Search Results

This chapter examines applications drawn from perturbation theory. The main topic in perturbation theory is the energy and spontaneous decay rate of the 21-cm hyperfine line in atomic hydrogen. Before there were electronic computers, people had quite an accurate theoretical understanding of the energy levels in helium and more complicated systems. The trick was (and is) to find approximation schemes that treat unimportant parts of a physical system in quite crude approximations while reducing the interesting parts to a problem simple enough that it is feasible to compute but yet detailed enough to yield accurate results. The approximation methods in the chapter deal with the effects of small changes in the Hamiltonian, resulting for example from the application of a static or time variable electric or magnetic field. This may cause small changes in energy levels, and it may induce transitions among eigenstates of the original Hamiltonian.

This chapter illustrates atomic structures of increasing complexity by using energy level diagrams of atoms or ions, for which fluorescence and natural laser radiation have been proposed. It defines the terminology used in connection with atomic structure, transitions, and various processes due to the spectroscopic language difference between atomic and stellar spectroscopy, which may constitute a barrier in scientific communication between the two fields. To address the two different systems of labelling spectral lines, a compromise is made with the use of LS notation for energy levels and transitions except for those lines which are better known as members of well-known line series, e.g., Lyα, Lyβ, Hα (Balmer α) etc. in hydrogen.

This chapter discusses the structure of atoms. Topics covered include electrons, electron waves, the nuclear atom, atomic structure, energy levels, interpretation of quantum numbers, spectra and atomic structure in the light of quantum rules, atom building, quantum mechanical rules, the wave function, the transition of systems from one energy state to another, and the orthogonal property of wave functions.

The electronic structure of molecules includes electronic (2-10 eV, UV-Vis absorption), vibrational (10^-2 - 2 eV, infrared spectroscopy & Raman scattering), and rotational (10^–5 – 10^–3 eV, microwave spectroscopy) energy levels that are probed by appropriate spectroscopy methods. Light, incident on a molecule or molecular solid, is either absorbed (IR, single photon, non-zero derivative of dipole moment), or elastically (Rayleigh) or inelastically (Raman, two-photon, non-zero derivative of the polarizability) scattered. Fourier transform infrared (FTIR) spectroscopy finds much use in materials characterization, including in studying the curing of polymer composites now incorporated in aircraft structures. When atoms form solids their electronic structure, particularly the energy levels of the outer electrons involved in the bonding, are significantly altered. Both occupied and unoccupied levels in solids are probed. Photoemission spectroscopy (PES) with X-rays (XPS) or ultraviolet light (UPS) incidence, and inverse PES probe occupied and unoccupied energy levels of surfaces, respectively. X-ray absorption spectroscopy (XAS) complements XPS, and probes unoccupied energy levels of solids. X-ray absorption near-edge structure (XANES) provides information on the final density of unoccupied states, the transition probabilities, and many body effects. Extended X-ray absorption fine structure (EXAFS) provides element-specific nearest neighbor distances and their coordination number.

This chapter examines energy levels in semiconductors, first by discussing effective-mass approximation and electron dynamics as well as Zener-Bloch oscillations. It then discusses Landau levels, plasma oscillations, excitons, hydrogenic impurities, hydrogen molecule centres, core effects, deep-level impurities, scattering states, and impurity bands. Impurities not only introduce localised states into the forbidden gap or semilocalised states associated with upper minima, but also modify the states in the conduction and valence bands. The defect affects an electron moving in a band, and this process is described by positive energy solutions of the Schrödinger equation containing the impurity potential.

This chapter shows how to measure the Hölder regularity of the weak solutions that are constructed when the scheme is executed more carefully. For this aspect of the convex integration scheme, a notion of frequency energy levels is introduced. This notion is meant to accurately record the bounds which apply to the (v, p, R) coming from the previous stage of the construction. The chapter presents an example of a candidate definition for frequency and energy levels. Based on this definition, the effect of one iteration of the convex integration procedure can be summarized in a single lemma, which states that there is a solution to the Euler-Reynolds equations with new frequency and energy levels. The chapter also considers the High–Low Interaction term and the Transport term.

This chapter checks frequency energy levels for the velocity and pressure. It begins by comparing the different estimates obtained for the corrections to the velocity and the pressure with the Main Lemma. It then considers bounds that will be established for a particular constant C once the constant Bsubscript Greek Small Letter Lamda has been chosen. It also checks whether the frequency and energy levels of the new velocity and pressure are consistent with the claims of the Main Lemma (10.1). To complete the proof of the Main Lemma (10.1), it now only remains to choose a constant Bsubscript Greek Small Letter Lamda so that (243) and (244) can be verified for the new energy levels. This choice of Bλ‎ is the last step of the proof.

Electrons in solids and how materials interact with solar energy (radiation) are presented. Electrons absorb solar radiation and are promoted to a higher energy level, which is how matter interacts with solar energy. Details of the energy levels that they naturally move to and why these levels exist are presented in a comprehensive and understandable manner to introduce the reader to semiconductor physics. A concise discussion of quantum mechanics is given to bring a seemingly esoteric science into use in the development of energy bands allowable to the electron. Then how the electrons (and holes) move within this energy landscape is presented, ultimately yielding their concentration, a quantity required to understand how a solar cell works.

This chapter properly formalizes the Main Lemma, first by discussing the frequency energy levels for the Euler-Reynolds equations. Here the bounds are all consistent with the symmetries of the Euler equations, and the scaling symmetry is reflected by dimensional analysis. The chapter proceeds by making assumptions that are consistent with the Galilean invariance of the Euler equations and the Euler-Reynolds equations. If (v, p, R) solve the Euler-Reynolds equations, then a new solution to Euler-Reynolds with the same frequency energy levels can be obtained. The chapter also states the Main Lemma, taking into account dimensional analysis, energy regularity, and Onsager's conjecture. Finally, it introduces the main theorem (Theorem 10.1), which states that there exists a nonzero solution to the Euler equations with compact support in time.

The Hamiltonian for the Coulomb problem in n space dimensions is given. It involves the two n-vectors R and P , which form a Heisenberg algebra. It follows that the n(n–1)/2 components of orbital angular momentum L_ij formed out of R and P generate the algebra so(n). The Lenz–Runge n-vector is introduced and shown to be a constant of the motion. A properly rescaled Lenz–Runge vector is shown to generate, together with the L_ij , so(n+1). The Hamiltonian can be expressed in terms of the quadratic Casimir operator of this so(n+1). This yields the Balmer formula for the energy levels of the hydrogen atom. Biographical notes on Coulomb, Heisenberg, Lenz and Runge are given.

Although age criterion is commonly used to define who young people are, a criterion that concerns mentality allows people who are outside the usual age bracket to be considered as young people, provided that they also think as young people. Another criterion to be considered is a person's energy levels since those who have energy and physical stamina and strength may also be considered young. The period of youth—adolescence—is often seen as a period of transition from childhood to adulthood. In order to know how to deal with various future events, young people utilize this period for testing out different behaviors. Adolescence has been prolonged because of the years added to the formal education system. This chapter emphasizes the need to adopt an integrated perspective that incorporates the psychoanalytical, physiological, socio-psychological, and other such aspects in examining young people.

“LS coupling; Hund's rules” describes LS coupling as it gives an account of our understanding of atomic states. Two “energy-level” diagrams commonly printed are shown to be wrong and highly misleading. Accompanying “explanations” are debunked. Correct diagrams, with explanations, are given, and compared with theoretical formulae. Hund's rules, when properly formulated, are shown to hold in all cases to which they can legitimately be applied. Hund's rules must be stated as given here (often they are not). This claim is validated because all possible variants are excluded by counterexamples.

This chapter begins by explaining that in its first stages, the diagnosis of personality consists of crudely quantitative estimates of the attributes which successively attract attention. It then examines the aspects of diagnosis of personality — diagnosis of needs and the estimation of strength of needs. It discusses that in judging the strength of needs, it is necessary to keep constant, if possible, or make allowances for, the factors which affect the phenomenon measured such as the level of diffuse energy, general intelligence, special abilities, degree of inhibition, and knowledge of the presenting situation. This chapter also examines the estimation of manifest and latent needs. It also provides a brief summary of certain criteria of quantity.

In atoms, the electron orbits around the nucleus are quantized, occur for discrete energies. Transitions between energy levels require absorption or emission of photons. In the case of an unusual occupation of excited levels, individual photons can trigger a coherent transition of all such levels to the ground state, thus leading to a beam of coherent photons, a laser beam.

Chapter 3 discusses the interaction Hamiltonian, which determines the way that light interacts with matter. Simple perturbative analysis is applied to see if basic dynamics of atoms can be explained. The partial successes of perturbation theory are compared with predictions of an “exact” method of calculating the occupation probabilities of various atomic states. The “exact” method is shown to fail, establishing the need for improved approaches that yield correct results in later chapters. The density matrix is introduced as a tool for describing not only the populations of atomic energy levels but also the coherence that can be created and lost during the dynamic evolution of atoms in time. A vector model based on the Bloch vector is presented as a useful way of picturing coherent atom–field interactions in an optical “spin” space, which proves to be particularly useful in understanding multiple-pulse interactions. Mechanisms are described that cause line broadening in optical spectroscopy, such as the Doppler effect. In preparation for the extensive use in later chapters of models based on only two or three energy levels, it is also shown that multi-level real atoms can experimentally be converted into two-level systems for strict comparisons with theory.

Investigates the energy levels in a configuration when a heavy positive particle (proton) and a light negative particle (electron) are present. The wave functions and permissible energy levels are derived from Schrödinger's equation. The role of quantum numbers is discussed. Electron spin and Pauli’s exclusion principle are introduced. The properties of the elements in the periodic table are discussed, based on the properties of the hydrogen atom. Exceptions when such a simple approach does not work are further discussed.

This chapter provides an overview of the book's structure. Section 3 deals with the error terms which need to be controlled, whereas Part III explains some notation of the book and presents a basic construction of the correction. The goal is to clarify how the scheme can be used to construct Hölder continuous weak solutions—continuous in space and time—to the incompressible Euler equations that fail to conserve energy. Part IV shows how to iterate the construction of Part III to obtain continuous solutions to the Euler equations. It then discusses the concept of frequency energy levels, along with the Main Lemma. It also highlights some additional difficulties which arise as one approaches the optimal regularity and illustrates how these difficulties can be overcome. Parts V–VII verify all the estimates needed for the proof of the Main Lemma.

The properties of the ideal Bose gas are calculated from the integral equations for the energy and the number of particles as a function of the temperature and chemical potential. It is shown that the integral equations break down below the Einstein temperature that corresponds to the transition to the low-temperature state. The lowest single-particle energy level must be treated explicitly to get the proper equations. With the inclusion of the lowest single-particle energy level, the low-temperature behavior is calculated. The occupation of the lowest level becomes comparable to the total number of particles in the system below the Einstein temperature, and equal to the total number of particles at zero temperature. A numerical solution to the properties of the Bose gas is discussed, and the detailed calculations are assigned to the problems at the end of the chapter.

A hybrid system composed of a spin coupled to a mechanical degree of freedom is interesting for several reasons: enhancing the functionality of the spin and of the mechanical system, and in the quantum and semiclassical regimes. This chapter gives an overview of some of the goals of spin-coupled mechanical systems and then introduces the nitrogen-vacancy (NV) centre, its energy level structure, and its sensitivity to external fields, with a discussion on what makes NVs exciting for hybrid quantum technologies and in particular integration with mechanical oscillators. Simple group theory arguments will be introduced to explain the NV’s energy level structure and its coupling to strain. It concludes with a discussion of different NV centre-mechanical coupling mechanisms, specifically magnetic- and strain-based ones, with realistic values for the coupling strength and protocols for coupling, as well as technical challenges to achieving a functional NV-mechanical coupled system, particularly in the quantum regime.