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This chapter presents systematic methods to evaluate the tunneling matrix elements in the Bardeen tunneling theory. A key problem in applying the Bardeen tunneling theory to STM is the evaluation of the tunneling matrix elements, which is a surface integral of the wavefunctions of the tip and the sample, roughly in the middle of the tunneling gap. By expanding the tip wavefunction in terms of spherical harmonics and spherical modified Bessel functions, very simple analytic expressions for the tunneling matrix elements are derived: the tunneling matrix elements are proportional to the amplitudes or the corresponding x-, y-, or z-derivatives of the sample wavefunction at the center of the tip. Two proofs are presented. The first proof is based on the Green's function of the Schrödinger's equation in vacuum. The second proof is based on a power-series expansion of the tip wavefunctions.

This chapter covers the rudiments of quantum theory, including the dual nature of light and matter, the Uncertainty Principle and probability concept, the electronic wavefunction, and the probability density function. Numerical examples are given to show that given the electronic wavefunction of a system, the probability of finding an electron in a volume element around a certain point in space can be readily calculated. Finally, the electronic wave equation, the Schrödinger equation, is introduced. This discussion is followed by the solution of a few particle-in-a-box problems, with the ‘box’ having the shape of a wire (one-dimensional), a cube, a ring, or a triangle. Where possible, the solutions of these problems are then applied to a chemical system.

This chapter examines the major assumptions and approximations in electronic structure calculations for perfect solids. It emphasizes those issues that are important in defect studies, such as the one-electron approximation, Koopman's approximation, Coulomb correlation, wavefunctions in band structures, and correspondence between electrons and holes.

In the early decades of the 20th century, a large and convincing body of experimental evidence pointed to the fact that atomic particles share some properties of massive bodies and some properties of waves. The founders of the new mechanics of the microscopic world, in search of the appropriate equations for the description of what seemed then a very weird universe, turned to the equations of vibrating strings — material waves whose state and energy change in leaps with the number of nodes. There is a certain resemblance between the quantum mechanical Schrodinger equation and the classical dynamic equation of vibrating strings. This chapter discusses the fundamentals of quantum mechanics, dynamic variables, wavefunctions, operators, the Schrödinger equation and stationary states, hydrogen atom and atomic orbitals, approximate quantum chemical methods, evolution of quantum chemical calculations, dimerisation energies and basis set superposition error, and early experiences in quantum chemistry.

Electronic states in disordered conductors on the verge of localization are predicted to exhibit critical spatial characteristics indicative of the proximity to a metal- insulator phase transition. In this chapter, we describe how the scanning tunneling microscopy (STM) measurements on doped semiconductors near the metal-insulator transition can be used to access these critical wavefunctions. Specifically, we use the STM to probe to visualize electronic states in Ga1-xMnxAs samples close to the metal-insualtor transition. Our measurements show that doping-induced disorder produces strong spatial variations in the local tunnelling conductance across a wide range of energies. Near the Fermi energy, where spectroscopic signatures of electron-electron interaction are the most prominent, the electronic states exhibit a diverging spatial correlation length. Power-law decay of the spatial correlations is accompanied by log-normal distributions of the local density of states and multifractal spatial characteristics. Our method can be used to explore critical correlations in other materials close to a quantum critical point.

This chapter presents systematic methods to evaluate the tunneling matrix elements in the Bardeen tunneling theory. A key problem in applying the Bardeen tunneling theory to STM is the evaluation of the tunneling matrix elements, which is a surface integral of the wavefunctions of the tip and the sample, roughly in the middle of the tunneling gap. By expanding the tip wavefunction in terms of spherical harmonics and spherical modified Bessel functions, very simple analytic expressions for the tunneling matrix elements are derived: the tunneling matrix elements are proportional to the amplitudes or the corresponding x-, y-, or z-derivatives of the sample wavefunction at the center of the tip. Two proofs are presented. The first proof is based on the Green’s function of the Schrödinger’s equation in vacuum. The second proof is based on a power-series expansion of the tip wavefunctions.

Henry has successfully tested the Split button of his quantum suit that enables a quantum object to be in several places at the same time. This bizarre phenomenon is explained by the superposition principle, which reflects Henry’s wavelike properties. Yet Henry remains a single entity as he splits into two superposed versions. In order to understand such behavior, the emergence of the relevant concepts, from classical wave theory, through the quanta hypothesis and the subsequent development of QM, are reviewed. The possibility of such momentous changes in human thinking as the QM scientific revolution is disconcerting if one seeks the truth of the world through physics. A retrospective of the views on the rapport of physics with reality and its roots in mythical or symbolic pictures of the world suggests that the QM revolution eludes straightforward explanation. The QM superposition principle is mathematically formulated in the appendix to this chapter.

This chapter shows that ultracold atoms placed in an optical disordered potential form an excellent system to study Anderson localization (AL) experimentally. The discussion covers AL for the beginner, ultracold atoms in optical speckle, one-dimensional Anderson localization, direct observation of Anderson localized 1D wavefunctions, what happens beyond the 1D effective mobility edge, and towards 2D and 3D experimental studies of AL.

This chapter derives computational chemistry methods for calculating molecular properties. It approximates the exact unperturbed wavefunctions and the complete set of operators in the expression for the linear response function or polarization propagator, which were derived in Chapter 2 via perturbation and response theory for exact states. This leads to methods like the random phase approximation (RPA), also called the time-dependent Hartree-Fock method (TDHF), and its multiconfigurational generalization MCRPA as well as to the second order polarization propagator approximation (SOPPA). Explicit working equations for some of these computational chemistry methods are presented. The chapter includes a discussion of recent extension and approximations to the SOPPA method like SOPPA(CCSD), SOPPA(CC2), and RPA(D).

This chapter analyzes issues that deal with quantum statistics (symmetry of wavefunctions); elements of the second quantization formalism (the occupation-number representation); and the simplest systems with a large number of particles.

This chapter deals with problems related to quasi-classical energy quantization; quasi-classical wavefunctions, probabilities, and mean values; penetration through potential barriers; and 1/N-expansion in quantum mechanics.

An electromagnetic field with a Lorentz-gauge vector potential and a scalar potential equal to zero can induce disturbance in an electron. In general, this disturbance can be considered a small perturbation which gives rise to transitions between unperturbed states at a rate expressed by first-order perturbation theory. A transition rate cannot be defined unless there is a spread of states in energy or, where the transition is between discrete states, there is a finite bandwidth of incident radiation. Semiconductors exhibit transitions to and from conduction bands, and a spread of final states always exists in the case of absorption. This chapter explores radiative transitions in semiconductors, first by describing the transition rate, local field correction, and photon drag. It then discusses photo-ionisation and radiative capture cross-sections, wavefunctions, direct interband transitions, photo-deionisation of a hydrogenic acceptor, photo-ionisation of a hydrogenic donor, photo-ionisation of quantum-defect impurities and deep-level impurities, indirect transitions, and indirect interband transitions. It also looks at free-carrier absorption as well as free-carrier light scattering.

Excitons are the primary photo-excited states in conjugated polymers. This chapter develops effective-particle models of excitons in the weak and strong-coupling limits. Excitons are classified as being Mott-Wannier in the weak-coupling limit. This class includes Frenkel, charge-transfer, and Wannier excitons. An exciton wavefunction mapping is introduced that transforms configuration interaction amplitudes into real-space wavefunctions. Excitons are classified as being Mott-Hubbard in the strong-coupling limit. Finally, numerical calculations of the intermediate coupling regime are described.

This chapter introduces the various programs that have been devised to accompany the text, using both Python and Fortran programs, some of which are interactive. The topics that are programmed and discussed include inter alia graphing, contouring, HMO calculations, Madelung constants, linear least squares, matrix operations, radial and angular wavefunctions, quadrature and roots of polynomials. In many cases, example data sets are provided in order to ensure correct working of the programs.

The concept of wavefunction was introduced in the first 1926 paper by Erwin Schrödinger as the central object of the atomic world and the cornerstone of quantum mechanics. It is a mathematical representation of de Broglie’s postulate that the electron is a material wave. It was defined as everywhere real, single-valued, finite, and continuously differentiable up to the second order. Nevertheless, for many decades, wavefunction has not been characterized as an observable. First, it is too small. The typical size is a small fraction of a nanometer. Second, it is too fragile. The typical bonding energy of a wavefunction is a few electron volts. The advancement of STM and AFM has made wavefunctions observable. The accuracy of position measurement is in picometers. Both STM and AFM measurements are non-destructive, which leaves the wavefunctions under observation undisturbed. Finally, the meaning of direct experimental7 observation and mapping of wavefunctions is discussed.

The problem of electron spin was in some way connected with special relativity. Dirac’s fascination with relativity and his already burgeoning reputation made the search for a fully relativistic form of the new quantum mechanics irresistible. An important clue was available in a treatment that Pauli had published in 1927, in which he had represented the spin angular momemtum operators as 2 × 2 Pauli spin matrices. Dirac presumed that a proper relativistic wave equation could be derived simply by extending the spin matrices to a fourth member, but quickly realized this couldn’t be the answer. As he played around with the equations, in 1928 he found that he needed 4 × 4 matrices, instead. This allowed him to derive a relativistic wave equation, and to show that electron spin was indeed the result. The two extra solutions were subsequently shown to belong to the positron. Dirac had discovered antimatter.

Despite the success of Schrödinger’s description of the H-atom, it became apparent that the spectrum of the simplest multi-electron atom—helium—could not be so readily explained. And the spectra of other atoms showed ‘anomalous’ splitting in a magnetic field. In 1920 Sommerfeld introduced a fourth quantum number. A few years later Pauli was led to the inspired conclusion that the electron must have a curious ‘two-valuedness’ characterized by a quantum number of ½, and went on to discover the exclusion principle. Perhaps this is because the electron possesses a self-rotation, leading to the notion of electron spin, potentially explaining why each orbital can accommodate only two electrons. Heisenberg traced this behaviour back to the symmetry properties of the wavefunctions. By observing which transitions in the spectrum of helium are allowed and which are forbidden, we can deduce the generalized Pauli principle, from which the exclusion principle follows.

Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ‎ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function<ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G₂ in the G₁-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.

“The periodic table of the elements is one of the most powerful icons in science: a single document that captures the essence of chemistry in an elegant pattern.” This statement taken from Eric Scerri’s marvelous book The Periodic Table: Its Story and Its Significance (Scerri 2007) grasps in one simple sentence the status that the periodic table has acquired in chemistry, but not only in chemistry: every person all over the world who took high school chemistry remembers for the rest of his/her life at least one thing from it (and from science courses in general): this mysterious table of the elements omnipresent in all books and documents, often decorating the classroom. Why? Although the answer is not easy it is probably because a whole discipline of science is condensed in a simple table or scheme offering at one glimpse the essence (and the beauty) of a part of science which for the rest of the lives of many students will remain unexplored—and even for which these students will create an aversion among others in view of the “polluting role” of chemistry. Nevertheless at some moment in their lives these students felt “something” that was remarkable and which they always remember, as witnessed by their comments when visiting their old school of university with their children. In the hierarchy of the sciences chemistry is often considered, mainly by physicists, as the “physics of the outer shell.” It is sometimes said that chemistry, as also quoted by Scerri (2007), has no deep ideas, not a few fundamental laws like in physics or biology (such as those governing quantum mechanics, relativity and evolution), but the not-so-distant observer will disagree: chemical periodicity as precisely reflected in the periodic table is in my view not only the most fundamental law of chemistry, but is a law or if you want an “organizing principle” with the same status as these famous laws in adjacent disciplines! Chemical periodicity is at the heart of reducing an astonishing amount of experimental data (and nowadays theoretical data as well) to a limited number of patterns often with common origin, enabling one to understand and interpret the properties of the now more-than 50 million compounds registered in the databases of the Chemical Abstracts Services (Toussant, 2009).