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Bulletin of the American Physical Society

Classical calculation of the lifetimes and branching ratios for radiative decays of hydrogenic atoms

Marko Horbatsch, Eric A. Hessels

Published May 19, 2005• 0 citations
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Abstract

The correspondence principle for atomic radiation is extended to all hydrogenic states n and l . 0. Lifetimes and branching ratios are obtained using analytic calculations of the classical radiated spectrum for the elliptical orbit corresponding to a particular quantum state. The polarization of the radiation is used to separate out the angular momentum decreasing and increasing transitions. The lifetimes show excellent agreement with quantum mechanics for all principal quantum numbers n and l o 1 se.g., ,100 parts per million for l o 30, ,0.1% for l o 9, ,1% for l o 3d. The calculated branching ratios are in reasonable agreement with quantum results for all n8, l8, n, and l o 1.

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Classical calculation of the lifetimes and branching ratios for radiative decays of hydrogenic atoms

Marko Horbatsch, Eric A. Hessels, Marko Horbatsch

The correspondence principle for atomic radiation is extended to all hydrogenic states $n$ and $l>0$. Lifetimes and branching ratios are obtained using analytic calculations of the classical radiated spectrum for the elliptical orbit corresponding to a particular quantum state. The polarization of the radiation is used to separate out the angular momentum decreasing and increasing transitions. The lifetimes show excellent agreement with quantum mechanics for all principal quantum numbers $n$ and $l\ensuremath{\geqslant}1$ (e.g., $<100\phantom{\rule{0.3em}{0ex}}\text{parts}\phantom{\rule{0.3em}{0ex}}\text{per}\phantom{\rule{0.3em}{0ex}}\text{million}$ for $l\ensuremath{\geqslant}30$, $<0.1%$ for $l\ensuremath{\geqslant}9$, $<1%$ for $l\ensuremath{\geqslant}3$). The calculated branching ratios are in reasonable agreement with quantum results for all ${n}^{\ensuremath{'}}$, ${l}^{\ensuremath{'}}$, $n$, and $l\ensuremath{\geqslant}1$.

Physical Review A•Feb 10, 2005•10 citations

Classical calculation of radiative decay rates of hydrogenic Stark states

Michael Horbatsch, Marko Horbatsch

Abstract The Kepler–Coulomb problem is solved in parabolic coordinates and the Larmor radiation problem is analyzed to complement a previous study performed for the usual representation in spherical polar coordinates. The problem is then extended to include a weak electric field and a solution in terms of action-angle variables is provided. A comparison with quantum spontaneous decay rates shows that for azimuthal quantum number m = 0 states only transitions to nearby n −Δ n principal quantum number states are described properly by the Wentzel–Kramers–Brillouin (WKB) quantized classical motions, but that for m > 0 reasonable results emerge for many values of Δ n . A simple approximate expression for the lifetime of m ≠ 0 states emerges from the semi-classical analysis.

arXiv:2011.02645•Journal of Physics B Atomic Molecular and Optical Physics•Jan 27, 2021•1 citation

How are quantum eigenfunctions of hydrogen atom related to its classical elliptic orbits?

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Abstract We show that a highly-excited energy eigenfunction ψ nlm ( r ) of hydrogen atom can be approximated as an equal-weight superposition of classical elliptic orbits with energy E n and angular momentum L = l ( l + 1 ) ħ , and z component of angular momentum L z = mħ . This correspondence is established by comparing the quantum probability distribution | ψ nlm ( r )| 2 and the classical probability distribution p c ( r ) of an ensemble of such orbits. This finding illustrates a general principle: in the semi-classical limit, an energy eigenstate of a quantum system is in general reduced to a collection of classical orbits, rather than a single classical orbit.

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Blackbody-induced decay, excitation and ionization rates for Rydberg states in hydrogen and helium atoms

Igor L. Glukhov, E A Nekipelov, Vitalii D Ovsiannikov

New features of the blackbody-induced radiation processes on Rydberg atoms were discovered on the basis of numerical data for the blackbody-induced decay P d nl ( T ), excitation P e nl ( T ) and ionization P ion nl ( T ) rates of n S, n P and n D Rydberg states calculated together with the spontaneous decay rates P sp nl in neutral hydrogen, and singlet and triplet helium atoms for some values of the principal quantum number n from 10 to 500 at temperatures from T = 100 K to 2000 K. The fractional rates R d ( e , ion) nl ( T ) = P nl d ( e , ion) ( T )/ P sp nl equal to the ratio of the induced decay (excitation, ionization) rates to the rate of spontaneous decay were determined as functions of T and n in every series of states with a given angular momentum l = 0, 1, 2. The calculated data reveal an essential difference between the asymptotic dependence of the ionization rate P ion nl ( T ) and the rates of decay and excitation P d ( e ) nl ( T )∝ T / n 2 . The departures appear in each Rydberg series for n > 100 and introduce appreciable corrections to the formula of Cooke and Gallagher. Two different approximation formulae are proposed on the basis of the numerical data, one for R d ( e ) nl ( T ) and another one for R ion nl ( T ), which reproduce the calculated values in wide ranges of principal quantum number from n = 10 to 1000 and temperatures between T = 100 K and T = 2000 K with an accuracy of 2% or better. Modified Fues' model potential approach was used for calculating matrix elements of bound–bound and bound–free radiation transitions in helium.

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Stark Lifetimes for the Hydrogen Atom

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The radiative transition probabilities in hydrogen have been calculated for all Stark transitions through $n=25$. In general, the total transition probability and lifetime of a particular state are a function of the parabolic quantum numbers ${n}_{1}$, ${n}_{2}$, and $m$. For $m>0$ and to within an error of the order of one percent the radiative lifetimes are functions only of $n$, $m$. The radiative lifetimes are presented graphically for all states from $n=2$ through $n=25$.

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The Bohr-Maxwell Description of the Hydrogen L-to-K Transition

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This paper develops and verifies a physics-first description of the hydrogen-atom transition from the L shell (n = 2) to the K shell (n = 1) by combining the historical Bohr model of atomic structure with classical Maxwell electrodynamics. Beginning with the Coulomb force, centripetal dynamics, and Bohr’s angular-momentum quantization condition, the analysis derives the allowed orbital radii, electron velocities, angular momenta, kinetic energies, electrostatic potential energies, and total bound-state energies of the hydrogen atom. For the n = 2 → n = 1 transition, the calculated energy difference is approximately 10.2 eV, corresponding through the Planck–Einstein relation ΔE = hν to a frequency of approximately 2.47 × 10¹⁵ Hz and a wavelength near 121.6 nm, consistent with the hydrogen Lyman-α transition. The emitted radiation is then examined through Maxwell’s equations and the vacuum constitutive relations. The resulting electromagnetic-wave description yields the transverse electric and magnetic fields together with the associated electromagnetic energy density, Poynting energy flux, momentum density, radiation pressure, and Maxwell stress tensor. In this way, the atomic energy difference obtained from the Bohr model is connected quantitatively to the classical propagation properties of the outgoing electromagnetic radiation. The derivation is mathematically consistent as a historical Bohr–Maxwell reconstruction, but it does not constitute a closed dynamical theory of spontaneous emission. The essential bridge between discrete atomic energy levels and the continuous Maxwell field is supplied by the quantum frequency condition ΔE = hν. Neither classical Maxwell electrodynamics nor Newtonian orbital mechanics independently derives this relation, and classical Maxwell theory does not by itself quantize the emitted radiation energy into individual photons. A complete treatment of spontaneous emission, transition probabilities, radiative lifetimes, and photon-field quantization requires quantum mechanics and ultimately quantum electrodynamics. Accordingly, the framework is presented as a verified correspondence between established limiting descriptions rather than as a replacement for Schrödinger wave mechanics, Dirac theory, or quantum electrodynamics. By clearly separating the atomic bound-state calculation, the quantum energy-frequency bridge, and the classical electromagnetic propagation problem, the analysis provides a rigorous conceptual and mathematical foundation for subsequent Maxwell–Schrödinger, Dirac–Maxwell, quantum-electrodynamic, Maxwell–Scretching, and Scretching Quantum Chain formulations of hydrogenic radiative transitions.

•Jul 16, 2026•0 citations
The correspondence principle for atomic radiation is extended to all hydrogenic states n and l . 0. Lifetimes and branching ratios are obtained using analytic calculations of the classical radiated spectrum for the elliptical orbit corresponding to a particular quantum state. The polarization of the radiation is used to separate out the angular momentum decreasing and increasing transitions. The lifetimes show excellent agreement with quantum mechanics for all principal quantum numbers n and l o 1 se.g., ,100 parts per million for l o 30, ,0.1% for l o 9, ,1% for l o 3d. The calculated branching ratios are in reasonable agreement with quantum results for all n8, l8, n, and l o 1.
The correspondence principle for atomic radiation is extended to all hydrogenic states nnn and l>0l>0l>0. Lifetimes and branching ratios are obtained using analytic calculations of the classical radiated spectrum for the elliptical orbit corresponding to a particular quantum state. The polarization of the radiation is used to separate out the angular momentum decreasing and increasing transitions. The lifetimes show excellent agreement with quantum mechanics for all principal quantum numbers nnn and $l\ensuremath{\geqslant}1$ (e.g., <100partspermillion<100\phantom{\rule{0.3em}{0ex}}\text{parts}\phantom{\rule{0.3em}{0ex}}\text{per}\phantom{\rule{0.3em}{0ex}}\text{million}<100partspermillion for $l\ensuremath{\geqslant}30$, <0.1<0.1%<0.1 for $l\ensuremath{\geqslant}9$, <1<1%<1 for $l\ensuremath{\geqslant}3$). The calculated branching ratios are in reasonable agreement with quantum results for all ${n}^{\ensuremath{'}}$, ${l}^{\ensuremath{'}}$, nnn, and $l\ensuremath{\geqslant}1$.
Abstract The Kepler–Coulomb problem is solved in parabolic coordinates and the Larmor radiation problem is analyzed to complement a previous study performed for the usual representation in spherical polar coordinates. The problem is then extended to include a weak electric field and a solution in terms of action-angle variables is provided. A comparison with quantum spontaneous decay rates shows that for azimuthal quantum number m = 0 states only transitions to nearby n −Δ n principal quantum number states are described properly by the Wentzel–Kramers–Brillouin (WKB) quantized classical motions, but that for m > 0 reasonable results emerge for many values of Δ n . A simple approximate expression for the lifetime of m ≠ 0 states emerges from the semi-classical analysis.
Abstract We show that a highly-excited energy eigenfunction ψ nlm ( r ) of hydrogen atom can be approximated as an equal-weight superposition of classical elliptic orbits with energy E n and angular momentum L = l ( l + 1 ) ħ , and z component of angular momentum L z = mħ . This correspondence is established by comparing the quantum probability distribution | ψ nlm ( r )| 2 and the classical probability distribution p c ( r ) of an ensemble of such orbits. This finding illustrates a general principle: in the semi-classical limit, an energy eigenstate of a quantum system is in general reduced to a collection of classical orbits, rather than a single classical orbit.
New features of the blackbody-induced radiation processes on Rydberg atoms were discovered on the basis of numerical data for the blackbody-induced decay P d nl ( T ), excitation P e nl ( T ) and ionization P ion nl ( T ) rates of n S, n P and n D Rydberg states calculated together with the spontaneous decay rates P sp nl in neutral hydrogen, and singlet and triplet helium atoms for some values of the principal quantum number n from 10 to 500 at temperatures from T = 100 K to 2000 K. The fractional rates R d ( e , ion) nl ( T ) = P nl d ( e , ion) ( T )/ P sp nl equal to the ratio of the induced decay (excitation, ionization) rates to the rate of spontaneous decay were determined as functions of T and n in every series of states with a given angular momentum l = 0, 1, 2. The calculated data reveal an essential difference between the asymptotic dependence of the ionization rate P ion nl ( T ) and the rates of decay and excitation P d ( e ) nl ( T )∝ T / n 2 . The departures appear in each Rydberg series for n > 100 and introduce appreciable corrections to the formula of Cooke and Gallagher. Two different approximation formulae are proposed on the basis of the numerical data, one for R d ( e ) nl ( T ) and another one for R ion nl ( T ), which reproduce the calculated values in wide ranges of principal quantum number from n = 10 to 1000 and temperatures between T = 100 K and T = 2000 K with an accuracy of 2% or better. Modified Fues' model potential approach was used for calculating matrix elements of bound–bound and bound–free radiation transitions in helium.
The radiative transition probabilities in hydrogen have been calculated for all Stark transitions through n=25n=25n=25. In general, the total transition probability and lifetime of a particular state are a function of the parabolic quantum numbers n1{n}_{1}n1​, n2{n}_{2}n2​, and mmm. For m>0m>0m>0 and to within an error of the order of one percent the radiative lifetimes are functions only of nnn, mmm. The radiative lifetimes are presented graphically for all states from n=2n=2n=2 through n=25n=25n=25.
This paper develops and verifies a physics-first description of the hydrogen-atom transition from the L shell (n = 2) to the K shell (n = 1) by combining the historical Bohr model of atomic structure with classical Maxwell electrodynamics. Beginning with the Coulomb force, centripetal dynamics, and Bohr’s angular-momentum quantization condition, the analysis derives the allowed orbital radii, electron velocities, angular momenta, kinetic energies, electrostatic potential energies, and total bound-state energies of the hydrogen atom. For the n = 2 → n = 1 transition, the calculated energy difference is approximately 10.2 eV, corresponding through the Planck–Einstein relation ΔE = hν to a frequency of approximately 2.47 × 10¹⁵ Hz and a wavelength near 121.6 nm, consistent with the hydrogen Lyman-α transition. The emitted radiation is then examined through Maxwell’s equations and the vacuum constitutive relations. The resulting electromagnetic-wave description yields the transverse electric and magnetic fields together with the associated electromagnetic energy density, Poynting energy flux, momentum density, radiation pressure, and Maxwell stress tensor. In this way, the atomic energy difference obtained from the Bohr model is connected quantitatively to the classical propagation properties of the outgoing electromagnetic radiation. The derivation is mathematically consistent as a historical Bohr–Maxwell reconstruction, but it does not constitute a closed dynamical theory of spontaneous emission. The essential bridge between discrete atomic energy levels and the continuous Maxwell field is supplied by the quantum frequency condition ΔE = hν. Neither classical Maxwell electrodynamics nor Newtonian orbital mechanics independently derives this relation, and classical Maxwell theory does not by itself quantize the emitted radiation energy into individual photons. A complete treatment of spontaneous emission, transition probabilities, radiative lifetimes, and photon-field quantization requires quantum mechanics and ultimately quantum electrodynamics. Accordingly, the framework is presented as a verified correspondence between established limiting descriptions rather than as a replacement for Schrödinger wave mechanics, Dirac theory, or quantum electrodynamics. By clearly separating the atomic bound-state calculation, the quantum energy-frequency bridge, and the classical electromagnetic propagation problem, the analysis provides a rigorous conceptual and mathematical foundation for subsequent Maxwell–Schrödinger, Dirac–Maxwell, quantum-electrodynamic, Maxwell–Scretching, and Scretching Quantum Chain formulations of hydrogenic radiative transitions.