Introduction The emission and absorption of radiation, as well as the conversion of radiation into other modes of energy such as heat or electricity, all involve interaction between electromagnetic waves and atoms, molecules, or free electrons. Such daily phenomena as the radiative emission by the sun, the shielding of earth from harmful UV radiation by the ozone layer, the blue color of the sky, and red sunsets are all – despite their celestial magnitude – generated by microscopic particles. Most lasers depend on emission by excited atoms (e.g. the He-Ne laser), ionized atoms (the Ar + laser), molecules (CO or CO 2 lasers), impurities trapped in crystal structures (Nd: YAG or Ti:sapphire lasers), or semiconductors (GaAs diode lasers). Similarly, many scattering processes of interest (e.g., Rayleigh or Mie scattering) result from the exchange of energy and momentum between incident radiation and atomic or molecular species. In the previous chapter we saw that the energy of microscopic particles is quantized: their energy can be acquired, stored, or released only in fixed lumps called quanta . The example of the “particle in the box” (eqn. 8.19) illustrated that these energy quanta are specific not only to the particle itself but to the system to which it belongs. Thus, in the box, the energy of the particle is specified by its own mass and by the dimension of the box; in a different box, the same particle will have an entirely different system of energy levels and the quanta will have different magnitudes.
Introduction The emission and absorption of radiation, as well as the conversion of radiation into other modes of energy such as heat or electricity, all involve interaction between electromagnetic waves and atoms, molecules, or free electrons. Such daily phenomena as the radiative emission by the sun, the shielding of earth from harmful UV radiation by the ozone layer, the blue color of the sky, and red sunsets are all – despite their celestial magnitude – generated by microscopic particles. Most lasers depend on emission by excited atoms (e.g. the He-Ne laser), ionized atoms (the Ar + laser), molecules (CO or CO 2 lasers), impurities trapped in crystal structures (Nd: YAG or Ti:sapphire lasers), or semiconductors (GaAs diode lasers). Similarly, many scattering processes of interest (e.g., Rayleigh or Mie scattering) result from the exchange of energy and momentum between incident radiation and atomic or molecular species. In the previous chapter we saw that the energy of microscopic particles is quantized: their energy can be acquired, stored, or released only in fixed lumps called quanta . The example of the “particle in the box” (eqn. 8.19) illustrated that these energy quanta are specific not only to the particle itself but to the system to which it belongs. Thus, in the box, the energy of the particle is specified by its own mass and by the dimension of the box; in a different box, the same particle will have an entirely different system of energy levels and the quanta will have different magnitudes.
Introduction Until now, our discussion of the interaction between radiation and matter has concentrated only on the spectral aspects of radiation. The results could determine the wavelengths for absorption and emission or the selection rules for such transitions, but could not be used to determine the actual extent of emission or absorption. These too are important considerations which are needed to fully quantify radiative energy transfer. Unfortunately, none of the classical theories can predict the extent of emission from an excited medium, or even the extent of absorption. Although the discussion in Section 4.9 (on the propagation of electromagnetic waves through lossy media) touched briefly on the concept of attenuation by absorption, it failed to show the reasons for the spectral properties of the absorption or to accurately predict its extent. We will see later that the classical results are useful only as a benchmark against which the actual absorber is compared. The objective of this chapter is therefore to present an introduction to quantum mechanical processes that control the emission and absorption by microscopic systems consisting of atoms and molecules. The results will then be used to predict the extent of emission by media when excited by an external energy source and to evaluate the absorption of incident radiation. It is now well recognized that all emission or absorption processes are the result of transitions between quantum mechanical energy levels.
At the beginning of the century doubts were growing about the whole edifice of classical physics, including the laws of mechanics in their Lagrangian and Hamiltonian forms, of statistical thermodynamics, as formulated by Boltzmann and Gibbs, and of optics, namely Maxwell’s and Fresnel’s theory. Einstein proposed that electromagnetic radiation exists in discrete quanta whilst propagating in space as well as during the emission or absorption processes. Scattering by free electrons is a common interaction connected with the absorption of electromagnetic radiation. If the electromagnetic energy is large compared with the ionization energy, the atomic electron can be treated as free. The characteristic features of the scattered radiation have been explained by Compton, on the basis of an elastic collision of a photon with a free electron, where both energy and momentum are conserved. The concept of the matter wavelength holds for both relativistic and non-relativistic particles, provided that the appropriate form of momentum is used.
Light emission occurs as a result of interaction between light and matter. Let us assume a two-level atom with levels 1 (ground state) and 2 (excited state). The energies of the ground and excited states are E 1 and E 2 , respectively, and the energy difference is therefore given by the difference ΔE 12 =E 2 −E 1 . When light with photons of energy equal to this difference is absorbed by the atomic or molecular system, an electron will be excited from level 1 to level 2. The energy of the photons is given by ΔE 12 =E 12 =hν 12 (h is Planck's constant and 1→2 is the frequency of light resulting from the level 1→2 transition). This energy is absorbed by an atom or a molecule that has energy levels, separated by ΔE, where these energy levels are the ground and the excited states. This kind of “quantum jump” of an electron between two states occurs in atomic systems between electronic levels; it can be extended to molecular systems, where vibrational and translational energy levels participate in the quantum jump. The interaction between light and matter involves the transition of electrons between different states. This interaction results in the absorption of photons (stimulated absorption) as well as spontaneous and stimulated emission. These processes can be described using Einstein's A and B coefficients, as will be described in the next paragraphs.
Introduction Electromagnetic interactions play a central role in low-energy physics, chemistry, and biology. They are responsible for the cohesion of atoms and molecules and are at the origin of the emission and absorption of light by such systems. They can be described in terms of absorptions and emissions of photons by charged particles or by systems of charged particles like atoms and molecules. Photons are the energy quanta associated with a light beam. Since the discoveries of Planck and Einstein at the beginning of the last century, we know that a plane light wave with frequency ν , propagating along a direction defined by the unit vector u , can also be considered as a beam of photons with energy E = hν and linear momentum p = ( h ν/ c ) u . We shall see later on that these photons also have an angular momentum along u depending on the polarization of the associated light wave. Conservation laws are very useful for understanding the consequences of atom–photon interactions. They express that the total energy, the total linear momentum, and the total angular momentum are conserved when the atom emits or absorbs a photon. Consider for example the conservation of the total energy. Quantum mechanics tells us that the energy of an atom cannot take any value. It is quantized, the possible values of the energy forming a discrete set E a , E b , E c , . . . In an emission process, the atom goes from an upper energy level E b to a lower one E a and emits a photon with energy h ν. Conservation of the total energy requires The energy lost by the atom on going from E b to E a is carried away by the photon.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.
Many processes, including absorption and stimulated emission occurring in lasers, can be handled using a semi-classical model for the atom–radiation interaction, in which the matter is given a quantum description, but the radiation is represented as a classical electromagnetic field (see Chapter 2). There are other phenomena that cannot be adequately described without quantizing the radiation. For example, it has been known since the 1930s that spontaneous emission can only be treated correctly using a fully quantum framework for the interaction, in which both the matter and the radiation are quantized, as we shall see in Chapter 6. However, it was not until the 1970s that situations were found in which a free electromagnetic field , far from sources, exhibited properties and behaviour that could not be described by a classical field, but which could be perfectly well interpreted in terms of a quantized field. This chapter is devoted to the quantization of the free electromagnetic field, far from the charges and currents sourcing it. This free electromagnetic field will be called radiation , and in Chapter 6 we shall specify exactly what is meant by radiation when sources are present. The canonical quantization procedure used here starts from a description of the classical dynamics of the field in the framework of the Hamiltonian formalism, the basic features of which are discussed in Section 4.1.