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arXiv:2011.02645 · Journal of Physics B Atomic Molecular and Optical Physics

Classical calculation of radiative decay rates of hydrogenic Stark states

Michael Horbatsch, Marko Horbatsch

Published Jan 27, 2021• 1 citation
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Abstract

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.

DOI
10.1088/1361-6455/abe0a5

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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 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 $l\ensuremath{\geqslant}1$.
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.
Hydrogenic spectra A good starting point is photoexcitation from the ground state of H. The problem naturally divides itself into two regimes: below the energy of classical ionization limit, where the states are for all practical purposes stable against ionization, and above it where the spectrum is continuous. As an example, we consider first the excitation of the n = 15 Stark states from the ground state in a field too low to cause significant ionization of n = 15 states. From Chapter 6 we know the energies of the Stark states, and we now wish to calculate the relative intensities of the transitions to these levels. One approach is to calculate them in parabolic coordinates. This approach is an efficient way to proceed for the excitation of H; however, it is not easily generalized to other atoms. Another, which we adopt here, is to express the n = 15 nn 1 n 2 m Stark states in terms of their nℓm components using Eqs. (6.18) or (6.19) and express the transition dipole moments in terms of the more familiar spherical n ℓ m states. In the excitation of the Stark states of principal quantum number n from the ground state only p state components are accessible via dipole transitions, so the relative intensities for light polarized parallel and perpendicular to the static field, π and σ polarizations, are proportional to the squared transformation coefficients |〈 nn 1 n 2 m | n ℓ m 〉| 2 from the nn 1 n 2 m parabolic states to the n ℓ m states for ℓ = 1 and m = 0 and 1. In Fig. 8.1 we show the relative intensities by means of the squared transformation coefficients |〈15 n 1 n 2 m |15 p m 〉| 2 for m = 0 and 1.
It is well known that for the nonrelativistic hydrogen atom it is possible to separate the Schrödinger equation in parabolic as well as spherical coordinates. The eigenfunctions obtained in these coordinate systems are each a suitable basis set in the absence of an electric field, but only the parabolic states, the Stark eigenfunctions, retain their character in the presence of a weak field. The properties of coherent superpositions of these Stark states are investigated and the motion of the resulting wave packet described. It is shown that a properly constituted superposition will mimic classical motion. It is also shown that the constant energy separation between adjacent Stark states of a given principal quantum number leads to periodic motion. In general, they split into distinct “clumps” of probability, but revive to form a single packet after each period. It is, however, possible to construct a packet that maintains its shape for all time.
We have simulated the motion of a Rydberg hydrogen atom in electric and magnetic fields that have a strong position dependence. The orientation and strengths of the fields depend strongly on the spatial position of the atom. The angle between the fields could take on any value. We use the quantum energy levels for a Rydberg state in electric and magnetic fields to derive a spatially dependent potential energy for the centre-of-mass motion of the atom. We compute the radiative decay rate into states with low principal quantum numbers. After the decay, the atom could be trapped using the magnetic moment of the electron. We present results for a specific trap geometry which could be experimentally realized.
The initial value problem and the field dressing associated with the time-dependent Stark effect are treated, which are of importance in applications. An exact solution of time-dependent field mixing of degenerate states as an initial value problem is obtained in the restricted subspace of a given principal quantum number of hydrogenic ions, with the field-dependent phases explicitly included. The field mixing in terms of the transformation from spherical to parabolic representation is clarified. The distinction between these two problems is stressed and previous work on the field-perturbed dielectronic recombination is re-examined.