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Oxford University Press eBooks

Radiation in Classical and Quantal Atoms

J. Weiner, Frederico Dias Nunes

Published Feb 2, 2017• 0 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780198796664.003.0012

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Radiation in Classical and Quantal Atoms

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This chapter begins with a discussion of radiation by a microscopic classical oscillator with radiative damping. It then turns to the Schrödinger equation for the hydrogen atom, and an extended analysis of the solutions to the Schrödinger equation for the hydrogen atom in polar coordinates is then presented. Real orbitals are constructed from linear combinations of the complex Schrödinger equation solutions. The significance of real orbitals for molecular structure and the consequent introduction of rotating and vibrating dipoles are then introduced. A semiclassical model (quantal atom and classical E-M field) for the absorption and emission of radiation is developed and compared to the Einstein A and B coefficients. It is shown how the time-dependence of the radiatively coupled non-stationary states of the hydrogen atom are analogous to the classical harmonic oscillator. The essentials of atomic transitions, as well as molecular rotational, vibrational, and electronic spectroscopy complete the chapter.

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The classical theory of the interaction of light with the electron clouds of atoms and molecules will be discussed in this chapter. The discussion will begin with the interaction of a steady electric field with a collection of point charges, leading to the development of terms describing the electric dipole and quadrupole moments. The classical Lorentz model is then introduced to describe interaction of an oscillating electric field with the electron cloud of an atom, and the concepts of absorption and emission are introduced. The propagation of a light wave through a medium with electric dipoles is then discussed. Finally, the classical theory of radiation from an oscillating dipole is discussed.

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Complement 2A: Classical model of the atom-field interaction: the Lorentz model

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This first complement is devoted to a completely classical approach of lightmatter interaction which was proposed by Lorentz at the end of the nineteenth century, before the advent of quantum mechanics, but after the discovery of the electron. Lorentz' phenomenological model is based on the experimental fact that atoms have well-defined and sharp absorption lines: he assumed that atoms behaved like harmonic oscillators, in which the electrons are bound to the atomic nucleus by a restoring force which varies linearly with its displacement (from its equilibrium point close to a nucleus), and makes them oscillate at a given frequency ω 0 equal to the experimentally determined absorption frequency. Within the frame of this model, we first calculate the electromagnetic field radiated by an oscillating electron. We show that in the absence of an externally applied force the free oscillations of the electron are damped, because electromagnetic energy is radiated at the expense of mechanical energy. We then study the characteristics of the radiation that is emitted when the oscillations are forced by the application of an external oscillatory electromagnetic field of angular frequency ω. We characterize the different regimes of this scattering of the incident electromagnetic wave and finally determine the polarization induced in the atomic medium by the incident electromagnetic wave. The Lorentz model can be considered as a lowest order approximation to a description of the light–matter interaction, a better approximation being the semi-classical treatment presented in Chapter 2, and the rigorous treatment being the completely quantum mechanical model presented in Chapter 6.

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Quantum mechanics

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Abstract This chapter reviews quantum mechanics of atoms, ions and molecules and their associated spectra. We have already invoked empirically Planck’s minimum quantum of action ℏ, and Heisenberg’s uncertainty principle. We were able to get so far, but the dynamics of electrons bound to atoms, ions and molecules requires wave or matrix mechanics of Schrödinger and Heisenberg developed in 1925. The present chapter reviews the elements of quantum theory necessary to understand the interaction of classical EM waves with electrons, bound and free. Symmetries are related to conservation equations, to find constants of the motion. The radiation field is treated mostly as a classical EM wave. Spin-orbitals and the quantum harmonic oscillator are identified as quantum building blocks. An explanation of spontaneous emission of radiation is given using the occupation number representation. Different kinds of radiative transitions found in dilute atmospheres and plasmas are discussed, quantum collisions are described, and regularities in spectra are highlighted. In typical plasmas, the important influence of electrons excited into two, not just one, spin-orbitals, is highlighted.

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Antennas, quantum optics and near-field microscopy

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Introduction Quantum optics The atom is the most elementary constituent of any model that describes the quantum nature of light–matter interaction. Because atoms emit and absorb light at well-defined frequencies, nineteenth century scientists thought of them as collections of harmonically oscillating electric dipole moments or EHDs. In the language of modern physics, the latter represent dipolar transitions among the various quantum mechanical states of an atom. In a strict definition, the field of quantum optics deals with problems that not only require the quantization of matter but also of the electromagnetic field, with examples such as (i) generation of squeezed light or Fock states, (ii) strong coupling of an atom and a photon, (iii) entanglement of a photon with an atom and (iv) Casimir and van der Waals forces. There are also many other important topics that have been discussed within the quantum optics community but do not necessarily require a full quantum electrodynamic (QED) treatment. Examples are (i) cooling and trapping of atoms, (ii) precision spectroscopy and (iii) modification of spontaneous emission. The simple picture of a TLS as an EHD remains very insightful and valuable to this day. Indeed, much of what we discuss in this chapter has to do with the interplay between the quantum and classical mechanical characters of dipolar oscillators. For instance, the extinction cross-section of a TLS, given by 3λ 2 /2π, can be derived just as well using quantum mechanics [70] or classical optics [234]. Another example, albeit more subtle, concerns the spontaneous emission rate.

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This chapter considers an atom as a quantum mechanical system. But the electromagnetic radiation will be treated as classical waves described by the Maxwell equations. The treatment is called semi-classical. Quantum mechanics describes an atom as a quantized system with a set of discrete energy levels. Each energy level is designated by a set of quantum numbers like the principal quantum of the Rydberg level and the quantum numbers representing the orbital, spin and total angular momentum. A molecule will have a different set of quantum numbers arising from its rotational and vibrational motions. Atoms in a solid have energy levels in the form of a band. The observed spectrum due to spontaneous emission from a collection of atoms is not monochromatic, but it is spread over a certain range of frequencies. The absorption spectrum also shows that the atom does not absorb at a single frequency, but it is spread over a band of frequencies around the resonance frequency.

CRC Press eBooks•Feb 19, 2018•0 citations
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.
This chapter begins with a discussion of radiation by a microscopic classical oscillator with radiative damping. It then turns to the Schrödinger equation for the hydrogen atom, and an extended analysis of the solutions to the Schrödinger equation for the hydrogen atom in polar coordinates is then presented. Real orbitals are constructed from linear combinations of the complex Schrödinger equation solutions. The significance of real orbitals for molecular structure and the consequent introduction of rotating and vibrating dipoles are then introduced. A semiclassical model (quantal atom and classical E-M field) for the absorption and emission of radiation is developed and compared to the Einstein A and B coefficients. It is shown how the time-dependence of the radiatively coupled non-stationary states of the hydrogen atom are analogous to the classical harmonic oscillator. The essentials of atomic transitions, as well as molecular rotational, vibrational, and electronic spectroscopy complete the chapter.
The classical theory of the interaction of light with the electron clouds of atoms and molecules will be discussed in this chapter. The discussion will begin with the interaction of a steady electric field with a collection of point charges, leading to the development of terms describing the electric dipole and quadrupole moments. The classical Lorentz model is then introduced to describe interaction of an oscillating electric field with the electron cloud of an atom, and the concepts of absorption and emission are introduced. The propagation of a light wave through a medium with electric dipoles is then discussed. Finally, the classical theory of radiation from an oscillating dipole is discussed.
This first complement is devoted to a completely classical approach of lightmatter interaction which was proposed by Lorentz at the end of the nineteenth century, before the advent of quantum mechanics, but after the discovery of the electron. Lorentz' phenomenological model is based on the experimental fact that atoms have well-defined and sharp absorption lines: he assumed that atoms behaved like harmonic oscillators, in which the electrons are bound to the atomic nucleus by a restoring force which varies linearly with its displacement (from its equilibrium point close to a nucleus), and makes them oscillate at a given frequency ω 0 equal to the experimentally determined absorption frequency. Within the frame of this model, we first calculate the electromagnetic field radiated by an oscillating electron. We show that in the absence of an externally applied force the free oscillations of the electron are damped, because electromagnetic energy is radiated at the expense of mechanical energy. We then study the characteristics of the radiation that is emitted when the oscillations are forced by the application of an external oscillatory electromagnetic field of angular frequency ω. We characterize the different regimes of this scattering of the incident electromagnetic wave and finally determine the polarization induced in the atomic medium by the incident electromagnetic wave. The Lorentz model can be considered as a lowest order approximation to a description of the light–matter interaction, a better approximation being the semi-classical treatment presented in Chapter 2, and the rigorous treatment being the completely quantum mechanical model presented in Chapter 6.
Abstract This chapter reviews quantum mechanics of atoms, ions and molecules and their associated spectra. We have already invoked empirically Planck’s minimum quantum of action ℏ, and Heisenberg’s uncertainty principle. We were able to get so far, but the dynamics of electrons bound to atoms, ions and molecules requires wave or matrix mechanics of Schrödinger and Heisenberg developed in 1925. The present chapter reviews the elements of quantum theory necessary to understand the interaction of classical EM waves with electrons, bound and free. Symmetries are related to conservation equations, to find constants of the motion. The radiation field is treated mostly as a classical EM wave. Spin-orbitals and the quantum harmonic oscillator are identified as quantum building blocks. An explanation of spontaneous emission of radiation is given using the occupation number representation. Different kinds of radiative transitions found in dilute atmospheres and plasmas are discussed, quantum collisions are described, and regularities in spectra are highlighted. In typical plasmas, the important influence of electrons excited into two, not just one, spin-orbitals, is highlighted.
Introduction Quantum optics The atom is the most elementary constituent of any model that describes the quantum nature of light–matter interaction. Because atoms emit and absorb light at well-defined frequencies, nineteenth century scientists thought of them as collections of harmonically oscillating electric dipole moments or EHDs. In the language of modern physics, the latter represent dipolar transitions among the various quantum mechanical states of an atom. In a strict definition, the field of quantum optics deals with problems that not only require the quantization of matter but also of the electromagnetic field, with examples such as (i) generation of squeezed light or Fock states, (ii) strong coupling of an atom and a photon, (iii) entanglement of a photon with an atom and (iv) Casimir and van der Waals forces. There are also many other important topics that have been discussed within the quantum optics community but do not necessarily require a full quantum electrodynamic (QED) treatment. Examples are (i) cooling and trapping of atoms, (ii) precision spectroscopy and (iii) modification of spontaneous emission. The simple picture of a TLS as an EHD remains very insightful and valuable to this day. Indeed, much of what we discuss in this chapter has to do with the interplay between the quantum and classical mechanical characters of dipolar oscillators. For instance, the extinction cross-section of a TLS, given by 3λ 2 /2π, can be derived just as well using quantum mechanics [70] or classical optics [234]. Another example, albeit more subtle, concerns the spontaneous emission rate.
This chapter considers an atom as a quantum mechanical system. But the electromagnetic radiation will be treated as classical waves described by the Maxwell equations. The treatment is called semi-classical. Quantum mechanics describes an atom as a quantized system with a set of discrete energy levels. Each energy level is designated by a set of quantum numbers like the principal quantum of the Rydberg level and the quantum numbers representing the orbital, spin and total angular momentum. A molecule will have a different set of quantum numbers arising from its rotational and vibrational motions. Atoms in a solid have energy levels in the form of a band. The observed spectrum due to spontaneous emission from a collection of atoms is not monochromatic, but it is spread over a certain range of frequencies. The absorption spectrum also shows that the atom does not absorb at a single frequency, but it is spread over a band of frequencies around the resonance frequency.