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In this work, we investigate the Shannon entropy of four recently proposed hyperbolic potentials through studying position and momentum entropies. Our analysis reveals that the wave functions of the single-well potentials U 0 , 3 exhibit greater localization compared to the double-well potentials U 1 , 2 . This difference in localization arises from the depths of the single- and double-well potentials. Specifically, we observe that the position entropy density shows higher localization for the single-well potentials, while their momentum probability density becomes more delocalized. Conversely, the double-well potentials demonstrate the opposite behavior, with position entropy density being less localized and momentum probability density showing increased localization. Notably, our study also involves examining the Bialynicki–Birula and Mycielski (BBM) inequality, where we find that the Shannon entropies still satisfy this inequality for varying depths u ¯ . An intriguing observation is that the sum of position and momentum entropies increases with the variable u ¯ for potentials U 1 , 2 , 3 , while for U 0 , the sum decreases with u ¯ . Additionally, the sum of the cases U 0 and U 3 almost remains constant within the relative value 0.01 as u ¯ increases. Our study provides valuable insights into the Shannon entropy behavior for these hyperbolic potentials, shedding light on their localization characteristics and their relation to the potential depths. Finally, we extend our analysis to the Fisher entropy F ¯ x and find that it increases with the depth u ¯ of the potential wells but F ¯ p decreases with the depth. [ABSTRACT FROM AUTHOR]

In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number (0 < n ≤ 2) by calculating position and momentum entropy. We find that the wave function will move towards the origin as the fractional derivative number n decreases and the position entropy density becomes more severely localized in more fractional system, i.e., for smaller values of n, but the momentum probability density becomes more delocalized. And then we study the Beckner Bialynicki-Birula–Mycieslki (BBM) inequality and notice that the Shannon entropies still satisfy this inequality for different depth u even though this inequality decreases (or increases) gradually as the depth u of the hyperbolic potential U 1 (or U 2 ) increases. Finally, we also carry out the Fisher entropy and observe that the Fisher entropy increases as the depth u of the potential wells increases, while the fractional derivative number n decreases. [ABSTRACT FROM AUTHOR]

In this work, we study the quantum information entropies for two different types of hyperbolic single potential wells. We first study the behaviors of the moving particle subject to two different hyperbolic potential wells through focusing on their wave functions. The shapes of these hyperbolic potentials are similar, but we notice that their momentum entropy densities change along with the width of each potential and the magnitude of position entropy density decreases when the momentum entropy magnitude increases. On the other hand, we illustrate the behaviors of their position and momentum entropy densities. Finally, we show the variation of position and momentum entropies S x and S p with the change of the potential well depth u and verify that their sum still satisfies the BBM inequality relation. [ABSTRACT FROM AUTHOR]

We solve the quantum system with the symmetric Razavy cosine type potential and find that its exact solutions are given by the confluent Heun function. The eigenvalues are calculated numerically. The properties of the wave functions, which depend on the potential parameter a, are illustrated for a given potential parameter ξ. It is shown that the wave functions are shrunk to the origin when the potential parameter a increases. We note that the energy levels ϵi (i∈[1,3]) decrease with the increasing potential parameter a but the energy levels ϵi (i∈[4,7]) first increase and then decrease with the increasing a. [ABSTRACT FROM AUTHOR]

In this work, we study the quantum system with the symmetric Razavy potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun functions. The eigenvalues have to be calculated numerically. The properties of the wave functions depending on m are illustrated graphically for a given potential parameter ξ. We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter m increases. This arises from the fact that the parity, which is a defined symmetry for very small m, is completely violated for large m. We also notice that the energy levels ϵi decrease with the increasing potential parameter m. [ABSTRACT FROM AUTHOR]

In this work, we study the Shannon information entropies Sr and Sp of an infinite spherical well. The Shannon entropy Sp is calculated numerically in terms of the analytical result of the wave function in momentum space. Some typical features of the position and momentum probability densities ρ(r) and ρ(p) as well as the information entropy densities ρs(r) and ρs(p) are demonstrated. We find that the position entropy Sr increases with the radius a of the spherical well for given quantum numbers l, m and n. It is interesting to note that the position entropy Sr decreases with the quantum numbers l and n for a fixed radius a and quantum number m. The position entropy Sr is almost independent of the quantum numbers l, m and n. The momentum entropy Sp first increases and then decreases with respect to the radius a. We also note that the Sr increases with the radius a and finally arrives at a constant. In addition, the Bialynicki–Birula–Mycielski (BBM) inequality is verified and also hold for this confined system. [ABSTRACT FROM AUTHOR]

We present exact solutions of solitonic profile mass Schrödinger equation with a modified Pöschl-Teller potential. We find that the solutions can be expressed analytically in terms of confluent Heun functions. However, the energy levels are not analytically obtainable except via numerical calculations. The properties of the wave functions, which depend on the values of potential parameter are illustrated graphically. We find that the potential changes from single well to a double well when parameter changes from minus to positive. Initially, the crest of wave function for the ground state diminishes gradually with increasing and then becomes negative. We notice that the parities of the wave functions for also change. [ABSTRACT FROM AUTHOR]

Using improved approximate schemes for centrifugal term and the singular factor 1/r appearing in potential itself, we solve the Schrödinger equation with the screen Coulomb potential for arbitrary angular momentum state l. The bound state energy levels are obtained. A closed form of normalization constant of the wave functions is also found. The numerical results show that our results are in good agreement with those obtained by other methods. The key issue is how to treat two singular points in this quantum system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

New type shift operators for three-dimensional infinite well potential are identified to connect those quantum systems with different radials R but with the same energy spectrum. It should be pointed out that these shift operators depend on all variables contained in wave functions. Thus they establish a novel relation between wave functions ψlm(r) and ψ(l±1)(m±1)(r). [ABSTRACT FROM AUTHOR]

The arbitrary l-state solutions of the Schrödinger equation with Scarf II potential are presented by taking a new approximate scheme to the centrifugal term. This is realized by expanding the variable around the minimum equilibrium point of the potential. The wave functions can be expressed by hypergeometric functions. It is shown that the calculated energy levels are in good agreement with accurate numerical ones. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010 [ABSTRACT FROM AUTHOR]

We propose proper quantization rule, [formula], where [formula]. 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 [lowercase_phi_synonym](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. [ABSTRACT FROM AUTHOR]

Abstract: In the case of spin symmetry we solve the Dirac equation with scalar and vector deformed generalized Pöschl–Teller (DGPT) potential and obtain exact energy equation and spinor wave functions for s-wave bound states. We find that there are only positive energy states for bound states in the case of spin symmetry based on the strong regularity restriction condition for the wave functions. The energy eigenvalue approaches a constant when the potential parameter α goes to zero. Two special cases such as generalized PT potential and standard PT potential are also briefly discussed. [Copyright &y& Elsevier]

Abstract: In this Letter the approximately analytical bound state solutions of the Dirac equation with the Manning–Rosen potential for arbitrary spin–orbit coupling quantum number k are carried out by taking a properly approximate expansion for the spin–orbit coupling term. In the case of exact spin symmetry, the associated two-component spinor wave functions of the Dirac equation for arbitrary spin–orbit quantum number k are presented and the corresponding bound state energy equation is derived. We study briefly two special cases; the general s-wave problem and the equal scalar and vector Manning–Rosen potential. [Copyright &y& Elsevier]

By using the exact quantization rule, we present analytical solutions of the Schrödinger equation for the deformed harmonic oscillator in one dimension, the Kratzer potential and pseudoharmonic oscillator in three dimensions. The energy levels of all the bound states are easily calculated from this quantization rule. The normalized wavefunctions are also obtained. It is found that the present approach can simplify the calculations. [ABSTRACT FROM AUTHOR]

The eigenfunctions and eigenvalues of the Schrödinger equation with a ring-shaped non-spherical oscillator are obtained. A realization of the ladder operators for the radial wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1,1) group. The closed analytical expressions for the matrix elements of different functions ρ and ρd/dρ with ρ=r2 are evaluated. [Copyright &y& Elsevier]

The controllability of a quantum system for the Morse potential with the bound states are investigated. The ladder operators are constructed directly from the wave functions with the factorization method and associated to an su(2) algebra. This quantum system with the discrete bound states can be strongly completely controlled, i.e., the eigenstates can be guided by the external field to approach arbitrarily close to a selected target state at any chosen time, which can be theoretically realized by the actions of the ladder operators on the ground state. [Copyright &y& Elsevier]

Abstract We propose two new type sine hyperbolic potentials V (x) = a 2 sinh 2 ⁡ (x) − k tanh 2 ⁡ (x) and V (x) = c 2 sinh 4 ⁡ (x) − k tanh 2 ⁡ (x). They may become single- or double-well potentials depending on the potential parameters a , c and k. We find that its exact solutions can be written as the confluent Heun functions H c (α , β , γ , δ , η ; z) , in which the energy level E is involved inside the parameter η. The properties of the wave functions, which is strongly relevant for the potential parameters a , c and k , are illustrated. Highlights • We propose two new type sine hyperbolic potentials. • The solutions are known as confluent Heun functions. • The wave functions with various parameter values are illustrated. [ABSTRACT FROM AUTHOR]

We present analytical matter-wave solutions to a generalized Gross–Pitaevskii (GGP) equation with several new time and space varying nonlinearity coefficients and external fields. This is realized by taking a suitable similarity transformation to the GGP equation which makes the original partial differential equation into a stationary and ordinary differential equation. We report a few families of analytical solutions of the GGP equation with several new time and space varying nonlinearity interactions, in which some physically relevant soliton solutions are found. The profile features of the evolution wave functions depend on the different choices of the composite functions ξ . [ABSTRACT FROM AUTHOR]

Abstract: We present the exact solutions of the Schrödinger equation with the double ring-shaped oscillator (DRSO) potential. By introducing a new variable and constructing super-universal associated Legendre polynomials we express the polar angular wave functions explicitly. We observe that the present DRSO has caused the symmetry breaking from the original spherical oscillator symmetries to the present symmetry due to the surrounded two ring-shaped inversed square potentials. Some special cases are also discussed. [Copyright &y& Elsevier]

• The 1D Schrödinger equation with the Mathieu potential is considered. • Its exact solutions are expressed in terms of the confluent Heun function. • Its eigenvalues are calculated by Wronskian determinant. • The properties of the wave functions with variation to the parameters are displayed. The exact solutions of the 1D Schrödinger equation with the Mathieu potential V (x) = a 2 sin 2 ⁡ (b x) − a b (2 c + 1) cos ⁡ (b x) + v b 2 (a > 0 , b > 0) are presented as a confluent Heun function H C (α , β , γ , δ , η ; z). The eigenvalues are calculated precisely by solving the Wronskian determinant. The wave functions for the positive and negative parameter c , which correspond to two different potential wells with symmetric axis x = 0 and x = π are plotted. It is found that the wave functions are shrunk to the origin for given values of the parameters a = 1 , b = 1 and v = 2 when the potential parameter | c | increases. [ABSTRACT FROM AUTHOR]