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

European Journal of Physics

Realistic pictures of the electron density in eigenstates of the hydrogen atom

J. Dlouhá, Václav Frei, Václav Pihera

Published Jan 1, 1989• 1 citation
Publisher

Abstract

The probability density of selected stationary states (up to n=4) of the hydrogen atom is visualised by means of lines of equal probability in Cartesian coordinate planes. The graphs computed from eigenfunctions of the Schrodinger equation are plotted in a unique scale in order to give a quantitatively true picture of the eigenstates. A comparison with similar published pictures is outlined.

DOI
10.1088/0143-0807/10/1/005

Related papers

ANALYSIS OF WAVE FUNCTIONS FOR HYDROGEN ATOMS IN CLASSICAL LIMIT

HUANG XIANG-YOU, 北京大学物理系,北京,100871;中国高等科学技术中心(世界实验室)

Basing on the classical trajectory motion for an electron circling an atomic nucleus and averaging the motion over initial coordinate parameters of ellipse trajectories for the electron, we get a density distribution in space. This distribution is consistent with probability density of stationary state for hydrogen atom in the classical limit. This comparison shows that a wave function of hydrogen atom in stationary state does not describe an individual atom but an ensemble.

Acta Physica Sinica•Jan 1, 1991•1 citation

The hydrogen atom according to wave mechanics, parts I, II, III, IV, V

John F. Ogilvie

Based on an exposition of the underlying physics and applied mathematics in arXiv:1603.00899, this paper in five separate parts presents a description of the properties of the amplitude functions of the hydrogen atom according to wave mechanics in the coordinate representation in all four systems of coordinates in which Schroedinger's equations are separable, explicitly I spherical polar coordinates, II paraboloidal coordinates, III ellipsoidal coordinates and IV spheroconical coordinates; part V discusses the implications of these multiple systems in which these atomic orbitals can be directly calculated. Most published plots of orbitals are inaccurate. In particular, parts I - IV contain many accurate plots of the surfaces of amplitude functions in the various systems with a common criterion for the amplitude at which the surface is constructed. Although the plots are generated from amplitude functions in the specified systems of coordinates, the plots are translated to cartesian coordinates that the eye can recognise. The surfaces of the amplitude functions in ellipsoidal and spheroconical functions are previously unreported, with the accurate surfaces in spherical polar and paraboloidal coordinates for a direct comparison.

arXiv:1612.05098•Nov 29, 2016•0 citations

Pictorial Representations of the Dirac Electron Cloud for Hydrogen-Like Atoms

Harvey E. White

Previous attempts to represent graphically the Schrodinger electron cloud for hydrogen-like atoms have been so successful that similar graphical representations have been looked for on the Dirac theory. With a single electron specified on Dirac's theory by a set of four wave functions instead of one, the probability density $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}={\ensuremath{\psi}}_{I}{{\ensuremath{\psi}}_{I}}^{*}+{\ensuremath{\psi}}_{2}{{\ensuremath{\psi}}_{2}}^{*}+{\ensuremath{\psi}}_{3}{{\ensuremath{\psi}}_{3}}^{*}+{\ensuremath{\psi}}_{4}{{\ensuremath{\psi}}_{4}}^{*}$ is treated by considering, as on the Schrodinger theory, the independent angular and radial factors separately. Tables and graphs are given for $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of $\ensuremath{\theta}$ alone. The graphs are compared with and shown to correspond closely to the space quantization of the classical electron orbits. Graphs representative of $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of $r$ alone are given and compared with the electronnucleus distances of the corresponding electron orbits. The radial and angular factors are brought together by means of a mechanical device described previously, which when photographed represents closely the electron-cloud $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$. Photographs are given for the states $1^{2}S_{\frac{1}{2}}$, $2^{2}P_{\frac{1}{2}, \frac{3}{2}}$, $3^{2}D_{\frac{3}{2}, \frac{5}{2}}$ $4^{2}F_{\frac{5}{2}, \frac{7}{2}}$, and $5^{2}G_{\frac{7}{2}}$, with $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$

Physical Review•Aug 1, 1931•34 citations

Wavefunction localisation for the hydrogen atom in parallel electric and magnetic fields

Saverio Bivona, Wolfgang Schweizer, Patrick F. O’Mahony +1 more

A quantum mechanical study of a simple two-dimensional problem identifies lines in coordinate space about which the probability densities are localised. These lines in general do not correspond to extrema in the system's static potential energy surface. The subject of this study has been the hydrogen atom in externally applied parallel electric and magnetic and magnetic fields. Probability density plots of eigenstates, in which the electron experiences comparable influences from all three fields: Coulomb, electric and magnetic, are each found to display a localisation at a polar angle related to an invariant previously obtained in a classical study of the problem. The line about which localisation occurs is found to vary with energy and is in coincidence with a periodic trajectory found within a classical framework.

Journal of Physics B Atomic Molecular and Optical Physics•Oct 28, 1988•11 citations

The Momentum Distribution in Hydrogen-Like Atoms

Boris Podolsky, Linus Carl Pauling

The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})$, in which $P$, $\ensuremath{\Theta}$, and $\ensuremath{\Phi}$ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates $r$, $\ensuremath{\theta}$, and $\ensuremath{\varphi}$ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})=\left\{\frac{1}{{(2\ensuremath{\pi})}^{\frac{1}{2}}}{e}^{\ifmmode\pm\else\textpm\fi{}\mathrm{im}\ensuremath{\Phi}}\right\} \left\{{\left(\frac{(2l+1)(l\ensuremath{-}m)!}{2(l+m)!}\right)}^{\frac{1}{2}}{{P}_{l}}^{m}(cos\ensuremath{\Theta})\right\}$ $\left\{\frac{\ensuremath{\pi}{2}^{2l+4}l!}{{(\ensuremath{\gamma}h)}^{\frac{3}{2}}}{\left(\frac{n(n\ensuremath{-}l\ensuremath{-}1)!}{(n+l)!}\right)}^{\frac{1}{2}}\frac{{\ensuremath{\zeta}}^{l}}{{({\ensuremath{\zeta}}^{2}+1)}^{l+2}}{C}_{n\ensuremath{-}l\ensuremath{-}1}^{l+1}\left(\frac{{\ensuremath{\zeta}}^{2}\ensuremath{-}1}{{\ensuremath{\zeta}}^{2}+1}\right)\right\}$ in which $\ensuremath{\zeta}=(\frac{2\ensuremath{\pi}}{\ensuremath{\gamma}h})P$, with $\ensuremath{\gamma}=(\frac{4{\ensuremath{\pi}}^{2}\ensuremath{\mu}{e}^{2}Z}{n{h}^{2}})=(\frac{Z}{n{a}_{0}})$. The probability ${\ensuremath{\Xi}}_{\mathrm{nl}}(P)\mathrm{dP}$ that the electron have a total momentum lying within the limits $P$ and $P+dP$ is also evaluated, and it is shown that the root mean square of the total momentum is equal to the momentum of the electron in a circular Bohr orbit with the same total quantum number.

Physical Review•Jul 1, 1929•262 citations

A POINT-LIKE PICTURE OF THE HYDROGEN ATOM

Farshad Faghihi, A. JANGJOO, M. KHANI

A point-like picture of the Schrödinger solution for hydrogen atom is worked to emphasize that "point-like particles" may describe as "probability wave function". In each case, the three-dimensional shape of the |Ψnlm(rn, cos θ)|2 is plotted and the paths of the point-like electron (it is better to say reduced mass of the pair particles) are described in each closed shell. Finally, the orbital shape of the molecules are given according to the present simple model. In our opinion, "interpretations of the Correspondence Principle", which is a basic principle in all elementary quantum text, seems to be reviewed again!

International Journal of Modern Physics C•Dec 1, 2004•0 citations
The probability density of selected stationary states (up to n=4) of the hydrogen atom is visualised by means of lines of equal probability in Cartesian coordinate planes. The graphs computed from eigenfunctions of the Schrodinger equation are plotted in a unique scale in order to give a quantitatively true picture of the eigenstates. A comparison with similar published pictures is outlined.
Based on an exposition of the underlying physics and applied mathematics in arXiv:1603.00899, this paper in five separate parts presents a description of the properties of the amplitude functions of the hydrogen atom according to wave mechanics in the coordinate representation in all four systems of coordinates in which Schroedinger's equations are separable, explicitly I spherical polar coordinates, II paraboloidal coordinates, III ellipsoidal coordinates and IV spheroconical coordinates; part V discusses the implications of these multiple systems in which these atomic orbitals can be directly calculated. Most published plots of orbitals are inaccurate. In particular, parts I - IV contain many accurate plots of the surfaces of amplitude functions in the various systems with a common criterion for the amplitude at which the surface is constructed. Although the plots are generated from amplitude functions in the specified systems of coordinates, the plots are translated to cartesian coordinates that the eye can recognise. The surfaces of the amplitude functions in ellipsoidal and spheroconical functions are previously unreported, with the accurate surfaces in spherical polar and paraboloidal coordinates for a direct comparison.
Previous attempts to represent graphically the Schrodinger electron cloud for hydrogen-like atoms have been so successful that similar graphical representations have been looked for on the Dirac theory. With a single electron specified on Dirac's theory by a set of four wave functions instead of one, the probability density $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}={\ensuremath{\psi}}_{I}{{\ensuremath{\psi}}_{I}}^{*}+{\ensuremath{\psi}}_{2}{{\ensuremath{\psi}}_{2}}^{*}+{\ensuremath{\psi}}_{3}{{\ensuremath{\psi}}_{3}}^{*}+{\ensuremath{\psi}}_{4}{{\ensuremath{\psi}}_{4}}^{*}$ is treated by considering, as on the Schrodinger theory, the independent angular and radial factors separately. Tables and graphs are given for $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of $\ensuremath{\theta}$ alone. The graphs are compared with and shown to correspond closely to the space quantization of the classical electron orbits. Graphs representative of $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of rrr alone are given and compared with the electronnucleus distances of the corresponding electron orbits. The radial and angular factors are brought together by means of a mechanical device described previously, which when photographed represents closely the electron-cloud $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$. Photographs are given for the states 12S121^{2}S_{\frac{1}{2}}12S21​​, 22P12,322^{2}P_{\frac{1}{2}, \frac{3}{2}}22P21​,23​​, 32D32,523^{2}D_{\frac{3}{2}, \frac{5}{2}}32D23​,25​​ 42F52,724^{2}F_{\frac{5}{2}, \frac{7}{2}}42F25​,27​​, and 52G725^{2}G_{\frac{7}{2}}52G27​​, with $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$
A quantum mechanical study of a simple two-dimensional problem identifies lines in coordinate space about which the probability densities are localised. These lines in general do not correspond to extrema in the system's static potential energy surface. The subject of this study has been the hydrogen atom in externally applied parallel electric and magnetic and magnetic fields. Probability density plots of eigenstates, in which the electron experiences comparable influences from all three fields: Coulomb, electric and magnetic, are each found to display a localisation at a polar angle related to an invariant previously obtained in a classical study of the problem. The line about which localisation occurs is found to vary with energy and is in coincidence with a periodic trajectory found within a classical framework.
The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})$, in which PPP, $\ensuremath{\Theta}$, and $\ensuremath{\Phi}$ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates rrr, $\ensuremath{\theta}$, and $\ensuremath{\varphi}$ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})=\left\{\frac{1}{{(2\ensuremath{\pi})}^{\frac{1}{2}}}{e}^{\ifmmode\pm\else\textpm\fi{}\mathrm{im}\ensuremath{\Phi}}\right\} \left\{{\left(\frac{(2l+1)(l\ensuremath{-}m)!}{2(l+m)!}\right)}^{\frac{1}{2}}{{P}_{l}}^{m}(cos\ensuremath{\Theta})\right\}$ $\left\{\frac{\ensuremath{\pi}{2}^{2l+4}l!}{{(\ensuremath{\gamma}h)}^{\frac{3}{2}}}{\left(\frac{n(n\ensuremath{-}l\ensuremath{-}1)!}{(n+l)!}\right)}^{\frac{1}{2}}\frac{{\ensuremath{\zeta}}^{l}}{{({\ensuremath{\zeta}}^{2}+1)}^{l+2}}{C}_{n\ensuremath{-}l\ensuremath{-}1}^{l+1}\left(\frac{{\ensuremath{\zeta}}^{2}\ensuremath{-}1}{{\ensuremath{\zeta}}^{2}+1}\right)\right\}$ in which $\ensuremath{\zeta}=(\frac{2\ensuremath{\pi}}{\ensuremath{\gamma}h})P$, with $\ensuremath{\gamma}=(\frac{4{\ensuremath{\pi}}^{2}\ensuremath{\mu}{e}^{2}Z}{n{h}^{2}})=(\frac{Z}{n{a}_{0}})$. The probability ${\ensuremath{\Xi}}_{\mathrm{nl}}(P)\mathrm{dP}$ that the electron have a total momentum lying within the limits PPP and P+dPP+dPP+dP is also evaluated, and it is shown that the root mean square of the total momentum is equal to the momentum of the electron in a circular Bohr orbit with the same total quantum number.
A point-like picture of the Schrödinger solution for hydrogen atom is worked to emphasize that "point-like particles" may describe as "probability wave function". In each case, the three-dimensional shape of the |Ψnlm(rn, cos θ)|2 is plotted and the paths of the point-like electron (it is better to say reduced mass of the pair particles) are described in each closed shell. Finally, the orbital shape of the molecules are given according to the present simple model. In our opinion, "interpretations of the Correspondence Principle", which is a basic principle in all elementary quantum text, seems to be reviewed again!