Energy Eigenvalue Equation Research Articles

Exact Solutions of the Klein-Gordon Equation for the Scarf-Type Potential via Nikiforov-Uvarov Method

In this paper we present the exact solutions of the one-dimensional Klein-Gordon equation for the Scarf-type potential with equal scalar and vector potentials. Exact solutions and corresponding energy eigenvalues equation are obtained using Nikiforov-Uvarov mathematical method for the s-wave bound state. The PT-symmetry and Hermiticity for this potential are also considered. It will be shown that the obtained results of the Scarf-type potential are reduced to the results of the well-known potentials in the special cases.

A generalized bosonic oscillator in the presence of a minimal length

We present an exact solution of the three-dimensional Duffin–Kemmer–Petiau oscillator for spins 1 and 0 in the momentum space with the presence of minimal length uncertainty by the technique of vector spherical harmonics. The eigenfunctions are determined for both cases and the energy eigenvalues equation are obtained. The limiting case is then deduced for a small parameter of deformation.

Exact Pseudospin Symmetric Solution of the Dirac Equation for Pseudoharmonic Potential in the Presence of Tensor Potential

Under the pseudospin symmetry, we obtain exact solution of the Dirac equation for the pseudoharmonic potential in the presence of the tensor potential with arbitrary spin–orbit coupling quantum number κ. The energy eigenvalue equation of the Dirac particles is found and the corresponding radial wave functions are presented in terms of confluent hypergeometric functions. We investigate the tensor potential dependence of the energy of the each state in the pseudospin doublet. It is shown that degeneracy between members of the pseudospin doublet is removed by tensor interaction. Furthermore, the radial node structure of the Dirac spinor is discussed.

Manifestly covariant classical correlation dynamics I. General theory

The solvability of The Dirac equation is studied for the exponential-type potentials with the pseudospin symmetry by using the parametric generalization of the Nikiforov–Uvarov method. The energy eigenvalue equation, and the corresponding Dirac spinors for Morse, Hulthen, and q-deformed Rosen–Morse potentials are obtained within the framework of an approximation to the spin-orbit coupling term, so the solutions are given for any value of the spin-orbit quantum number κ = 0, or κ ≠ 0.

Approximate analytical solutions of the pseudospin symmetric Dirac equation for exponential-type potentials

The solvability of The Dirac equation is studied for the exponential-type potentials with the pseudospin symmetry by using the parametric generalization of the Nikiforov-Uvarov method. The energy eigenvalue equation, and the corresponding Dirac spinors for Morse, Hulthen, and q-deformed Rosen-Morse potentials are obtained within the framework of an approximation to the spin-orbit coupling term, so the solutions are given for any value of the spin-orbit quantum number $\kappa=0$, or $\kappa \neq 0$.

Approximate analytical solutions of the pseudospin symmetric Dirac equation for exponential‐type potentials

AbstractThe solvability of The Dirac equation is studied for the exponential‐type potentials with the pseudospin symmetry by using the parametric generalization of the Nikiforov–Uvarov method. The energy eigenvalue equation, and the corresponding Dirac spinors for Morse, Hulthen, and q‐deformed Rosen–Morse potentials are obtained within the framework of an approximation to the spin‐orbit coupling term, so the solutions are given for any value of the spin‐orbit quantum number κ = 0, or κ ≠ 0.

Comment on ‘Approximate analytical solutions of the Dirac equation with the Pöschl–Teller potential including spin-orbit coupling’

In this comment we show that the energy eigenvalue equations and the wave functions given by a previous article on the solutions of the Dirac equation with Pöschl–Teller potential must be corrected.

Approximate analytical solutions of the Klein–Gordon equation with scalar and vector Eckart potentials

By using the functional analysis method, we solve approximately the Klein–Gordon equation with equal scalar and vector Eckart potentials for the arbitrary orbital angular momentum number l. The bound state energy eigenvalue equation and the unnormalized radial wavefunctions have been approximately obtained by considering a proper approximation to the centrifugal potential term.

Entangled photon–electron states and the number-phase minimum uncertainty states of the photon field

The exact analytic solutions of the energy eigenvalue equation of the system consisting of a free electron and one mode of the quantized radiation field are used for studying the physical meaning of a class of number-phase minimum uncertainty states. The states of the mode which minimize the uncertainty product of the photon number and the Suskind and Glogower (1964 Physics 1 49–61) cosine operator have been obtained by Jackiw (1968 J. Math. Phys. 9 339–46). However, these states have so far remained mere mathematical constructions without any physical significance. It is proved that the most fundamental interaction in quantum electrodynamics—namely the interaction of a free electron with a mode of the quantized radiation field—leads quite naturally to the generation of the mentioned minimum uncertainty states. It is shown that from the entangled photon–electron states developing from a highly excited number state, due to the interaction with a Gaussian electronic wave packet, the minimum-uncertainty states of Jackiw's type can be constructed. In the electron's coordinate representation, the physical meaning of the expansion coefficients of these states is the joint probability amplitudes of simultaneous detection of an electron and of a definite number of photons. The photon occupation probabilities in these states preserve their functional form as time elapses, but they depend on the location in space-time of the detected electron. An analysis of the entanglement entropies derived from the photon number distribution and from the electron's density operator is given.

DEFORMATION QUANTIZATION OF CONFINED SYSTEMS

The Weyl–Wigner formulation of quantum confined systems poses several interesting problems. The energy stargenvalue equation as well as the dynamical equation do not display the expected solutions. In this paper, we review some previous results on the subject and add some new contributions. We reformulate the confined energy eigenvalue equation by adding to the Hamiltonian a new (distributional) boundary potential. The new Hamiltonian is proved to be globally defined and self-adjoint. Moreover, it yields the correct Weyl–Wigner formulation of the confined system.

The equation for energy eigenvalues in MQW structures with particles having different effective masses in different layers

The equation for energy eigenvalues in MQW structures with particles having different effective masses in different layers

The equation for energy eigenvalues in MQW structures with an arbitrary number of quantum wells

The equation for energy eigenvalues in MQW structures with an arbitrary number of quantum wells

Comment on “A single equation for finite rectangular well energy eigenvalues,” by B. Cameron Reed [Am. J. Phys. 58 (5), 503–504 (1990)

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Pauli equation for spin-1/2 dyonium in non-Abelian gauge theory

The study of the behaviour of a spin-1/2 non-Abelian dyon moving in the field of an Abelian dyon has been undertaken by solving Dirac equation for energy eigenvalues. The Hamiltonian of this system has been shown to involve spin as well as the contribution of massive fields associated with these particles. By introducing suitable spinors, the Pauli equation for a spin-1/2 dyon moving in the field of another dyon has been solved in non-Abelian gauge theory. It is shown that the introduction of massive fields perceptibly modifies the energy eigenvalue and eigenfunction of the bound state of the system.

The Hamiltonian constraint in the teleparallel equivalent of general relativity

In the teleparallel equivalent of general relativity the integral form of the Hamiltonian constraint contains explicitly theadm energy in the case of asymptotically flat space-times. We show that such expression of the constraint leads to a natural and straightforward construction of a Schrodinger equation for time-dependent physical states. The quantized Hamiltonian constraint is thus written as an energy eigenvalue equation. We further analyse the constraint equations in the case of a space-time endowed with a spherically symmetric geometry. We find the general functional form of the time-dependent solutions of the quantized Hamiltonian and vector constraints.

PHENOMENOLOGICAL RELATIVISTIC STUDY OF THE ENERGY OF A Λ IN ITS GROUND AND EXCITED STATES IN HYPERNUCLEI

The binding energy BΛ of a Λ-particle in hypernuclei is studied by means of the Dirac equation with attractive and repulsive potentials of rectangular shapes and of the same radius. The energy eigenvalue equation in this case is obtained analytically for every bound state, as well as the large and small component of the wave function (for given BΛ). A first attempt is also made to investigate the possibility of deriving in particular cases approximate analytic expressions of BΛ for the excited states. Using various least squares fittings, numerical results for the binding energy of the Λ are given and comparisons and comments are also made.

Quantum statistical foundation to the fermi liquid model and Ginzburg-Landau wave function

An energy eigenvalue equation for a quasi-particle is derived, starting with the Heisenberg equation of motion for an annihilation operator. An elementary derivation of the Fermi liquid model having a sharply defined Fermi surface in the k-space is given, starting with a realistic model of a metal including the Coulomb interaction among and between electrons and lattice-ions. The Ginzburg-Landau wave function [Psi][sub [sigma]](r), where [sigma] represents the superconducting pairon (Cooper-pair) state, is shown to be connected with the one-pairon density operator n by [Psi][sub [sigma]](r) = [l angle]r[vert bar]n[sup 1/2][vert bar][sigma][r angle]. A close analogy between supercurrent and laser is indicated. 31 refs.

Theory of the proximity effect in [formula omitted] magnetic field I

The Proximity Effect phenomenon in non-zero parallel magnetic field can be well explained by solving the complete L-G equation, including the cubic term. The L-G equation becomes the equation of the one-dimensional harmonic-oscillator. The energy eigenvalue equation naturally produces an explicit functional relation between the upper critical magnetic field H and the transition temperature T over a wide range of T near T c: T=T c{1− 4ed sξ 2 sB C n [ (k y− 2ed sB n ) 2+ 2eB n +(k y− 2ed sB n )]} Where B=μ oH, T c = T s for a single superconductor layer, T c = T csn for a S-N lamina, T c=T css for a S-S lamina, k y is the equivalent wave number, and C is a constant. The agreement between the theory and the experimental results is very good.

Three-fermion relativistic bound states and cluster decomposition

We write a system ofN coupled compatible Dirac equations. The energy eigenvalue equation is obtained by simply adding the free Hamiltonians and the mutual interaction potentials. We write also a system of three coupled compatible Dirac equations, which is more complicated but completely separable (we obtain three free equations if we «switch off» the three mutual interactions, but we obtain also two coupled and one free equations if we «switch off» two of the three mutual interactions only). As an example, we present a three-fermion symmetric relativistic harmonic-oscillator model.

A single equation for finite rectangular well energy eigenvalues

A single equation for finite rectangular well energy eigenvalues