On the hydrogen-oscillator connection: Passage formulas between wave functions

Kibler, M.; Ronveaux, A.; Négadi, Tidjani

doi:10.1063/1.527064

Cited by 43 publications

(30 citation statements)

References 40 publications

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“…The latter result provides us with the quantization, in polar coordinates, of the two-dimensional isotropic harmonic oscillator with an inverse square potential. The limiting case = y = 0 gives back the well-known result for the two-dimensional isotropic harmonic oscillator [44,45].…”

Section: Quantizing the Oscillatorsmentioning

confidence: 92%

Classical and quantum study of a generalized Kepler–Coulomb system

Kibler

Campigotto

1993

Int J of Quantum Chemistry

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The nonrelativistic potential V = a/r + p cos O/(r sin + y/(r sin 0)2, which constitutes an extension of the Kepler-Coulomb potential and of the Hartmann potential, is investigated from both a classical and a quantum mechanical viewpoint. The corresponding Schrodinger equation is solved in parabolic coordinates by using the Kustaanheimo-Stiefel transformation. The accidental degeneracy for the discrete spectrum is explained through the introduction of an su(2) dynamical invariance algebra. The Hamilton-Jacobi equation is solved in parabolic coordinates. The planarity and the periodicity of the trajectories are investigated. All finite trajectories are found to be quasi-periodic (rather than periodic as in the case of the Kepler-Coulomb system).

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Section: Quantizing the Oscillatorsmentioning

confidence: 92%

Classical and quantum study of a generalized Kepler–Coulomb system

Kibler

Campigotto

1993

Int J of Quantum Chemistry

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“…Therefore, we have recovered the well-known result that the KS-map leads to the Hopffiberin9 S 3 -* S 2 of the sphere S 3, result that, in the light of the general theory of space bundles [21], allows us to 'pre-quantize the symplectic manifold S 2', i.e., to construct a quantum principal SX-fiber bundle (S 3, S 2, S ~, rr) with projection rr given by the KS-map and quantum contact manifold given by the sphere S a endowed with a contact real 1-form defined by the left-hand side of the bilinear relation (6).…”

Section: (R" -{0}) X R" ~ (R a -{0}) X Rs: (U V) ~-~ (X Y)mentioning

confidence: 99%

“…In this perspective the Euler angles are considered as special variables, suited for particular mechanical problems such as the rotator problem. As an application we examine the quantistic mathematical connection of Ikeda and Miyachi and show that the Euler angles appear in a natural way; there is no need of 'ad hoc' devices nor of suitable rewritings up to S 4 permutations ( [5,6]). …”

Section: Introductionmentioning

confidence: 99%

On the connection among three classical mechanical problems via the hypercomplex KS-transformation

Vivarelli

1991

Celestial Mech Dyn Astr

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With reference to our previous notes [23][24][25][26][27][28][29][30] on the hypercomplex regularizations of the classical Kepler problem and on the hypercomplex descriptions of the classical rotational mechanics, we arrive at an intimate mathematical relationship among the three classical mechanical problems: the elliptic Kepler problem, the spherical Euler-Poinsot problem and the four-dimensional isotropic harmonic oscillator. A key role is played: (i) by the Kustaanheimo-Stiefel (KS) regularizing transformation viewed, in the light of Souriau's quantization procedure, as the projection map in the Hopf fibering of the contact 3-sphere; (ii) by a novel Lagrangian description of rotational kinematics expressed in terms of the Euler-Rodrigues parameters; (iii) by the unit vector a (characterizing both the attitude frame of the rotator and the direction of the major axis of the Kepler orbit) and shown to link 'fi la Cartan' the KS-theory with the rotation theory. The Kepler-Poinsot-Oscillator connection is examined also in a quantistic Schroedinger approach and is related to the results of Ikeda and Miyachi.

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“…This is somewhat surprising, because there are numerous fundamental and applied questions related to this problem, such as the linearization of products of basis-set functions associated to shape-invariant potentials [10], the transformation formulae between quantum mechanical wavefunctions in different coordinate systems (e.g. the bound-state wavefunctions on L 2 (R 3 ) in spherical and parabolic coordinates [22]), the interbasis expansions for potentials of equal [23] and different [24] dimensionality (e.g. the passage formulae from the R 3 hydrogen wavefunctions to R 4 oscillator wavefunctions [22]), the determination of the Talmi-Brody-Moshinsky coefficients [25] so widely used in nuclear structure, and the evaluation of two-centre, two-and three-electron integrals in variational atomic analysis [13].…”

Section: F [Y](x) = σ (X)y (X) + τ (X)y (X)mentioning

confidence: 99%

“…the bound-state wavefunctions on L 2 (R 3 ) in spherical and parabolic coordinates [22]), the interbasis expansions for potentials of equal [23] and different [24] dimensionality (e.g. the passage formulae from the R 3 hydrogen wavefunctions to R 4 oscillator wavefunctions [22]), the determination of the Talmi-Brody-Moshinsky coefficients [25] so widely used in nuclear structure, and the evaluation of two-centre, two-and three-electron integrals in variational atomic analysis [13].…”

Section: F [Y](x) = σ (X)y (X) + τ (X)y (X)mentioning

confidence: 99%

General linearization formulae for products of continuous hypergeometric-type polynomials

Sánchez-Ruiz

Artés

Martínez–Finkelshtein

et al. 1999

J. Phys. A: Math. Gen.

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Abstract. The linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials (HPs) of a continuous variable, which consists of the expansion of the product of two arbitrary HPs in series of an orthogonal HP set. Here, this problem is algebraically solved directly in terms of the coefficients of the second-order differential equations satisfied by the involved polynomials. General expressions for the expansion coefficients are given in integral form, and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis (Hermite, Laguerre and Jacobi), as well as several generalized linearization formulae involving these families. The connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions, which often reduce to a single function of the same type; when possible, these functions are evaluated in closed form. In some cases, sign properties of the coefficients such as positivity or non-negativity conditions are derived as a by-product from their resulting explicit representations.

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On the hydrogen-oscillator connection: Passage formulas between wave functions

Cited by 43 publications

References 40 publications

Classical and quantum study of a generalized Kepler–Coulomb system

Classical and quantum study of a generalized Kepler–Coulomb system

On the connection among three classical mechanical problems via the hypercomplex KS-transformation

General linearization formulae for products of continuous hypergeometric-type polynomials

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