Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Le Journal de Physique Colloques

GROUP PROPERTIES OF RADIAL WAVEFUNCTIONS

L. L. ARMSTRONG, JR.

Published Nov 1, 1970• 7 citations
Publisher

Abstract

The non-compact group O (2, 1) is used in an investigation of hydrogenic radial wavefunctions. These radial functions are shown to form bases for infinite dimensional representations of the algebra of O (2, 1). It is found that rN and r-N (N positive) transform as tensors with respect to this algebra. Analysis of matrix elements of rN and r-N shows that the Wigner-Eckart theorem is valid for this group and that the pertinent Clebsch-Gordan coefficients are proportional to the familiar R(3) Clebsch-Gordan coefficients. This proportionality provides simple explanations of the selection rules for hydrogenic radial matrix elements noted by Pasternack and Sternheimer, and the proportionality of hydrogenic expectation values of rN and r-N to 3 - j symbols.

DOI
10.1051/jphyscol:1970403

Related papers

Radial matrix elements and the angular momentum technique

Dmitry A. Varshalovich, A. V. Karpova

For hydrogen wave functions, it is shown that radial matrix elements of type 〈nl|r k |nl′〉 can be expressed in terms of the Clebsch-Gordan coefficients, which characterize the relation between angular momenta l and l′, but in a nonphysical region of their arguments.

Optics and Spectroscopy•Jan 1, 2015•4 citations

Expression of Relativistic Expectation Values of Powers of r in Terms of Clebsch–Gordan Coefficients

Jean‐Christophe Pain

It is shown that, in the relativistic hydrogenic approximation, expectation values of the type $$\left\langle {n\ell j} \right|{{r}^{k}}\left| {n\ell j} \right\rangle $$ can be expressed in terms of Clebsch–Gordan coefficients or $$3jm$$ symbols. This generalizes the results obtained by Varshalovich and Karpova [Opt. Spectrosc. 118, 1–5 (2015); Opt. Spektrosk. 118, 3–7 (2015)] in the non-relativistic hydrogenic case.

arXiv:2003.14197•Optics and Spectroscopy•Aug 1, 2020•3 citations

Expression of relativistic expectation values of powers of r in terms of Clebsch-Gordan coefficients-=SUP=-*-=/SUP=-

Jean‐Christophe Pain

It is shown that, in the relativistic hydrogenic approximation, expectation values of the type < nlj|rk|nlj> can be expressed in terms of Clebsch-Gordan coefficients or 3jm symbols. This generalizes the results obtained by Varshalovich and Karpova [Opt. Spectrosc. 118, 1-5 (2015); Opt. Spektrosk. 118, 3-7 (2015)] in the non-relativistic hydrogenic case. Keywords: relativistic hydrogenic approximation, expectation values of powers of r. Clebsch-Gordan coefficents.

Журнал технической физики•Jan 1, 2020•0 citations

Stark states and O(4) symmetry of hydrogenic atoms

J. W. B. Hughes

The O(4) symmentry of hydrogenic atoms is used to interpret angular momentum nlm and Stark [vertical line] n 1 n 2 m eigenstates as different realizations of irreducible representations of the O(4) group Differential expressions are obtained for [vertical line] n 1 n 2 m shift operators, and for the [vertical line] nlm shift operator O - 1 which changes l and m by -1 These are used to obtain the formula < n 1 n 2 m | nlm '> = (-1) 2 n 2 +| m |- m 2 C [1/2( n -1), 1/2( n -1), l , 1/2( m + n 2 - n 1 ), 1/2( m + n 1 - n 2 ), m '] for the transformation matrix elements connecting the two sets of states where, in the notation of Rose, C ( J 1 J 2 J , m 1 m 2 m ) is the angular momentum Clebsch-Gordan coefficient

Proceedings of the Physical Society•Aug 1, 1967•91 citations

Two-dimensional harmonic oscillator matrix elements, hyperbolic angular momentum and spherical harmonics

Wolfgang Witschel

Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice d mm' j ( theta ) and which are written as t mm jj' ( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.

Journal of Physics A Mathematical Nuclear and General•Oct 11, 1974•11 citations

O(3,1) symmetry of the hydrogen atom

S. Datta Majumdar, D Basu

The transformation coefficients connecting the Stark and the angular momentum states belonging to the positive spectrum of the hydrogen atom are calculated by a group- theoretical method which makes use of the O(3,1) symmetry of the states and also by a purely analytical method. The results of the two calculations agree except for an undetermined factor not containing the orbital quantum number l. Complete agreement between the results is achieved by taking the normalization factors for the continuum states to be analytic continuations of those for the bound states. The transformation coefficients turn out to be SU(2) Clebsch-Gordan coefficients (j 1 j 2 l; m 1 m 2 m) with complex j 1 j 2 m 1 m 2 , and physical l,m. From the general theory the well known expansion of the Coulomb scattering function is obtained by giving the magnetic quantum number m and one of the electric quantum numbers n 2 , the special values, 0 and -1, respectively.

Journal of Physics A Mathematical Nuclear and General•May 1, 1974•7 citations
The non-compact group O (2, 1) is used in an investigation of hydrogenic radial wavefunctions. These radial functions are shown to form bases for infinite dimensional representations of the algebra of O (2, 1). It is found that rN and r-N (N positive) transform as tensors with respect to this algebra. Analysis of matrix elements of rN and r-N shows that the Wigner-Eckart theorem is valid for this group and that the pertinent Clebsch-Gordan coefficients are proportional to the familiar R(3) Clebsch-Gordan coefficients. This proportionality provides simple explanations of the selection rules for hydrogenic radial matrix elements noted by Pasternack and Sternheimer, and the proportionality of hydrogenic expectation values of rN and r-N to 3 - j symbols.
It is shown that, in the relativistic hydrogenic approximation, expectation values of the type ⟨nℓj∣rk∣nℓj⟩\left\langle {n\ell j} \right|{{r}^{k}}\left| {n\ell j} \right\rangle ⟨nℓj∣rk∣nℓj⟩ can be expressed in terms of Clebsch–Gordan coefficients or 3jm3jm3jm symbols. This generalizes the results obtained by Varshalovich and Karpova [Opt. Spectrosc. 118, 1–5 (2015); Opt. Spektrosk. 118, 3–7 (2015)] in the non-relativistic hydrogenic case.
The O(4) symmentry of hydrogenic atoms is used to interpret angular momentum nlm and Stark [vertical line] n 1 n 2 m eigenstates as different realizations of irreducible representations of the O(4) group Differential expressions are obtained for [vertical line] n 1 n 2 m shift operators, and for the [vertical line] nlm shift operator O - 1 which changes l and m by -1 These are used to obtain the formula < n 1 n 2 m | nlm '> = (-1) 2 n 2 +| m |- m 2 C [1/2( n -1), 1/2( n -1), l , 1/2( m + n 2 - n 1 ), 1/2( m + n 1 - n 2 ), m '] for the transformation matrix elements connecting the two sets of states where, in the notation of Rose, C ( J 1 J 2 J , m 1 m 2 m ) is the angular momentum Clebsch-Gordan coefficient
Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice d mm' j ( theta ) and which are written as t mm jj' ( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.
The transformation coefficients connecting the Stark and the angular momentum states belonging to the positive spectrum of the hydrogen atom are calculated by a group- theoretical method which makes use of the O(3,1) symmetry of the states and also by a purely analytical method. The results of the two calculations agree except for an undetermined factor not containing the orbital quantum number l. Complete agreement between the results is achieved by taking the normalization factors for the continuum states to be analytic continuations of those for the bound states. The transformation coefficients turn out to be SU(2) Clebsch-Gordan coefficients (j 1 j 2 l; m 1 m 2 m) with complex j 1 j 2 m 1 m 2 , and physical l,m. From the general theory the well known expansion of the Coulomb scattering function is obtained by giving the magnetic quantum number m and one of the electric quantum numbers n 2 , the special values, 0 and -1, respectively.