in a good agreement with those obtained by other approximate methods.

Ma–Xu quantization rule and exact JWKB condition for translationally shape invariant potentials

For translationally shape invariant potentials, the exact quantization rule proposed by Ma and Xu results from the exactness of the modified JWKB quantization condition proved by Barclay. We propose here a very direct alternative way to calculate the appropriate correction for the whole class of translationally shape invariant potentials.

Qiang–Dong proper quantization rule and its applications to exactly solvable quantum systems

We propose proper quantization rule, ∫xAxBk(x)dx−∫x0Ax0Bk0(x)dx=nπ, where k(x)=2M[E−V(x)]/ℏ. The xA and xB are two turning points determined by E=V(x), and n is the number of the nodes of wave function ψ(x). We carry out the exact solutions of solvable quantum systems by this rule and find that the energy spectra of solvable systems can be determined only from its ground state energy. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. The beauty and simplicity of the rule come from its meaning—whenever the number of the nodes of ϕ(x) or the number of the nodes of the wave function ψ(x) increases by 1, the momentum integral ∫xAxBk(x)dx will increase by π. We apply this proper quantization rule to carry out solvable quantum systems such as the one-dimensional harmonic oscillator, the Morse potential and its generalization, the Hulthén potential, the Scarf II potential, the asymmetric trigonometric Rosen–Morse potential, the Pöschl–Teller type potentials, the Rosen–Morse potential, the Eckart potential, the harmonic oscillator in three dimensions, the hydrogen atom, and the Manning–Rosen potential in D dimensions.

Energy spectra for modified Rosen–Morse potential solved by the exact quantization rule

The exact quantization rule is efficient for calculating the energy spectra of all exactly solvable quantum systems. In this work, we calculate the energy spectrum of the Schrödinger equation with the modified Rosen–Morse potential by this quantization rule.

A new quantization rule to the energy spectra for modified hyperbolic‐type potentials

AbstractA new exact quantization rule simplifies the calculation of the energy levels for exactly solvable quantum systems. The energy spectra of the modified hyperbolic‐type potentials are carried out by this rule. These potentials include the modified Scarf, Rosen–Morse, second Pöschl‐Teller and Eckart potentials. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Application of the exact quantization rule for some noncentral separable potentials

The energy spectrum of some noncentral separable potentials are obtained using the exact quantization rule in r and θ dimensions. The results are consistent with those obtained by other methods. PACS Nos.: 03.65.Ca, 03.65.Ge

Exact quantization rule to the Kratzer-type potentials: an application to the diatomic molecules

For arbitrary values of n and l quantum numbers, we present a simple exact analytical solution of the D-dimensional (D ≥ 2) hyperradial Schrödinger equation with the Kratzer and the modified Kratzer potentials within the framework of the exact quantization rule (EQR) method. The exact bound state energy eigenvalues (E nl ) are easily calculated from this EQR method. The corresponding normalized hyperradial wave functions (ψ nl (r)) are also calculated. The exact energy eigenvalues for these Kratzer-type potentials are calculated numerically for a few typical LiH, CH, HCl, CO, NO, O2, N2 and I2 diatomic molecules for various values of n and l quantum numbers. Numerical tests using the energy calculations for the inter dimensional degeneracy (D = 2 − 4) for I2, LiH, HCl, O2, NO and CO are also given. Our results obtained by EQR are in exact agreement with those obtained by other methods.

Geometric Proof of Exact Quantization Rules in One Dimensional Quantum Mechanics

In terms of the construction of vector field with momentum and logarithmic derivative of wavefunction as its components, a geometric proof of an exact quantization rule in one dimensional quantum mechanics systems is given. The quantization rule arises from the SO(2) gauge transformation. In addition, the quantization rule is generalized to the case when the potential function is piecewise continuous between the two turning points.

Arbitrary /-state approximate solutions of the Hulthén potential through the exact quantization rule

Abstract In this paper, using the Exact Quantization Rule, we present approximate analytical solutions of the radial Schrödinger equation with non-zero l values for the Hulthén potential in the frame of an approximation to the centrifugal potential for any l states. The energy levels of all bound states can be easily calculated from the Exact Quantization Rule. Specifically, the normalized analytical wave functions are also obtained. Some energy eigenvalues are numerically calculated and compared with those obtained by other methods such as asymptotic iteration, supersymmetry, numerical integration methods, and the schroedinger Mathematica package.

Energy Spectrum of the Relativistic Duffin-Kemmer-Petiau Equation

The bound state energy eigenvalues for the relativistic DKP oscillator and DKP Coulomb potentials are determined by using an exact quantization rule. The corresponding eigenfunctions are also obtained. The results are consistent with those obtained by others methods.