Abstract We are now equipped with the necessary mathematical background to analyse NMR in quantum-mechanical terms. In this chapter, the analysis is restricted to standard NMR, essentially the free-induction decay following a single r.f. pulse. The first section deals with the case when the only interaction experienced by the spins is the Zeeman interaction with an external field. It illustrates the way in which quantum mechanics is used in practice and its main result is a quantum justification of the equation of motion of the magnetization derived in Chapter 1 by a purely classical approach. The second section is a simplified description of chemical shifts and indirect interactions and of their influence on the NMR spectra. For such a simple problem, it will already become apparent that the formal machinery developed in the previous chapters is far superior to the classical or naive quantum methods, as regards the ease and the simplicity with which quantitative results are obtained.
Abstract We are now equipped with the necessary mathematical background to analyse NMR in quantum-mechanical terms. In this chapter, the analysis is restricted to standard NMR, essentially the free-induction decay following a single r.f. pulse. The first section deals with the case when the only interaction experienced by the spins is the Zeeman interaction with an external field. It illustrates the way in which quantum mechanics is used in practice and its main result is a quantum justification of the equation of motion of the magnetization derived in Chapter 1 by a purely classical approach. The second section is a simplified description of chemical shifts and indirect interactions and of their influence on the NMR spectra. For such a simple problem, it will already become apparent that the formal machinery developed in the previous chapters is far superior to the classical or naive quantum methods, as regards the ease and the simplicity with which quantitative results are obtained.
Abstract As stated in the preface, it is only through the use of quantum mechanics that one can get a real understanding of the working of the sophisticated methods that turned high-resolution NMR into a fundamental and powerful tool for the investigation of molecules, especially in the liquid state. However useful, the classical and naive quantum descriptions given in Chapter 1 suffice only for interpreting the simplest version of NMR spectroscopy, namely the resonance spectra obtained either by the CW method or by the Fourier transform of the FID following a single r.f. pulse. As regards methods such as 2D spectroscopy, these descriptions are fundamentally inadequate and provide at best a qualitative and more-or-less intuitive feeling of what happens, very often with serious conceptual difficulties.
Abstract Quantum mechanics was created mainly to account for microscopic phenomena that occur on a scale of order IO−10 m. It has turned out to be an astonishingly successful theory which applies on a scale down to at least 10−18 m. Quantum theory has become established as part of the essential core of physics. The purpose of this introductory chapter is twofold. Firstly, in Sections 1.1–1.7 we review the formalism of elementary nonrelativistic quantum mechanics. In this review we attempt to give enough of the terminology and basics of the subject so that a reader who is not already familiar with quantum mechanics will be able to follow the applications considered later in the book. For the reader who finds our discussion too brief, the literature on quantum mechanics is vast and we recommend in particular Refs 1–5. The second purpose of this chapter is to present the basic tools of operator methods as they are used in this book: thus in Sections 1.8–1.11 we discuss commutation relations, shift operators, and the virial, hypervirial, and Hellmann Feynman theorems.
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.
This chapter gives a summary of the mathematical formalism used in quantum mechanics, with a short bibliography put directly at its end. It should be seen as a complement, to be used by readers who wish to know more than what has been recalled about the mathematical tools in the other chapters. Some results are given without explicit proofs; they can be found for instance in Chapters II and IV of [1] . Many quantum mechanics textbooks introduce its general formalism in a more complete way, for instance Chapter VII of [2] , Chapter 3 of [3] , or Chapter 2 of [4] . We first summarize the general formalism of quantum mechanics for any physical system (§11.1), with the Dirac notation; we then study how this formalism treats the grouping of several physical systems into one single quantum system (§11.2); finally, we study a few simple special cases (§11.3), for instance a single particle in an external potential, with or without spin; the reader who prefers wave functions to more abstract reasonings in spaces of states may begin with this section. General physical system The general formalism of quantum mechanics applies to all physical systems, whether they contain a single particle, many particles of various sorts, one or several fields, etc. Quantum space of states The physical state of a system at each time is defined in quantum mechanics by a state vector which, in Dirac notation, is written ∣ ψ 〈 – or ∣ ψ(t) 〈 if one wishes to make the time dependence explicit.
The aim of relativistic quantum mechanics is to describe the finer details of the structure of atoms and molecules, where relativistic effects become nonnegligible. It is a sort of intermediate realm, between the familiar nonrelativistic quantum mechanics and fully relativistic quantum field theory, and thus it lacks the simplicity and elegance of both. Yet it is a necessary tool, mostly for quantum chemists. Pilkuhn's book offers to this audience an up-to-date survey of these methods, which is quite welcome since most previous textbooks are at least ten years old. The point of view of the author is to start immediately in the relativistic domain, following the lead of Maxwell's equations rather than classical mechanics, and thus to treat the nonrelativistic version as an approximation. Thus Chapter 1 takes off from Maxwell's equations (in the noncovariant Coulomb gauge) and gradually derives the basic aspects of Quantum Mechanics in a rather pedestrian way (states and observables, Hilbert space, operators, quantum measurement, scattering,. Chapter 2 starts with the Lorentz transformations, then continues with the Pauli spin equation and the Dirac equation and some of their applications (notably the hydrogen atom). Chapter 3 is entitled `Quantum fields and particles', but falls short of treating quantum field theory properly: only creation/annihilation operators are considered, for a particle in a box. The emphasis is on two-electron states (the Pauli principle, the Foldy--Wouthuysen elimination of small components of Dirac spinors, Breit projection operators. Chapter 4 is devoted to scattering theory and the description of relativistic bound states. Chapter 5, finally, covers hyperfine interactions and radiative corrections. As we said above, relativistic quantum mechanics is by nature limited in scope and rather inelegant and Pilkuhn's book is no exception. The notation is often heavy (mostly noncovariant) and the mathematical level rather low. The central topic is the description of atoms and molecules, including relativistic effects. The author fulfils this program in a reasonable way and offers a valuable tool to the targeted audience. I am not overly enthusiastic about the end result, but I might be prejudiced. Clearly, going further would require the full power of quantum field theory, but this is clearly beyond the scope of the book.