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1
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Weakly Bound States in Heterogeneous Waveguides: A Calculation to Fourth Order

100%
Amore P.
Acta Physica Polonica A
|
2017
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vol. 132
|
issue 4
1351-1357
EN
We have extended a previous calculation of the energy of a weakly heterogeneous waveguide to fourth order in the density perturbation, deriving its general expression. For particular configurations where the second and third orders both vanish, we discover that the fourth order contribution lowers in general the energy of the state, below the threshold of the continuum. In these cases the waveguide possesses a localized state. We have applied our general formula to a solvable model with vanishing second and third orders reproducing the exact expression for the fourth order.
2
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Arbitrary l-state solutions of the Schrödinger equation for the Eckart potential with an improved approximation of the centrifugal term

100%
Stanek J.
Open Physics
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2011
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vol. 9
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issue 6
1503-1508
EN
Applying an improved approximation scheme to the centrifugal term, the approximate analytical solutions of the Schrödinger equation for the Eckart potential are presented. Bound state energy eigenvalues and the corresponding eigenfunctions are obtained in closed forms for the arbitrary radial and angular momentum quantum numbers, and different values of the screening parameter. The results are compared with those obtained by the other approximate and numerical methods. It is shown that the present method is systematic, more efficient and accurate.
3
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Threshold conditions and bound states for locally periodic delta potentials

100%
Dharani M., Sahu B., Shastry C.
Open Physics
|
2014
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vol. 12
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issue 10
755-766
EN
We present a systematic study of the conditions for the generation of threshold energy eigen states and also the energy spectrum generated by two types of locally periodic delta potentials each having the same strength λV and separation distance parameter a: (a) sum of N attractive potentials and (b) sum of pairs of attractive and repulsive potentials. Using the dimensionless parameter g = λV a in case (a) the values of g = g n, n = 1, 2, …, N at which threshold energy bound state gets generated are shown to be the roots of Nth order polynomial D 1(N, g) in g. We present an algebraic recursive procedure to evaluate the polynomial D 1(N, g) for any given N. This method obviates the need for the tedious mathematical analysis described in our earlier work to generate D 1(N, g). A similar study is presented for case (b). Using the properties of D 1(N, g) we establish that in case (a) the critical minimum value of g which guarantees the generation of the maximum possible number of bound states is g = 4. The corresponding result for case (b) is g = 2. A typical set of numerical results showing the pattern of variation of g n as a function of n and several interesting features of the energy spectrum for different values of g and N are also described.
4
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Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions

100%
Dharani M., Sahu B., Shastry C.
Open Physics
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2013
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vol. 11
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issue 8
995-1005
EN
This paper proves that for N attractive delta function potentials the number of bound states (Nb) satisfies 1 ≤ N b ≤ N in one dimension (1D), and is 0 ≤ N b ≤ N in three dimensions (3D). Algebraic equations are obtained to evaluate the bound states generated by N attractive delta potentials. In particular, in the case of N attractive delta function potentials having same separation a between adjacent wells and having the same strength λV, the parameter g=λVa governs the number of bound states. For a given N in the range 1–7, both in 1D and 3D cases the numerical values of gn, where n=1,2,..N are obtained. When g=gn, Nb ≤ n where Nb includes one threshold energy bound state. Furthermore, gn are the roots of the Nth order polynomial equations with integer coefficients. Based on our numerical calculations up to N=40, even when N becomes large, 0 ≤ g n ≤ 4 and $\frac{{\Sigma g_n }} {N} \simeq 2 $ and this result is expected to be generally valid. Thus, for g > 4 there will be no threshold or zero energy bound state, and if g≈ 2 for a given large N, the number of bound states will be approximately N/2. The empirical formula gn = 4/[1+exp((N 0 − n)/β)] gives a good description of the variation of gn as a function of n. This formula is useful in estimating the number of bound states for any N and g both in 1D and 3D cases.
5
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A comment on the analogous confinement of a neutral particle to a quantum dot via noninertial effects

88%
Bakke K.
Open Physics
|
2012
|
vol. 10
|
issue 5
1089-1094
EN
In this contribution, we discuss the nonrelativistic limit of the Dirac equation for a neutral particle with a permanent electric dipole moment interacting with external fields in a noninertial frame. We show a case where the geometry of the manifold can play the role of a hard-wall confining potential due to noninertial effects, and can yield bound states analogous to a confinement of the spin-half neutral particle interacting with external fields to a quantum dot described by a hard-wall confining potential [33].
6
Content available remote

On solutions of the Schrödinger equation for some molecular potentials: wave function ansatz

75%
Ikhdair S., Sever R.
Open Physics
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2008
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vol. 6
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issue 3
697-703
EN
Making an ansatz to the wave function, the exact solutions of the D-dimensional radial Schrödinger equation with some molecular potentials, such as pseudoharmonic and modified Kratzer, are obtained. Restrictions on the parameters of the given potential, δ and ν are also given, where η depends on a linear combination of the angular momentum quantum number ℓ and the spatial dimensions D and δ is a parameter in the ansatz to the wave function. On inserting D = 3, we find that the bound state eigensolutions recover their standard analytical forms in literature.
7
Content available remote

Path integral treatment of a noncentral electric potential

75%
Ghoumaid A., Benamira F., Guechi L., Khiat Z.
Open Physics
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2013
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vol. 11
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issue 1
78-88
EN
We present a rigorous path integral treatment of a dynamical system in the axially symmetric potential $V(r,\theta ) = V(r) + \tfrac{1} {{r^2 }}V(\theta ) $ . It is shown that the Green’s function can be calculated in spherical coordinate system for $V(\theta ) = \frac{{\hbar ^2 }} {{2\mu }}\frac{{\gamma + \beta \sin ^2 \theta + \alpha \sin ^4 \theta }} {{\sin ^2 \theta \cos ^2 \theta }} $ . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number l ≥ 0, are recovered.
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