We investigate the interaction between the hydrogen atomic orbitals and the quantized modes of the electromagnetic field within the domain of nonrelativistic quantum electrodynamics in the Coulomb gauge. Contrary to the conventional dipole approximation and higher-order multipole approximations, which hold in the long-wavelength regime, we make no such approximations and investigate the rigorous wavelength dependence of the interaction matrix elements. Using analytical as well as numerical considerations, we establish the short-wavelength behaviour in the domain that is beyond the multipole approximation but still within the nonrelativistic regime. Our results differ significantly from the predictions of the dipole approximation. These features are graphically demonstrated in the case of the radiative 1 s -2 p z coupling in the hydrogen atom. We report a previously unnoticed short-wavelength contribution to the interaction that vanishes in the multipole expansion. We note that the convergence properties of radiative energy shifts are incorrectly predicted by the dipole approximation and that the unapproximated result is in fact quickly convergent.
We investigate the interaction between the hydrogen atomic orbitals and the quantized modes of the electromagnetic field within the domain of nonrelativistic quantum electrodynamics in the Coulomb gauge. Contrary to the conventional dipole approximation and higher-order multipole approximations, which hold in the long-wavelength regime, we make no such approximations and investigate the rigorous wavelength dependence of the interaction matrix elements. Using analytical as well as numerical considerations, we establish the short-wavelength behaviour in the domain that is beyond the multipole approximation but still within the nonrelativistic regime. Our results differ significantly from the predictions of the dipole approximation. These features are graphically demonstrated in the case of the radiative 1 s -2 p z coupling in the hydrogen atom. We report a previously unnoticed short-wavelength contribution to the interaction that vanishes in the multipole expansion. We note that the convergence properties of radiative energy shifts are incorrectly predicted by the dipole approximation and that the unapproximated result is in fact quickly convergent.
This paper presents a quantum-field-theoretic formulation of the hydrogen 2p → 1s Lyman-α transition, replacing the semiclassical Bohr–Maxwell description with the modern framework of quantum electrodynamics. The initial and final bound-electron states are described by the Dirac equation in the proton’s Coulomb field, with the Schrödinger equation recovered as the leading nonrelativistic limit. The electromagnetic radiation field is quantized, and photon emission is generated by the QED interaction Hamiltonian. The transition amplitude is evaluated from the atomic current and quantized electromagnetic field, leading through Fermi’s golden rule to the spontaneous-emission rate and Einstein coefficient A₂₁. Energy and momentum conservation require the emitted photon to carry the atomic transition energy, including the small correction associated with proton recoil. At leading order, the Lyman-α photon has ΔE ≈ 10.2 eV, ν ≈ 2.47 × 10¹⁵ Hz, and λ ≈ 121.6 nm. In the electric-dipole approximation, the field-theoretic calculation reduces to the standard matrix element of the interaction Hamiltonian H_int = −μ·E, thereby connecting QED directly to the transition dipole moment, oscillator strength, Einstein A coefficient, and radiative lifetime. The calculation reproduces the accepted numerical results for hydrogen while identifying their origin in the coupling between the quantized electron field and the quantized electromagnetic field. The paper also explains how classical electrodynamics is recovered from QED. Maxwell’s equations govern the expectation values of the electromagnetic-field operators in the appropriate classical-field limit, particularly for coherent states containing many photons. A single spontaneously emitted Lyman-α photon, however, remains a genuinely quantum excitation rather than a classical electromagnetic wave. Leading corrections beyond the nonrelativistic Coulomb approximation are then considered, including relativistic fine structure, the Lamb shift, and atomic recoil. The resulting treatment provides a rigorous bridge from the historical Bohr model and Schrödinger wave mechanics to Dirac theory and full QED, verifying the Lyman-α energy, frequency, wavelength, transition probability, and radiative decay structure within a unified quantum-field framework.
The photonic environment can significantly influence emission properties and interactions among atomic systems. In such scenarios, frequently the electric dipole approximation is assumed that is justified as long as the spatial extent of the atomic system is negligible compared to the spatial variations of the field. While this holds true for many canonical systems, it ceases to be applicable for more contemporary nanophotonic structures. To go beyond the electric dipole approximation, we propose and develop in this article an analytical framework to describe the impact of the photonic environment on emission and interaction properties of atomic systems beyond the electric dipole approximation. Particularly, we retain explicitly magnetic dipolar and electric quadrupolar contributions to the light-matter interactions. We exploit a field quantization scheme based on electromagnetic Green's tensors, suited for dispersive materials. We obtain expressions for spontaneous emission rate, Lamb shift, multipole-multipole shift and superradiance rate, all being modified with dispersive environment. The considered influence could be substantial for suitably tailored nanostructured photonic environments, as demonstrated exemplarily.
The photonic environment can significantly influence emission properties and interactions among atomic systems. In such scenarios, frequently the electric dipole approximation is assumed that is justified as long as the spatial extent of the atomic system is negligible compared to the spatial variations of the field. While this holds true for many canonical systems, it ceases to be applicable for more contemporary nanophotonic structures. To go beyond the electric dipole approximation, we propose and develop in this article an analytical framework to describe the impact of the photonic environment on emission and interaction properties of atomic systems beyond the electric dipole approximation. Particularly, we retain explicitly magnetic dipolar and electric quadrupolar contributions to the light-matter interactions. We exploit a field quantization scheme based on electromagnetic Green’s tensors, suited for dispersive materials. We obtain expressions for spontaneous emission rate, Lamb shift, multipole-multipole shift and superradiance rate, all being modified with dispersive environment. The considered influence could be substantial for suitably tailored nanostructured photonic environments, as demonstrated exemplarily.
The subject of this chapter is the quantum mechanical analysis of the interaction of electromagnetic radiation with atomic transitions. The analysis is based on the Schrödinger wave equation, and in the first section, the gauge-invariant form of the external electromagnetic field is introduced. The electric dipole interaction and the long-wavelength approximation for the analysis of this interaction are discussed. The perturbative analysis of both single-photon and two-photon electric dipole interactions is presented, and density matrix analysis is introduced. The interaction of radiation with the resonances of atomic hydrogen is then discussed. The analysis is performed for both coupled and uncoupled representations. In the last section of the chapter, the radiative interactions for multielectron atoms are discussed. The Wigner–Eckart theorem and selection rules for transitions between levels characterized by coupling are developed. The effect of hyperfine splitting on radiative transitions is also briefly discussed.
In view of the possibilities of laser spectroscopy with very short pulses and relatively high frequencies, we examine aspects of the theory of atom-field interactions that are related to violation of the condition of the long-wavelength approximation (LWA) according to which kr⪡1, where r is ``of the order of atomic dimensions'' and k is the magnitude of the wave vector. On- and off-resonance transitions are considered, with kr being larger than unity due to the large extent of the two wave functions involved in the coupling matrix element. The implementation of the analysis uses bound-free transition matrix elements with the n=50 and 80 hydrogenic functions as initial states and values of k up to 27eV, which is sufficient to produce kr>1, thereby rendering the LWA inoperative. In spite of this, it is shown that, for on-resonance transitions, the results from the use of the well-known multipole operators resulting from the LWA [e.g., the electric dipole approximation (EDA)] agree with those from the exact expressions derived here from the application of the multipolar Hamiltonian. This numerical agreement is proven analytically. As a test of the kind of convergence of the multipole series expansion for small values of k and large r, it is shown that the lowest-order ratio of electric dipole to quadrupole matrix elements decreases rapidly within 1.0 atomic unit above threshold. Finally, it is shown that off-resonance couplings lead to differences between the full-interaction operator and the EDA which cannot be neglected. In the extreme case of intrashell couplings for the n=50 shell, calculation shows that the 50p state is coupled to angular momentum states up to $\ensuremath{\ell}=21$, compared to the electric dipole coupling of $\ensuremath{\Delta}\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}1$.
A recent investigation of the interaction between the Maxwell radiation field and neutral atoms or molecules has led us to consider the relation between the straightforward interaction Hamiltonian between particle (k) and the radiation field, and the more physical, and explicitly gauge invariant, couplings of the form q. E and µ. H, etc., where q and µ are the electric and magnetic dipole moments of a system of particles. That these latter form of interaction energies follow from well-formulated approximations for atoms in external fields has been known for many years (Goeppert-Mayer(1); see also Rosenfeld(2)). When the Maxwell field is an internal and dynamical field further discussion is necessary, especially concerning the Coulomb interactions. Their advantage has recently been shown in dealing with dipole interactions in the long range Van der Waals force between two neutral systems where the complete coupling, in dipole approximation, is given by — q.ET for each neutral system, where ET is the transverse electric field. The short-range London limit is contained in this coupling despite the absence of longitudinal electric fields (Power and Zienau (3)). In this note we extend these arguments to higher electric multipoles and magnetic multipoles. We also demonstrate their use in the quantum theory of scattering of light, and in particular the derivations of Stephen (4) of the phase angle between emergent beams in optical double refraction phenomena are shown to follow immediately.