Proper Quantization Rule Research Articles

Energy and momentum eigenspectrum of the Hulthén-screened cosine Kratzer potential using proper quantization rule and SUSYQM method.

Using the Qiang-Dong proper quantization rule (PQR) and the supersymmetric quantum mechanics approach, we obtain the eigenspectrum of the energy and momentum for time-independent and time-dependent Hulth'en-screened cosine Kratzer potentials. For the suggested time-independent Hulthén-screened cosine Kratzer potential (HSCKP), we solve the Schrödinger equation in D dimensions. The Feinberg-Horodecki equation for time-dependent Hulth'en-screened cosine Kratzer potential (tHSCKP) also solves. To address the inverse square term in the time-independent and time-dependent equations, we employed the Greene-Aldrich approximation approach. We were able to extract time-independent and time-dependent potentials, as well as their accompanying energy and momentum spectra. In three-dimensional space, we estimate the rotational vibrational (RV) energy spectrum for many homodimers (H2, I2, O2) and heterodimers (MnH, ScN, LiH, HCl). We also use the recently introduced formula approach to obtain the relevant eigen function. We also calculate momentum spectra for the dimers MnH and ScN. The method is compared to prior methodologies for accuracy and validity using numerical data for heterodimer LiH, HCl and homodimer I2, O2, H2. The calculated energy and momentum spectra are tabulated and analyzed.

Determination of Energy Spectra By Using Proper Quantization Rule of Woods-Saxon Potential

In this study, the energy spectra of Schrodinger equation for non-zero l values considering Woods Saxon potential (WSP) is calculated using proper quantization rule, then the binding energies (BE) of random light nuclei is obtained and the optimized potential parameters such as potential depth (V0) and surface thickness (a) are found. In order to calculate the energy levels of the nuclei with WSP, the PQR method was used, which has not been considered before. In quantum mechanics, the exact solution of energy systems, momentum, and quantum states can be found using the proper quantization rule(PQR) method.Using the Matlab calculation program, we have achieved numerical values of the energy spectrum for random light nuclei and compared the result with the experimental Nuclear Data Center (NDC) values. In addition, we found potential depth and surface thickness for four light nuclei. Correlations between the light nuclei show the facts about the nuclear structure characteristics, origin, and energies of these nuclei. Pearson’s correlation coefficient is accepted as the most common correlation coefficient. According to the values of Pearson correlation coefficients, it is observed that there is a significant positive correlation between the nucleons examined. Finally, we plot the E-V0-a diagrams for those values to optimize and provide the appropriate coefficients. It is shown that there is a good agreement between the results of this work and experimental values.

Bound states of Dirac equation using the proper quantization rule

Using the proper quantization rule, we investigate the exact solution of Dirac equation for Hartmann and the ring-shaped non-spherical harmonic oscillator potentials under the condition of equal scalar and vector potentials. By considering the proper quantization condition within angular and radial variables, the exact relativistic energy spectra are obtained for each system. Then by the mean of suitable changes of variables, the corresponding spinor wave-functions are constructed where the normalization constants are exactly calculated. We also derived the non-relativistic limit of energy spectra.

J—state solutions and thermodynamic properties of the Tietz oscillator

In this work, we have solved the radial part of the Schrödinger equation with Tietz potential to obtain explicit expressions for bound state ro-vibrational energies and radial eigenfunctions. The proper quantization rule and ansatz solution technique were used to arrive at the solutions. In modeling the pseudo-spin–orbit term of the effective potential, the Pekeris-like and the Greene-Aldrich approximation recipes were applied. Using our equation for eigen energies, we have deduced expression for bound state energy eigenvalues of Deng-Fan oscillator. The result obtained agrees with available literature data for this potential. Also, for arbitrary values of rotational and vibrational quantum numbers, we have calculated bound state energies for the Tietz oscillator. Our computed results are in excellent agreement with those in the literature. Furthermore, the result showed that unlike Greene-Aldrich approximation, energies computed based on Pekeris-like approximation are better and almost indistinguishable from numerically obtained energies of the Tietz oscillator in the literature. With the help of our formula for ro-vibrational energy, analytical expressions for some important thermodynamic relations were also derived for the Tietz oscillator. The derived thermal functions which include ro-vibrational: partition function, free energy, mean energy, entropy and specific heat capacity were subsequently applied to the spectroscopic data of KI diatomic molecule. Studies of the thermal functions indicated that the partition function decreases monotonically as the temperature is raised and increases linearly for increase in the upper bound vibrational quantum number. On the other hand, increase in either temperature or upper bound vibrational quantum number amounts to monotonic rise in the entropy of the KI molecules

Deep Task-Based Quantization.

Quantizers play a critical role in digital signal processing systems. Recent works have shown that the performance of acquiring multiple analog signals using scalar analog-to-digital converters (ADCs) can be significantly improved by processing the signals prior to quantization. However, the design of such hybrid quantizers is quite complex, and their implementation requires complete knowledge of the statistical model of the analog signal. In this work we design data-driven task-oriented quantization systems with scalar ADCs, which determine their analog-to-digital mapping using deep learning tools. These mappings are designed to facilitate the task of recovering underlying information from the quantized signals. By using deep learning, we circumvent the need to explicitly recover the system model and to find the proper quantization rule for it. Our main target application is multiple-input multiple-output (MIMO) communication receivers, which simultaneously acquire a set of analog signals, and are commonly subject to constraints on the number of bits. Our results indicate that, in a MIMO channel estimation setup, the proposed deep task-bask quantizer is capable of approaching the optimal performance limits dictated by indirect rate-distortion theory, achievable using vector quantizers and requiring complete knowledge of the underlying statistical model. Furthermore, for a symbol detection scenario, it is demonstrated that the proposed approach can realize reliable bit-efficient hybrid MIMO receivers capable of setting their quantization rule in light of the task.

Approximate solution of the Schrödinger equation with Manning-Rosen plus Hellmann potential and its thermodynamic properties using the proper quantization rule

We obtained, by using the proper quantization rule, the approximate solutions of the Schrodinger equation with Manning-Rosen plus Hellmann potential for any l-state. Furthermore, thermodynamic properties, such as the vibrational mean U , specific heat C, free energy F and entropy S for the pure vibrational state in the classical limit for these energy eigenvalues, are studied.

Energy spectra and the expectation values of diatomic molecules confined by the shifted Deng-Fan potential

The approximate bound state solutions of the Schrodinger equation with the shifted Deng-Fan potential was obtained via a proper quantization rule. The energy spectra for the homogenous diatomic molecules (H2, I2); the heterogeneous diatomic molecules (CO, HCl, LiH); the neutral transition metal hydrides (ScH, TiH, VH, CrH); the transition-metal lithide (CuLi); the transition-metal carbides (TiC, NiC); the transition metal nitrite (ScN) and the transition metal fluoride (ScF) were calculated. By applying the Hellmann-Feynman theorem, the expression for the expectation values of the square of inverse of position r-2, potential energy V, kinetic energy T and the square of momentum p2 were derived and the numerical values for diatomic molecules were presented. The result is reliable and very consistent with the available ones in the literature.

Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential

A bound state solution is a quantum state solution of a particle subjected to a potential such that the particle’s energy is less than the potential at both negative and positive infinity. The particle’s energy may also be negative as the potential approaches zero at infinity. It is characterized by the discretized eigenvalues and eigenfunctions, which contain all the necessary information regarding the quantum systems under consideration. The bound state problems need to be extended using a more precise method and approximation scheme. This study focuses on the non-relativistic bound state solutions to the generalized inverse quadratic Yukawa potential. The expression for the non-relativistic energy eigenvalues and radial eigenfunctions are derived using proper quantization rule and formula method, respectively. The results reveal that both the ground and first excited energy eigenvalues depend largely on the angular momentum numbers, screening parameters, reduced mass, and the potential depth. The energy eigenvalues, angular momentum numbers, screening parameters, reduced mass, and the potential depth or potential coupling strength determine the nature of bound state of quantum particles. The explored model is also suitable for explaining both the bound and continuum states of quantum systems.

Spectroscopic study of some diatomic molecules via the proper quantization rule

Spectroscopic techniques are very essential tools in studying electronic structures, spectroscopic constants and energetic properties of diatomic molecules. These techniques are also required for parametrization of new method based on theoretical analysis and computational calculations. In this research, we apply the proper quantization rule in spectroscopic study of some diatomic molecules by solving the Schr\"odinger equation with two solvable quantum molecular systems-Tietz-Wei and shifted Deng-Fan potential models for their approximate nonrelativistic energy states via an appropriate approximation to the centrifugal term. We show that the energy levels can be determined from its ground state energy. The beauty and simplicity of the method applied in this study is that, it can be applied to any exactly as well as approximately solvable models. The validity and accuracy of the method is tested with previous techniques via numerical computation for H$_2$ and CO diatomic molecules. The result also include energy spectrum of 5 different electronic states of NO and 2 different electronic state of ICl.

Proper quantization rule approach to three-dimensional quantum dots

In this article, we develop a formalism to obtain the energy levels of the electron in a central force potential confined in a spherical quantum dot with radius rC by the proper quantization rule and the Wentzel-Kramers-Brillouin approximation. It is shown that the numerical results are in good agreement with exact solutions. To illustrate this method, we consider the linear harmonic oscillator and Coulomb potential confined within an impenetrable sphere of radius rC in three dimensions. © 2013 Wiley Periodicals, Inc.

Energy spectrum for a modified Rosen-Morse potential solved by proper quantization rule and its thermodynamic properties

We apply our recently proposed proper quantization rule, \({\int_{x_A}^{x_B}k(x) dx -\int_{x_{0A}}^{x_{0B}}k_0(x) dx=n\pi}\) , where \({k(x)=\sqrt{2 M [E-V(x) ]}/\hbar}\) to obtain the energy spectrum of the modified Rosen-Morse potential. The beauty and symmetry of this proper rule come from its meaning—whenever the number of the nodes of \({\phi(x)}\) or the number of the nodes of the wave function ψ(x) increases by one, the momentum integral \({\int_{x_A}^{x_B} k(x)dx}\) will increase by π. Based on this new approach, we present a vibrational high temperature partition function in order to study thermodynamic functions such as the vibrational mean energy U, specific heat C, free energy F and entropy S. It is surprising to note that the specific heat C (k = 1) first increases with β and arrives to the maximum value and then decreases with it. However, it is shown that the entropy S (k = 1) first increases with the deepness of potential well λ and then decreases with it.

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.

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.