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2010, arXiv: Mathematical Physics
Exploring the idea that equation for radial wave function must be compatible with the full Schrodinger equation, a boundary condition () 0 u 0 = is derived.
We show that equation for radial wave function in its traditional form is compatible with the full Schrodinger equation if and only if a definite additional constraint required. This constraint has a boundary condition form at the origin. Some of consequences are also discussed. Comment: 6 pages
American Journal of Physics, 2011
There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.
Open Physics, 2007
By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial Schrödinger equation for the pseudoharmonic and the Kratzer potentials in two dimensions. The bound-state solutions are easily calculated from this eigenfunction ansatz. The corresponding normalized wavefunctions are also obtained.
Computers & Mathematics with Applications, 2010
We propose a method of numerical integration of differential equations of the type x 2 y ′′ + f (x)y = 0 by approximating its solution with solutions of equations of the type x 2 y ′′ + (ax 2 + bx + c)y = 0. This approximation is performed by segmentary approximation on an interval. We apply the method to obtain approximate solutions of the radial Schrödinger equation on a given interval and test it for two different potentials. We conclude that our method gives a similar accuracy than the Taylor method of higher order.
Journal of Computational and Applied Mathematics, 1989
An algorithm previously introduced by Brown et al. (1963) for solving radial Schrodinger equations is revisited and implemented in a more accurate way. The method is firstly applied to equations where potentials are present which are finite at the origin and which have an asymptotic behaviour V(r) + 0 as r + 00. Typical examples of potentials belonging to that class are the Woods-Saxon and the Morse potential. Furthermore the method is also used for Coulomb-like potentials such as the Hulthen and the Hellmann potential. A comparison between the approximated numerical values and other available numerical and exact bound state energies is made.
1994
We present an inverse scattering construction of generalised point interactions (GPI) – point-like objects with non-trivial scattering behaviour. The construction is developed for single centre S-wave GPI models with rational S-matrices, and starts from an integral transform suggested by the scattering data. The theory of unitary dilations is then applied to construct a unitary mapping between Pontryagin spaces which extend the usual position and momentum Hilbert spaces. The GPI Hamiltonian is defined as a multiplication operator on the momentum Pontryagin space and its free parameters are fixed by a physical locality requirement. We determine the spectral properties and domain of the Hamiltonian in general, and construct the resolvent and Møller wave operators thus verifying that the Hamiltonian exhibits the required scattering behaviour. The physical Hilbert space is identified. The construction is illustrated by GPI models representing the effective range approximation. For negat...
Asian Journal of Physical and Chemical Sciences, 2020
In this paper, we applied the semi-classical quantization approximation method to solve the radial Schrödinger equation with a generalized Pseudoharmonic potential. The four turning points problem within the framework of the Wentzel-Kramers-Brillouin (WKB) method was transformed into two turning points and subsequently, the energy spectrum was obtained. Some special cases of the generalized Pseudoharmonic potential are presented. The WKB approximation approach reproduces the exact energy expression obtained with several analytical methods in the literature. The values of the energy levels for some selected diatomic molecules (N2, CO, NO, CH) obtained numerically are in excellent agreement with those from previous works in the literature
Differential and Integral Equations
In this note we complete the study of [3], where we got existence results for the quasilinear elliptic equation −∆w + V (|x|) w − w ∆w 2 = K(|x|)g(w) in R N , with singular or vanishing continuous radial potentials V (r), K(r). In [3] we assumed, for technical reasons, that K(r) was vanishing as r → 0, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables w = f (u) and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in R N \ {0}. The nonlinearity g has a double-power behavior, whose standard example is g(t) = min{t q 1 −1 , t q 2 −1 } (t > 0), recovering the usual case of a singlepower behavior when q1 = q2.
Uzhhorod University Scientific Herald. Series Physics. Issue45., 2019
Purpose. The authors of known for us textbooks on quantum mechanics pay attention only to the first regular solution of the Schrödinger equation for the hydrogen atom. To exclude the second linearly independent solution from the general solution, different textbooks give various arguments such as invalid boundary conditions in the coordinate origin, the appearance of Dirac delta function, or divergence of the kinetic energy in the origin. Methods. Using the power series method, we obtained an exact analytic expression for the second independent solution of the Schrödinger equation for the hydrogen atom. Results. The solution consists of a sum of two parts, one of which increases indefinitely over long distances, while the other is limited and contains a logarithmic term. This feature is peculiar to all values of the orbital angular momentums. Conclusions. On the example of the hydrogen atom, we demonstrated the mathematically correct algorithm of the construction of the independent solutions for the power series method. In particular, this algorithm is important in the case of quantum systems with coupled channels which are described by two or more coupled Schrödinger equations.
We obtain the extra delta-like singularity while reduction of the Laplace operator in spherical coordinates, elimination of which restricts the radial wave functions at the origin. This restriction has the form of boundary condition for the radial wave function. Comment: 7 pages
2011
By careful exploration of separation of variables into the Laplacian in spherical coordinates, we obtain the extra delta-like singularity, elimination of which restricts the radial wave function at the origin. This constraint has the form of boundary condition for the radial Schrodinger equation.
2022
The problem of boundary behavior at the origin of coordinates is discussed for D-dimensional Schrodinger equation in the framework of hyperspherical formalism, which have been often considered last time. We show that the naive (Dirichlet) condition, which seems as natural, is not mathematically well justified, on the contrary to the 3-dimensional case. The stronger argument in favor of Dirichlet boundary condition is the requirement of time independence of wave function's norm. The problem remains open for singular potentials.
Careful exploration of the idea that equation for radial wave function must be compatible with the full Schrodinger equation shows appearance of the delta-function while reduction of full Schrodinger equation in spherical coordinates. Elimination of this extra term produces a boundary condition for the radial wave function, which is the same both for regular and singular potentials. Comment: 5 pages
Journal of Computational Physics, 1997
We shall assume here that V(r) is continuous on (0, ȍ) and has the following behavior at the endpoints: it tends A new approach to the numerical solution of boundary value problems for differential equations, which originated in recent pa-to zero as fast or faster than 1/r 2 , as r Ǟ ȍ, and as r Ǟ 0 pers by Greengard and Rokhlin, is improved and adapted to the it does not grow faster than 1/r. Most of the physically numerical solution of the radial Schrö dinger equation. The approach meaningful potentials satisfy these conditions. The Couis based on the conversion of the differential equation into an intelomb potential is an exception, but it can also be handled gral equation together with the application of a spectral type Clenby the method described here. Under these conditions on shaw-Curtis quadrature method. Through numerical examples, the integral equation method is shown to be superior to finite difference V(r), the differential equation (1.1a) is of a limit-point type methods. ᮊ 1997 Academic Press (Coddington and Levinson [1, p. 256]) and the initial value problem (1.1a) and (1.1b) has a bounded solution on (0, ȍ) whose asymptotic behavior depends on normalization.
2002
In this work the Schrödinger equation of the hydrogen-like atom is analytically solved. Three sets of analytical solution are obtained if the factor r −l is not neglected. The first solution is the same as the traditional radial wave function; another one diverges; the last one is far different from the traditional solution. On the consideration of the finite size of the nucleus, the third wave function does not diverge while r approaches to zero. Its radial wave function has below characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nucleus by comparison with that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally.
Physics Letters A, 1991
Using an ansatz forthe eigenfunction, we have obtained an exact solution ofthe radial Schrodinger equation for inverse-power potentials in three dimensions.
International Journal of Theoretical Physics, 2009
The Schrödinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schrödinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.
Journal of Physics B: Atomic, Molecular and Optical Physics, 2007
We present the exact and iterative solutions of the radial Schrödinger equation for a class of potential, V (r) = A r 2 − B r + Cr κ , for various values of κ from -2 to 2, for any n and l quantum states by applying the asymptotic iteration method. The global analysis of this potential family by using the asymptotic iteration method results in exact analytical solutions for the values of κ = 0, −1 and −2. Nevertheless, there are no analytical solutions for the cases κ = 1 and 2. Therefore, the energy eigenvalues are obtained numerically. Our results are in excellent agreement with the previous works. PACS numbers: 03.65.Ge Keywords: asymptotic iteration method, eigenvalues and eigenfunctions, Kratzer, Modified Kratzer, Goldman-Krivchenkov, spiked harmonic oscillator, Coulomb plus linear and Coulomb plus harmonic oscillator potentials.
Pramana, 2014
Approximate solutions to the N-dimensional radial Schrödinger equation for the potential ar 2 + br − c/r are obtained by employing the formulation described in Ciftci et al, J. Phys. A 43, 415206 (2010). The problem, for some special cases, is solved numerically. Using this analysis, the energy spectra of a two-dimensional two-electron quantum dot (QD) in a magnetic field are also obtained. The results of this study are in good agreement with the other studies.
African Journal of Physics, 2020
In this paper, we used the WKB method to solve the radial Schrodinger equation for the Cornell potential (linear plus coulomb potential). The WKB method yields complex eigenvalue equations. We applied approximation schemes to obtain eigenenergies and mass spectrum of heavy mesons numerically. The eigenvalues and mass spectra fluctuations strongly depend on the values of the linear and coulomb potential parameters. The results obtained in this paper are in good agreement with the results obtained by other theoretical calculations and available experimental data.
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