Degeneracy of energy levels of excited hydrogen atoms is not fully lifted by external perpendicular electric and magnetic fields in the first order of perturbation theory. Within the submanifold of residual degeneracy the levels splitting is governed by the second-order effects. The related classical electron trajectories exhibit instabilities characterized by the Lyapunov exponent. The present study establishes quantum mechanical implications of these instabilities. In particular, we derive from a quantum framework a simple analytical approximation for the Lyapunov exponent. The classical instabilities are linked to the presence of a diabatic quasistationary state in the quantum mechanical problem. Its meaning is elucidated by considering a related non-stationary quantum system with time-dependent field parameter. Some of the state-to-state transition probabilities are approximately expressed via Lyapunov exponents. Exact formulae for these probabilities are derived in the framework of multistate generalization of the Landau–Zener model. Non-adiabatic transitions are interpreted in terms of analytical properties of adiabatic potential curves.
Degeneracy of energy levels of excited hydrogen atoms is not fully lifted by external perpendicular electric and magnetic fields in the first order of perturbation theory. Within the submanifold of residual degeneracy the levels splitting is governed by the second-order effects. The related classical electron trajectories exhibit instabilities characterized by the Lyapunov exponent. The present study establishes quantum mechanical implications of these instabilities. In particular, we derive from a quantum framework a simple analytical approximation for the Lyapunov exponent. The classical instabilities are linked to the presence of a diabatic quasistationary state in the quantum mechanical problem. Its meaning is elucidated by considering a related non-stationary quantum system with time-dependent field parameter. Some of the state-to-state transition probabilities are approximately expressed via Lyapunov exponents. Exact formulae for these probabilities are derived in the framework of multistate generalization of the Landau–Zener model. Non-adiabatic transitions are interpreted in terms of analytical properties of adiabatic potential curves.
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 nonstationary Schr\"odinger equation is considered in a finite basis of states. The model Hamiltonian matrix corresponds to a single diabatic potential curve with a Coulombic $\ensuremath{\sim}1/t$ time dependence. An arbitrary number of other diabatic potential curves are flat, i.e., time independent and have arbitrary energies. Related states are coupled by constant interactions with the Coulomb state. The resulting nonstationary Schr\"odinger equation is solved by the method of contour integral. Probabilities of transitions to any other state are obtained as $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ in a simple analytical form for the case when the Coulomb state is populated initially (at instant of time $\stackrel{\ensuremath{\rightarrow}}{t}+0).$ The formulas apply both to the cases when a horizontal diabatic potential curve is crossed by the Coulomb one and to a noncrossing situation. In the limit of weak coupling, the transition probabilities are interpreted in terms of a sequence of pairwise Landau-Zener-type transitions. Mapping of the Coulomb model onto an exactly solvable exponential multistate model is established. For the special two-state case, the well-known Nikitin model is recovered.
The mixing of substates of an isolated Rydberg manifold (constant n,m) is studied using time-evolution matrices U(t,t0). Coherent mixing is effected by a linearly ramped electric field, F(t)=F\ifmmode \dot{}\else \.{}\fi{}t. The spherical l basis, diabatic (hydrogenic) p basis, and adiabatic (nonhydrogenic) q basis are considered. In the linear Stark regime, the diabatic levels are coupled by F(t) through the atomic core, parametrized by quantum defects. The resulting adiabatic levels EqF in this model group into a submanifold degenerate at F=0 plus shifted levels split off from the adiabatic manifold. Analytical expressions for the q levels Eq\ensuremath{\tau} and their coupling \ensuremath{\Gamma}qq\ensuremath{'}\ensuremath{\tau} are derived as a function of rescaled field or time \ensuremath{\tau}. The time evolution of the levels' populations is studied first for the two-level case---the usual Landau-Zener effect (LZE)---which is generalized to a multilevel Landau-Zener effect (MLZE) for any n manifold. The matrices U(t,t0) are constructed in the Riemann product representation, including Magnus corrections up to fourth order. The probability Pq\ensuremath{\rightarrow}q\ensuremath{'}(Vl) of making a transition from state q to q\ensuremath{'} upon a one-way pass across the manifold depends on a single parameter Vl=[2\ensuremath{\mu}l2/(3F\ifmmode \dot{}\else \.{}\fi{}n9)]1/2 (when one \ensuremath{\mu}l\ensuremath{\ne}0). Numerical results for l=m=0 indicate that either diabatic (p\ensuremath{\rightarrow}p\ensuremath{'}=p) or adiabatic (q\ensuremath{\rightarrow}q\ensuremath{'}=q) transitions always predominate, no matter what the ramp rate F\ifmmode \dot{}\else \.{}\fi{}. Strong evidence supports the conjecture that outer diabatic transitions between edge states of a manifold obey Pndiab(V)=exp(-\ensuremath{\beta}n\ensuremath{\pi}V2) for l=m=0, with \ensuremath{\beta}n=1 for n=2 (the LZE) and \ensuremath{\beta}n\ensuremath{\approx}lnn for n\ensuremath{\gg}2 (the MLZE), and that a similar analytical result holds for arbitrary {\ensuremath{\mu}l} and m.
This study aimed to numerically investigate the impact of dipole-dipole interactions on the energy levels of a hydrogen atom, specifically the energy shift caused by the interaction of the atom's dipole moment with external electric fields. This research involved applying perturbation theory and numerical techniques to quantify the changes in energy levels due to dipole orientations and magnitudes, contributing to a deeper understanding of atomic behavior and its implications in various scientific disciplines. The results presented showed that the wavefunction of the hydrogen atom exhibits degeneracy, especially n=2, n=3 states. This implies that n=2, n=3, have different states but the same energy as a result of the subshells with angular momentum l, which contains 2l+1 distinct state corresponding to positive value of m (magnetic angular momentum quantum number) m=-l±1. The total degeneracy of the energy levels n is caused by n2. We find out that as the orientation of the electron charges (θ,ϕ) change the dipole moment also changes and thus has a perturbing effect on the energy levels of the hydrogen atom leading to a shift in its energy. This is confirmed, as the energy shift increases slightly from the ground state energy function Φ100 to the second exicted state Φ300, while the degeneracy state of Φ310 experiences infinitesimal energy shift above. This is because the probability of the electron been found in the lower energy level is greater than at higher energy level and so it is mostly affected there by energy shift. Also, the dipole moment could increase liberational energy of the atom and thus lead to more excitement even though its impact is quite negative. The outcome of this study could have implications in precision measurements, quantum computing, and the understanding of complex atomic systems.
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.
In the first step, the Ehrenfest reasoning concerning the adiabatic invariance of the angular orbital momentum is applied to the electron motion in the hydrogen atom. It is demonstrated that the time of the energy emission from the quantum level n+1 to level n can be deduced from the orbital angular momentum examined in the hydrogen atom. This time is found precisely equal to the time interval dictated by the Joule-Lenz law governing the electron transition between the levels n+1 and n. In the next step, the mechanical parameters entering the quantum systems are applied in calculating the time intervals characteristic for the electron transitions. This concerns the neighbouring energy levels in the hydrogen atom as well as the Landau levels in the electron gas submitted to the action of a constant magnetic field.