ABSTRACT The relativistic equations of motion are completely solved for a special configuration o... more ABSTRACT The relativistic equations of motion are completely solved for a special configuration of the external magnetic field, a case which seems to have never been considered before. It is also shown how the quaternionic formalism can replace the traditional Lagrange formalism in that kind of problem. The characteristics of the movement are briefly discussed in all possible cases.
Resume Nous indiquous comment le formalisme quaternionique permet de demontrer simplement que l&#... more Resume Nous indiquous comment le formalisme quaternionique permet de demontrer simplement que l'ensemble des etats propres de l'atome d'hydrogene relativiste est complet. Il semble que cette precision faisait defaut jusqu'a present.
Resume Nous montrons comment l'emploi des quaternions permet l'integration directe par se... more Resume Nous montrons comment l'emploi des quaternions permet l'integration directe par separation des variables de systemes d'equations aux derivees partielles. Le traitement des equations de Maxwell et de Dirac en sort consideralement simplifie et uniformise. A titre d'exemple, nous appliquons cette methode a l'atome hydrogenoide de Dirac et aux regles de selections correspondantes.
Journal of Physics A: Mathematical and General, 1975
Fourier expansions in modified Bessel functions are evaluated in terms of elementary functions. T... more Fourier expansions in modified Bessel functions are evaluated in terms of elementary functions. The results are obtained in the form of rapidly convergent series so that numerical computations are quite easy. This leads to the possibility of tabulating the modified Bessel functions in their critical domain. A physical application is also given: simple expansions giving the Madelung constant for cubic crystallographic structures are established; they involve only elementary functions and exhibit the greatest known accuracy.
A new method is presented which sums certain slowly convergent series. It is based on the use of ... more A new method is presented which sums certain slowly convergent series. It is based on the use of the Hankel integral transform and Schlömilch series. This method is applied with great success to the computation of lattice sums in ionic crystals. In particular, the Madelung constant is calculated with great accuracy through rather simple calculations: The final results only involve elementary functions so that the numerical evaluation is quite easy.
We study the problem of the motion of a charged particle in radial magnetic induction fields of t... more We study the problem of the motion of a charged particle in radial magnetic induction fields of the type B = h(θ)r/r3. A Coulomb electric field is also added. The mechanics of Newton, Einstein, Schrödinger, and Dirac are successively considered in view of exact solutions. The analogies between the four treatments are emphasized.
Dirac's equations are completely solved for four entirely new configurations of the external ... more Dirac's equations are completely solved for four entirely new configurations of the external magnetic field. The use of the quaternionic formalism simplifies the problem of separation of variables. For three of those new magnetic fields the radial equations just obtained are not classical. Section II studies them and gives their solutions in term of usual transcendental functions. When energy is quantized, energy levels are given.
We study the problem of the motion of a charged particle in noncentral potentials of the type f(θ... more We study the problem of the motion of a charged particle in noncentral potentials of the type f(θ)/r2 + V(r). Newton's and Schrödinger's mechanics are considered. Exact solutions exist if V(r)=−H/r or Kr2 (i.e., Coulomb or harmonic oscillator potentials) while f(θ) may have at least three different expressions as a function of θ if the problem is three-dimensional and seven expressions if it is two-dimensional. The classical trajectories are computed and the energy levels in the corresponding quantum problem are given. Analogies between the two treatments are discussed.
Note Application of Generalized Pad6 Approximants to the Special Function Evaluation Problem A ne... more Note Application of Generalized Pad6 Approximants to the Special Function Evaluation Problem A new approach to generalized Pad& approximants is presented. Its main interest lies in the fact that it avoids instability and/or round-off errors which occur in traditional algorithms for evaluating Pade approximants. Moreover, the approximants are readily obtained. Applications concerned with the tabulation of some transcendental functions in their critical domain (Weber and Bessel functions) illustrate the power of the method.
Abstract It is shown that the refined non-perturbative Hill's method solves in a very neat wa... more Abstract It is shown that the refined non-perturbative Hill's method solves in a very neat way the Schrodinger equation Φ″ + [ E − x 2 − λx 2 (1 + gx 2 ) ]Φ = 0 . The refinement of the standard Hill's method consists in finding a renormalized frequency to optimize the calculation. This method appears to be preferable to the variational one recently used by A. K. Mitra in the same context.
Journal of Computational and Applied Mathematics, 1979
The eigenfunctions of the one dimensional Schr6dinger equation q/" + [E V(x)]a/=0, where V(x) is ... more The eigenfunctions of the one dimensional Schr6dinger equation q/" + [E V(x)]a/=0, where V(x) is a polynomial, are represented by expansions of the form k~ Ck ~°k (6o x). The functions 0 Ck (6o, x) are chosen in such a way that recurrence relations hold for the coefficients c k • examples treated are D k (6o x) (Weber-Hermite functions), exp (-cox 2) x k, exp (-cxq) D k (6ox). From these recurrence relations, one considers an infinite bandmatrix whose finite square sections permit to solve approximately the original eigenproblem. It is then shown how a good choice of the parameter 6o may reduce dramatically the complexity of the computations, by a theoretical study of the relation holding between the error on an eigenvalue, the order of the matrix, and the value of 6o. The paper contains tables with 10 significant figures of the 30 first eigenvalues corresponding to V(x)-x 2m, m-2(1)7, and the 6 first eigenvalues corresponding to V(x) = x 2 + Xx 10 and x 2 + Lx 12 , ~= .01(.01).1(.1)1(1)10(10)100.
Abstract Schrodinger's equation is exactly soluble if one considers a central potential of th... more Abstract Schrodinger's equation is exactly soluble if one considers a central potential of the type Ar 2 + Br − D/r provided D takes particular well chosen values.
This paper investigates the convergence properties of continued fractions of PoincarC's type with... more This paper investigates the convergence properties of continued fractions of PoincarC's type with the aid of Pincherle's theorem, extends the investigation to the case where a tail is added to the approximants, and distinguishes between accelerative and decelerative tails. It shows that classical sequence transforms like 8 or Levin/u algorithms accelerate the sequence of the approximants modified by the optimal accelerative tail. A two-step procedure is deduced which performs the acceleration of the convergence of continued fractions of Poincar~s type. Lf the sequence cfk converges, then its limit is cf, = cf. The speed of the convergence is accurately described by the paraxeter p defined via the relative error (cf # 0) exp(-p)= )cfk-cf I/)cf 1. If Cf eqUdS zero, we may consider the absolute error eXp( -6) = 1 cfk -Cf I = 1 cfk I.
Volume 35A, number 2 PHYSICS LETTERS 17 May 1971 gaps at energy E, it is necessary that both v 1 ... more Volume 35A, number 2 PHYSICS LETTERS 17 May 1971 gaps at energy E, it is necessary that both v 1 In the case where the asymptotic form of and v2 be outside the interval (VL, vR); this con-Gc(71) isi/ij, there is in general only one soluclusion follows from the formula for the trace of tion and it is at large values of ii. For example, the Green function [6]:
ABSTRACT The relativistic equations of motion are completely solved for a special configuration o... more ABSTRACT The relativistic equations of motion are completely solved for a special configuration of the external magnetic field, a case which seems to have never been considered before. It is also shown how the quaternionic formalism can replace the traditional Lagrange formalism in that kind of problem. The characteristics of the movement are briefly discussed in all possible cases.
Resume Nous indiquous comment le formalisme quaternionique permet de demontrer simplement que l&#... more Resume Nous indiquous comment le formalisme quaternionique permet de demontrer simplement que l'ensemble des etats propres de l'atome d'hydrogene relativiste est complet. Il semble que cette precision faisait defaut jusqu'a present.
Resume Nous montrons comment l'emploi des quaternions permet l'integration directe par se... more Resume Nous montrons comment l'emploi des quaternions permet l'integration directe par separation des variables de systemes d'equations aux derivees partielles. Le traitement des equations de Maxwell et de Dirac en sort consideralement simplifie et uniformise. A titre d'exemple, nous appliquons cette methode a l'atome hydrogenoide de Dirac et aux regles de selections correspondantes.
Journal of Physics A: Mathematical and General, 1975
Fourier expansions in modified Bessel functions are evaluated in terms of elementary functions. T... more Fourier expansions in modified Bessel functions are evaluated in terms of elementary functions. The results are obtained in the form of rapidly convergent series so that numerical computations are quite easy. This leads to the possibility of tabulating the modified Bessel functions in their critical domain. A physical application is also given: simple expansions giving the Madelung constant for cubic crystallographic structures are established; they involve only elementary functions and exhibit the greatest known accuracy.
A new method is presented which sums certain slowly convergent series. It is based on the use of ... more A new method is presented which sums certain slowly convergent series. It is based on the use of the Hankel integral transform and Schlömilch series. This method is applied with great success to the computation of lattice sums in ionic crystals. In particular, the Madelung constant is calculated with great accuracy through rather simple calculations: The final results only involve elementary functions so that the numerical evaluation is quite easy.
We study the problem of the motion of a charged particle in radial magnetic induction fields of t... more We study the problem of the motion of a charged particle in radial magnetic induction fields of the type B = h(θ)r/r3. A Coulomb electric field is also added. The mechanics of Newton, Einstein, Schrödinger, and Dirac are successively considered in view of exact solutions. The analogies between the four treatments are emphasized.
Dirac's equations are completely solved for four entirely new configurations of the external ... more Dirac's equations are completely solved for four entirely new configurations of the external magnetic field. The use of the quaternionic formalism simplifies the problem of separation of variables. For three of those new magnetic fields the radial equations just obtained are not classical. Section II studies them and gives their solutions in term of usual transcendental functions. When energy is quantized, energy levels are given.
We study the problem of the motion of a charged particle in noncentral potentials of the type f(θ... more We study the problem of the motion of a charged particle in noncentral potentials of the type f(θ)/r2 + V(r). Newton's and Schrödinger's mechanics are considered. Exact solutions exist if V(r)=−H/r or Kr2 (i.e., Coulomb or harmonic oscillator potentials) while f(θ) may have at least three different expressions as a function of θ if the problem is three-dimensional and seven expressions if it is two-dimensional. The classical trajectories are computed and the energy levels in the corresponding quantum problem are given. Analogies between the two treatments are discussed.
Note Application of Generalized Pad6 Approximants to the Special Function Evaluation Problem A ne... more Note Application of Generalized Pad6 Approximants to the Special Function Evaluation Problem A new approach to generalized Pad& approximants is presented. Its main interest lies in the fact that it avoids instability and/or round-off errors which occur in traditional algorithms for evaluating Pade approximants. Moreover, the approximants are readily obtained. Applications concerned with the tabulation of some transcendental functions in their critical domain (Weber and Bessel functions) illustrate the power of the method.
Abstract It is shown that the refined non-perturbative Hill's method solves in a very neat wa... more Abstract It is shown that the refined non-perturbative Hill's method solves in a very neat way the Schrodinger equation Φ″ + [ E − x 2 − λx 2 (1 + gx 2 ) ]Φ = 0 . The refinement of the standard Hill's method consists in finding a renormalized frequency to optimize the calculation. This method appears to be preferable to the variational one recently used by A. K. Mitra in the same context.
Journal of Computational and Applied Mathematics, 1979
The eigenfunctions of the one dimensional Schr6dinger equation q/" + [E V(x)]a/=0, where V(x) is ... more The eigenfunctions of the one dimensional Schr6dinger equation q/" + [E V(x)]a/=0, where V(x) is a polynomial, are represented by expansions of the form k~ Ck ~°k (6o x). The functions 0 Ck (6o, x) are chosen in such a way that recurrence relations hold for the coefficients c k • examples treated are D k (6o x) (Weber-Hermite functions), exp (-cox 2) x k, exp (-cxq) D k (6ox). From these recurrence relations, one considers an infinite bandmatrix whose finite square sections permit to solve approximately the original eigenproblem. It is then shown how a good choice of the parameter 6o may reduce dramatically the complexity of the computations, by a theoretical study of the relation holding between the error on an eigenvalue, the order of the matrix, and the value of 6o. The paper contains tables with 10 significant figures of the 30 first eigenvalues corresponding to V(x)-x 2m, m-2(1)7, and the 6 first eigenvalues corresponding to V(x) = x 2 + Xx 10 and x 2 + Lx 12 , ~= .01(.01).1(.1)1(1)10(10)100.
Abstract Schrodinger's equation is exactly soluble if one considers a central potential of th... more Abstract Schrodinger's equation is exactly soluble if one considers a central potential of the type Ar 2 + Br − D/r provided D takes particular well chosen values.
This paper investigates the convergence properties of continued fractions of PoincarC's type with... more This paper investigates the convergence properties of continued fractions of PoincarC's type with the aid of Pincherle's theorem, extends the investigation to the case where a tail is added to the approximants, and distinguishes between accelerative and decelerative tails. It shows that classical sequence transforms like 8 or Levin/u algorithms accelerate the sequence of the approximants modified by the optimal accelerative tail. A two-step procedure is deduced which performs the acceleration of the convergence of continued fractions of Poincar~s type. Lf the sequence cfk converges, then its limit is cf, = cf. The speed of the convergence is accurately described by the paraxeter p defined via the relative error (cf # 0) exp(-p)= )cfk-cf I/)cf 1. If Cf eqUdS zero, we may consider the absolute error eXp( -6) = 1 cfk -Cf I = 1 cfk I.
Volume 35A, number 2 PHYSICS LETTERS 17 May 1971 gaps at energy E, it is necessary that both v 1 ... more Volume 35A, number 2 PHYSICS LETTERS 17 May 1971 gaps at energy E, it is necessary that both v 1 In the case where the asymptotic form of and v2 be outside the interval (VL, vR); this con-Gc(71) isi/ij, there is in general only one soluclusion follows from the formula for the trace of tion and it is at large values of ii. For example, the Green function [6]:
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Papers by Andre HAUTOT