This chapter provides the quantum mechanical definition of the multipole moments, and study those of their properties that follow from symmetry considerations. It offers both the power and limitations of symmetry arguments. The interaction between the electromagnetic moments of atoms and nuclei determines the hyperfine structure of the spectrum, and because of this hyperfine spectroscopy is a powerful technique for studying the electromagnetic properties of nuclei. The chapter describes the quantum theory of systems that have a Correspondence Principle limit in classical point mechanics. It explores the detailed connection between the abstract formalism and the wave functions of Schrodinger's theory by computing the x-p transformation function. The chapter offers the symmetry properties of two-nucleon systems. The statistical mechanics of systems obeying the exclusion principle was worked out by Fermi and Dirac and is called Fermi-Dirac statistics; particles that obey this type of statistics are called fermions.
This chapter provides the quantum mechanical definition of the multipole moments, and study those of their properties that follow from symmetry considerations. It offers both the power and limitations of symmetry arguments. The interaction between the electromagnetic moments of atoms and nuclei determines the hyperfine structure of the spectrum, and because of this hyperfine spectroscopy is a powerful technique for studying the electromagnetic properties of nuclei. The chapter describes the quantum theory of systems that have a Correspondence Principle limit in classical point mechanics. It explores the detailed connection between the abstract formalism and the wave functions of Schrodinger's theory by computing the x-p transformation function. The chapter offers the symmetry properties of two-nucleon systems. The statistical mechanics of systems obeying the exclusion principle was worked out by Fermi and Dirac and is called Fermi-Dirac statistics; particles that obey this type of statistics are called fermions.
Abstract Two concepts, of fundamental importance to quantum mechanics, will be discussed in this chapter. The first is that of symmetry. Even though the concept of symmetry is familiar in a wide range of natural sciences, the way symmetry constrains the consequences of quantum mechanics is rather subtle and, at the same time, far-reaching. The second is the symmetry property of the states under the exchange of two identical particles. Identical particles with a half-integer spin (fermions) obey the Fermi–Dirac statistics: the wave functions are totally antisymmetric under the exchange of the particles. Identical particles with an integer spin (bosons) are instead subject to the Bose–Einstein statistics. Their wave functions are totally symmetric.
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
Bohr’s correspondence principle is that associated with each observable in classical mechanics is an observable in quantum mechanics that follows certain quantization axioms which can be used to explain the energy spectra of atoms and quantum oscillators. In Schrodinger’s wave mechanics, the state of the system at any time is described by a density operator in a Hilbert space. The chapter describes how the Pauli equation can be used to calculate the average electric dipole moment and average magnetic dipole moment of an atom in an external electromagnetic field. This calculation enables to calculate the electric polarization field and the magnetization field of a material.
Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.
Our modern understanding of atoms, molecules, solids, atomic nuclei, and elementary particles is largely based on quantum mechanics. Quantum mechanics grew in the mid-1920s out of two independent developments: the matrix mechanics of Werner and the wave mechanics of Erwin Schrödinger. For the most part this chapter follows the path of wave mechanics, which is more convenient for all but the simplest calculations. The general principles of the wave mechanical formulation of quantum mechanics are laid out and provide a basis for the discussion of spin, identical particles. and scattering processes. The general principles are supplemented with the canonical formalism to work out the Schrödinger equation for charged particles in a general electromagnetic field. The chapter ends with the unification of the approaches of wave and matrix mechanics by Paul Dirac, and a modern approach, known as Hilbert space, is briefly described.
Because atoms and molecules are small, their behavior can only be understood using quantum mechanics, rather than classical mechanics. Quantum mechanics amounts to studying the behavior of electrons and nuclei and their interactions. We will limit ourselves to slowly moving electrons and nuclei, and thus will not need to discuss relativistic quantum mechanics. Large particles also obey quantum mechanics, but the relevant wavelengths are so small that we don't notice them. Small particles absolutely require the use of quantum mechanics for any meaningful understanding. This treatment is not merely an inconvenience: it leads to useful and interesting phenomena such as the Pauli exclusion principle, the Aufbau principle, Hund's rules, the Heisenberg uncertainty principle, wave-particle duality, the photoelectric effect, Ehrenfest's theorem, and the correspondence principle.