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 $l\ensuremath{\geqslant}1$.
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⩾1 (e.g., <100partspermillion for l\ensuremath⩾30, <0.1 for l\ensuremath⩾9, <1 for l\ensuremath⩾3). The calculated branching ratios are in reasonable agreement with quantum results for all n\ensuremath′, l\ensuremath′, n, and l\ensuremath⩾1.
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.
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.
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 n1, n2, 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.
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.
The branching ratios for radiative decay of H atoms excited in small electric fields (0--5 V/cm) have been calculated for the nl states up to n=6. A simple computational procedure was employed, allowing only for Stark-effect mixing of levels with the same values of the quantum numbers (n,j,mj). The results are compared with more detailed calculations available for 3l-state decay made using the density-matrix formalism, and new calculations of this type reported here for 4l-state decay. In conjunction with theory, this allowed the domain of validity of the simple computational procedure to be established as a function of n. The results show that the branching ratios depend quite strongly on electric-field magnitude, pointing to the need to exercise caution in measurements of H emissions, and in application of the available data to other problems.
Chapter 12 introduces light–matter interaction at the atomic scale and at interaction energies less than, or comparable to, the chemical bond. The classical electrodynamics treatment for fields is retained, but a very simple quantum atomic structure with electric dipole transitions among atomic and molecular internal states is introduced. The correspondence between classical and quantal electron orbital angular momentum states in atoms is developed in detail for the hydrogen atom, and the difference between quantised orbital states and real, space-directed orbitals is explained. A wave mechanical quantum representation brings out the analogies between classical light waves, quantum matter waves, classical dipole radiation, and atomic radiative emission. The intrinsic electron spin, that has no classical analogue, is introduced and discussed. The chapter ends with a discussion of the rigid rotor, harmonic oscillator model to interpret rotational and vibrational diatomic molecular spectra. Finally rovibronic states complete the picture with the introduction of diatomic molecular electronic transitions.