Time for another book review, this time volume three of the Course in Theoretical Physics by Landau and Lifshitz. This book is available for sale in the bookstore below. This volume is titled, "Quantum Mechanics (Non-relativistic theory)," and covers, in one volume, standard non-relativistic quantum mechanics, atomic and molecular physics, group theory, and nuclear physics.
This book has eighteen chapters and six mathematical appendices on special functions. There are 152 individual sections for a total of 677 pages.
Chapter one covers the basics of quantum mechanics, beginning with the uncertainty principle. This is one of the hardest concepts for people to get used to and it encompasses most of the weirdness of quantum mechanics, this section also develops the idea of the state of a system. These ideas are further developed into the notion of the superposition of states; it is this superposition that prevents the standard idea of the excluded middle from being true. Then the authors begin to develop the mathematical machinery of quantum mechanics by introducing several sections on operators and their spectrum. Then there is a statement that quantum mechanics must reduce to Newtonian mechanics when Planck's constant is small. This leads to a discussion of wave functions and observables.
The second chapter can be thought of as an attempt to extend the results of the first volume (chapter two) to quantum mechanics. The chapter begins by establishing the Hamiltonian operator. This is extended to the derivative of operators with respect to a parameter (in this case, time). The conservation of energy is then extended to stationary states including the ground state, energy levels, degenerate states, and bound states. There are two sections of the mathematics of matrices. Then the authors interoduce the density matrix. This is followed by a section on the momentum operator, extending the idea of conservation of momentum. The chapter ends with a section on the uncertainty relations.
The third chapter introduces the Schrödinger equation and discusses its properties. It then presents the probability current density. Then the chapter shifts gears a little bit and describes how to obtain the Schrödinger equation from a variational principle. The authors then discuss one-dimensional applications, potential wells, the quantum harmonic oscillator, motion in an external field, and reflection and transmission.
Chapter four extends angular momentum into its quantum mechanical incarnation by introducing the rotation operator and its quantization. The first section also includes a fair bit of tensor manipulation, particuarly along the lines of the finite-rotation tensor. This is then extended to finding the eigenvalues and eigenfunctions of the angular momentum. The authors then present the selection rules for angular momentum. A section on parity follows, and the chapter ends with adding angular momenta.
Chapter five covers motion in a central field that is radially symmetrical. The first section turns the two body problem into an equivalent one-body problem and introduces spherical harmonics, and the radial and magnetic quantum numbers. This is expanded into a discussion of spherical waves along with spherical Bessel functions, Hankel functions, asymptotic expansions, and the phase shift. The next section involves resolving a plane wave. The next section asks and answers the question, "What is the critical field where the fall of a particle to the center of the field becomes possible?" The authors then address the issue of motion in a Coulomb field, here they begin with the case of a discrete spectrum and introduce hypergeometric functions, the principal quantum number, and Laguerre polynomials; the continuous spectrum introduces the confluent hypergeometric function, and then they explore Coulomb degeneracy. The chapter ends with another approach to the Coulomb field, that of using parabolic coordinates.
Chapter six deals with one of the principle methods of approximation in quantum mechanics, perturbation theory. The chapter begins with a presentation of time-independent perturbation theory. Following this are sections on the secular equation and time-dependent perturbations. Then there are several examples of transitions under perturbations: for finite time, periodic perturbations, and for continuous spectra. The authors then discuss the energy-time undertainty relations (a subject that generates a lot of confusion). The chapter ends with potential energy treated as a perturbation.
Chapter seven discusses the semi-classical case of quantum mechanics, beginning with the semi-classical wave function. Then the authors present the boundary conditions that allow a semi-classical approach. The next section introduces the Bohr-Sommerfeld rule and connects to the first volume in terms of adiabatic invariants and the sepration of variables method for solving the Hamilton-Jacobi equation, and the second volume in the presentation of the number of characteristic vibrations of a wave. The next section links the discussion of the semi-classical approach to the fifth chapter, by describing semiclassical motion in a central field. The authors then discuss quantum tunneling. The next section is very practical in its discussion of how to calculate matrix elements for the semi-classical approach and introduces the bizarre notion of complex time. The next section deals with transition probabilities and the problem of reflection above the potential barrier. The final section of the chapter asks the quation as to what the transition probabilities are for an adiabatic invariant.
Chapter eight covers an extremely important, and often misunderstood, topic in quantum mechanics—spin. The chapter opens with a length section introducing the idea of the total angular momentum of a particle as the sum of its orbital angular moemntum and its spin. This leads into a discussion of the spin operator (and the famous Pauli spin matrices). Then the authors introduce you to the mathematical objects called spinors. Using these techniques the wave functions of particles of arbitrary spin are calculated and the irreducible tensor formulation is introduced. These methods are extended, and connected to the theory of rigid bodies, through the finite rotation operator. the authors then cover the issue of the polarization of particles. The chapter ends with the subject of time reversal.
Chapter nine covers the important topic of identical particles, that is fermions and bosons. These particles are introduced in the fisrt section along with the Pauli principle. The next section explains some of the limitations of the Schrödinger equation regarding spin and introduces the exchange interaction for fermions. This is followed by a very important section that lays the ground-work for group theory and introduces the so-called Youngs diagram for symmetry. The final two sections of the chapter lay the groundwork for quantum field theory with the second quantization, first of bosons then fermions.
Chapter ten is very long and introduces atomic physics beginning with atomic energy levels, fine structure, and spin-orbit coupling. This is then extended to the electronic states of atoms, including Hund's rule. These ideas are applied to hydrogen-like atomic energy levels (one-electron). The next section covers the self-consistent field method for calculating energies and wave functions of stationary states. The problem of similar calculations for complex atoms (which cause the self-consistent field method to become unweildy) motivates the Thomas-Fermi equation and utilizes the virial theorem (from volume one). The next section covers the problem of the variation of wave functions of the electron in an atom from long to short distances. This prefaces the next section on the fine structure of atomic energy levels, a subject requiring a knowledge of the electron density near the atomic nucleus. Gears shift and the periodic table is the next topic of disucssion. The next section covers the notion that the binding energy of the inner shell is so large that when an electron transitions to an outer shell, the resulting ion is unstable; this section introduces x-ray terms and the Auger effect. This is followed by a discussion of the quantum mechanical treatment of multipole moments where the dipole moment is a vector and the multipole moment is a tensor. Then the authors discuss the Stark effect, where atomic energy levels are altered in an external field. The last section of the chapter deals with the linear Stark effect, where the atomic energy levels of hydrogen are split proportionally to the field, unlike other atoms.
Chapter eleven extends the atomic theory to diatomic molecules, beginning with a discussion of electron terms in such molecules. This is then extended to cover intersection of the terms and again develops the foundations of group theory. Then the chapter covers the relationship between the atomic and molecular terms. Then the important issue of valence groups is developed. The authors then discuss a topic of great interest to physical chemists, the vibrational and rotational structures in the diatomic molecule, first of the singlet, and then the four multiplet cases. The symmetry of the molecular terms is then explored. The next section is a practical one, calculating the matrix terms for a diatomic molecule. This is then extended to the case of the interaction between the electron state and the rotation of the molecule. Then the topic of atomic interactions at long ranges is addressed, the so-called van der Waals force. The chapter ends with a phenomena where a diatomic molecule can disintegrate spontaneously by passing from one state to another, called pre-dissociation, and their relationship to perturbations in the molecular spectra.
Chapter twelve is the introduction to group theory. The chapter begins with a discussion of symmetry transformations and the notion that these transformations can be commutative. This is the underlying idea for transformation groups and then point groups. Then the authors develop the concepts of group representations and then irreducible representations. These ideas are then extended to classifying terms, the so-called character table. These ideas are further extended to selection rules. Then the focus shifts to continuous groups, specifically the rotation group. The chapter ends with a section describing the representations of finite point groups.
Chapter thirteen returns to molecules, this time polyatomic molecules. The chapter begins with the application of group theory to classify molecular vibrations, and then vibrational energy levels. Then the issue of the stability of symmetrical molecular configurations is taken up, including the Jahn-Teller theorem. Then the authors disuss the quantization of the rigid rotator, beginning with a spherical rotator, then the asymmetrical rotator. These results are then extended to interactions between vibrations and rotations in molecules. The last section of the chapter takes up the task of classifying the molecular terms of the wave function in total, the sum of electron, nuclear vibrations, and rotations.
Chapter fourteen completes the discussion of angular momentum, beginning with the 3j symbols, and the Clebsch-Gordon coefficients. Then the authors use these to form a spherical tensor. Then there is a section of 6-j symbols (which have a bizarre application in black hole physics). Then they discuss the matrix elements for adding angular momenta (the plural form of angular momentum). The chapter ends with an extension of this to axially symmetric systems.
Chapter fifteen begins what I like to think of as something of a cheat, it is the behavior of quantum systems in fields. Of course in quantum theory fields are mediated by force carriers (in the case of electromagnetism, by photons) and are thus the realm of quantum electrodynamics and quantum field theory. This can be thought of as a semi-classical case of quantum mechanics. The chapter begins with a section on the Schrödinger equation in a magnetic field, introducing the Bohr magneton, and depends heavily on volume two of the series. Then the emphasis shifts to motion in a uniform magnetic field and introduces Landau levels (yes, the same Landau). This is then extended to atoms in a magnetic field, the Landé factor, Langevin's formula, and the quantum mechanical origin of magnetism. The chapter ends with the current dnesity in a magnetic field.
Chapter sixteen forms a core of basic nuclear physics. This begins with a presentation of the similarities between neutrons and protons called isotopic invariance, and brinign some elements of symmetry anf group theory into the mix. This is followed by a section on nuclear forces and their tensor character. There is a lengthy presentation of the shell model of nuclear structure. This is then extended to the case of nonspherical nuclei allowing for energy levels of a fixed nucleus and a rotational energy. There are several nuclear effects on atoms, the isotopic shift of energy levels due to finite mass or finite radius, and the hyperfine structure of the atom. The hyperfine structure can even be applied to molecules.
Chapter seventeen and eighteen constitute a course in the quantum theory of scattering in their own right. This begins with the general theory of scattering, including calculating the cross-section and partial scattering amplitudes, and then the general properties of scattering are explored. This is followed by an introduction to the S-matrix. Then the Born approximation and the transport scattering cross section are presented. These ideas are extended to the semi-calssical case. The next section explores the scattering amplitude for complex energies and introduces some ideas of the theory of Riemann surfaces. These ideas are then simplified by dispersion relations. The next section explores the momentum representation formulation of the scattering problem. Then the problem of high energy scattering is addressed. This is followed scattering at slow speeds and resonance scattering at low energies. This result is then extended to quasi-statioanry states. This is followed by Rutherford scattering. This notion is in turn extended to a system of wave functions in a continuous Coulomb field. Then there is a section on the collision of identical particles. These ideas are extended to nuclear particles in resoance scattering of charged particles. The next section describes elastic collisions between fast electrons and atoms. These ideas are generalized to include spin-orbit coupling-dependent scattering. The final section of chapter seventeen is another application of complex analysis, the so-called Regge-pole theory.
The final chapter of the book is a lengthy treatment of inelastic collisions, beginning with elastic collision in the presence of inelastic processes, introducing the reaction cross section. Then it addresses full-inelastic scattering of slow particles. Then the authors address the S-matrix where reactions occur. These ideas are then applied to the problem of quasi-stationary states. The focus then shifts to the problem of interactions between particles formed as a result of a previous interaction. Then the problem of calculating cross sections near the reaction threshold. The authors then address the problem of inelastic scattering between fast electrons and atoms. the next section introduces the effective retardation, the energy loss due to an interaction. Then the issue of interactions between high mass particles and atoms. This is then applied to neutron scattering. The last section is then about high-energy inelastic scattering.
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