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

Annals of Mathematics

On the Wave Equation of the Hydrogen Atom

T. H. Gronwall

Published Jan 1, 1931• 1 citation
Publisher

Abstract

and those values of the constant E (energy levels or characteristic constants) are to be determined for which there exist solutions of (1) (characteristic functions) which are single-valued and finite everywhere. SchrodingerI solves this problem by introducing polar coordinates r, 0, 'p, whereby the equation becomes separable, and writing i = x (r) it(0, ( ), where u (0, Sp) is any one of the surface harmonics of the unit sphere

DOI
10.2307/1968413

Related papers

On the Complete Form of Solving the Problem of the Static Schrodinger Equation for the Hydrogen Atom

Ibratjon X. Aliyev, Obbozjon Qo’ldashov, Karimberdi Egamberdiyvich Onarkulov +3 more

Springer proceedings in physics•Jan 1, 2026•0 citations

Hydrogen-like Atom and Atomic Orbitals

Caio Lima Firme

This chapter physical properties of the hydrogen atom and its Hamiltonian function as a particle in a spherically symmetric potential having a wave function as a product of the radial function and spherical harmonic function. Hereafter, it comes to two independent wave equations: the radial equation and the angular equation. After some derivation, it is shown that the angular equation is a product of the azimuthal angle equation and polar angle equation (as discussed in the previous chapter). The azimuthal angle equation is similar to the particle in a ring problem from the previous chapter and after some derivation the polar angle equation becomes an associated Legendre equation with well-known polynomials as solutions. Next, the solution to the radial equation is presented which becomes after some derivation an associated Laguerre equation with well-known polynomials as solutions and from this result is obtained the energy equation for the hydrogen-like atom. Subsequently, it is derived the normalization constant of the radial function. Afterwards, the numerical analysis of the radial equation of the hydrogen atom along with its corresponding Fortran code is presented. Finally, the hydrogen atomic orbitals equations and plots are depicted followed by the fine structure of the hydrogen (198 words).

CRC Press eBooks•Mar 21, 2022•0 citations

Hidden symmetry and explicit spheroidal eigenfunctions of the hydrogen atom

Stella M. Sung, Dudley R. Herschbach

The Schrödinger equation for a hydrogenic atom is separable in prolate spheroidal coordinates, as a consequence of the ‘‘hidden symmetry’’ stemming from the fixed spatial orientation of the classical Kepler orbits. One focus is at the nucleus and the other a distance R away along the major axis of the elliptic orbit. The separation constant α is not an elementary function of Z or R or quantum numbers. However, for given principal quantum number n and angular momentum projection m, the allowed values of α and corresponding eigenfunctions in spheroidal coordinates are readily obtained from a secular equation of order n−m. We evaluate α(n,m;ZR) and the coefficients gl(α) that specify the spheroidal eigenfunctions as hybrids of the familiar ‖nlm〉 hydrogen-atom states with fixed n and m but different l values. Explicit formulas and plots are given for α and gl and for the probability distributions derived from the hybrid wave functions, ∑lgl(α)‖nlm〉, for all states up through n=4. In the limit R→∞ these hybrids become the solutions in parabolic coordinates, determined simply by geometrical Clebsch–Gordan coefficients that account for conservation of angular momentum and the hidden symmetry. We also briefly discuss some applications of the spheroidal eigenfunctions, particularly to exact analytic solutions of two-center molecular orbitals for special values of R and the nuclear charge ratio Za/Zb.

The Journal of Chemical Physics•Nov 15, 1991•21 citations

Solution of the Schrödinger Equation for the Hydrogen Atom in Rectangular Coordinates

G. Richard Fowles

American Journal of Physics•Apr 1, 1962•6 citations

The Schrödinger Wave Equation and the Hydrogen Atom

Philip R. Wallace

Springer eBooks•Jan 1, 1996•2 citations

The hydrogen atom according to wave mechanics in cartesian coordinates

John F. Ogilvie

A partial separation of the variables is practicable for the solution of Schroedinger's temporally independent equation in cartesian coordinates x,y,z, which yields moderately simple algebraic formulae for the amplitude functions involving quantum numbers k,l,m, the same as in spherical polar coordinates. The properties of angular momentum are thus achieved with no angular variable. Several plots of surfaces of constant psi(x,y,z) are presented to illustrate the resemblance of the shapes of these surfaces to the shapes of surfaces of psi(r,theta,phi) with the corresponding quantum numbers.

arXiv:1709.04759•Aug 30, 2017•2 citations
and those values of the constant E (energy levels or characteristic constants) are to be determined for which there exist solutions of (1) (characteristic functions) which are single-valued and finite everywhere. SchrodingerI solves this problem by introducing polar coordinates r, 0, 'p, whereby the equation becomes separable, and writing i = x (r) it(0, ( ), where u (0, Sp) is any one of the surface harmonics of the unit sphere
This chapter physical properties of the hydrogen atom and its Hamiltonian function as a particle in a spherically symmetric potential having a wave function as a product of the radial function and spherical harmonic function. Hereafter, it comes to two independent wave equations: the radial equation and the angular equation. After some derivation, it is shown that the angular equation is a product of the azimuthal angle equation and polar angle equation (as discussed in the previous chapter). The azimuthal angle equation is similar to the particle in a ring problem from the previous chapter and after some derivation the polar angle equation becomes an associated Legendre equation with well-known polynomials as solutions. Next, the solution to the radial equation is presented which becomes after some derivation an associated Laguerre equation with well-known polynomials as solutions and from this result is obtained the energy equation for the hydrogen-like atom. Subsequently, it is derived the normalization constant of the radial function. Afterwards, the numerical analysis of the radial equation of the hydrogen atom along with its corresponding Fortran code is presented. Finally, the hydrogen atomic orbitals equations and plots are depicted followed by the fine structure of the hydrogen (198 words).
The Schrödinger equation for a hydrogenic atom is separable in prolate spheroidal coordinates, as a consequence of the ‘‘hidden symmetry’’ stemming from the fixed spatial orientation of the classical Kepler orbits. One focus is at the nucleus and the other a distance R away along the major axis of the elliptic orbit. The separation constant α is not an elementary function of Z or R or quantum numbers. However, for given principal quantum number n and angular momentum projection m, the allowed values of α and corresponding eigenfunctions in spheroidal coordinates are readily obtained from a secular equation of order n−m. We evaluate α(n,m;ZR) and the coefficients gl(α) that specify the spheroidal eigenfunctions as hybrids of the familiar ‖nlm〉 hydrogen-atom states with fixed n and m but different l values. Explicit formulas and plots are given for α and gl and for the probability distributions derived from the hybrid wave functions, ∑lgl(α)‖nlm〉, for all states up through n=4. In the limit R→∞ these hybrids become the solutions in parabolic coordinates, determined simply by geometrical Clebsch–Gordan coefficients that account for conservation of angular momentum and the hidden symmetry. We also briefly discuss some applications of the spheroidal eigenfunctions, particularly to exact analytic solutions of two-center molecular orbitals for special values of R and the nuclear charge ratio Za/Zb.
A partial separation of the variables is practicable for the solution of Schroedinger's temporally independent equation in cartesian coordinates x,y,z, which yields moderately simple algebraic formulae for the amplitude functions involving quantum numbers k,l,m, the same as in spherical polar coordinates. The properties of angular momentum are thus achieved with no angular variable. Several plots of surfaces of constant psi(x,y,z) are presented to illustrate the resemblance of the shapes of these surfaces to the shapes of surfaces of psi(r,theta,phi) with the corresponding quantum numbers.