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Advanced modern physics. Solutions to problems. (English) Zbl 1333.81001

Hackensack, NJ: World Scientific (ISBN 978-981-4704-51-9/pbk). viii, 339 p. (2016).
The present excellent textbook provides solutions to over 180 problems published in the third and last book on modern physics “Advanced Modern Physics: Theoretical Foundations” by the second author [Advanced modern physics. Theoretical foundations. Hackensack, NJ: World Scientific (2010; Zbl 1206.81004)]. This seems to be a very challenging material, ranging across advanced quantum mechanics, angular momentum, scattering theory, Lagrangian field theory, symmetries, Feynman rules, quantum electrodynamics (QED), higher-order processes, path integrals, and canonical transformations for quantum systems.
After a short “Introduction”, Chapter 2 of the textbook deals with quantum mechanics in an abstract Hilbert space. Several problems on basic operator methods are solved. Ehrenfest’s theorem, which provides the direct path to classical correspondence, is derived and applied. It is shown that the usual many-body Hamiltonian commutes with the number operator. Another problem examines the translation operator, and the overlap of the eigenstates of position and momentum is then derived. Matrix mechanics is considered in one problem. The abstract time-independent Schrödinger equation is projected onto eigenstates of momentum in another. Finally, the transition from wave functions to abstract Hilbert space is examined in detail for one-dimensional rotations.
Chapter 3 contains an extensive overview on the angular momentum. The problems start with the veryfication of several commutation relations. Finite matrix representations of the angular momentum operator are then derived, including the Pauli matrices. Other problems deal with the calculation of the finite rotation matrices. Several problems focus on the properties of the Clebsch-Gordon (C-G) coefficients, used for the coupling of two angular momenta. One problem extends the derivation of these coefficients to a spin \(s=1\). The relation to Wigners \(3j\)-coefficients is studied and used to deduce the symmetry properties of the C-G coefficients. \(J\) descsribes an angular momentum. Other problems deduce essential properties of the \(6j\)-coefficients, employed in the coupling of three angular momenta. The definition of irreducible tensor operators (ITO) is examined in one problem, and, with the aid of the Wigner-Eckart theorem, valuable expressions for the reduced matrix elements of ITO’s working on one part of a coupled scheme are derived in another. A final problem derives the addition law for angular momenta from the weight diagram.
Chapter 4 deals with the scattering theory. After problems demonstrating the symmetric form and group property of the time-dependent scattering operator in leading order, there are a variety of problems on the time-independent scattering analysis. One problem derives the form of the incoming-wave scattering state, and another develops the analysis of potential scattering in term of these states. That the large semi-circle used to complete the contour for the scattering Green’s function in potential scattering makes a vanishing contribution as \(R\rightarrow\infty\) is demonstrated in a problem, while another works out the asymptotic form of the coordinate-space wave function in this same limit. A series of very useful problems then analyse the operator form of the general Green’s function, culminating in the closed expressions for the scattering states. A subsequent problem obtains a general expression for the \(S\)-matrix following from these states.
Chapter 5, entitled Lagrangian field theory, provides the basis for the transition from classical continuum mechanics to quantum field theory, where a problem on the string provides the paradigm. The stress tensor is extended to several generalized coordinates in a problem. A series of problems take the reader, with the aid of Ehrenfest’s theorem, to the quantum form of Hamilton’s equations. There are two problems on constructing hermitian forms of radial momentum operators. Three problems deal with the transition from the Schrödinger picture to the interaction picture with time-dependent operators. One problem shows that the canonical equal-time commutation relations are picture independent. Other problems deal with the massive scalar field, and the momentum in this field. An important series of problems considers the complex scalar field. Here, a conserved current exists, and the corresponding charge displays quanta of both signs, interpreted as particles and antiparticles. Several problems are concerned with the Dirac field theory.One obtains the momentum in the Dirac field. Another shows how to symmetrize the stress tensor in this case. Still another problem demonstrates the Lorenz invariance of the Dirac lagrangian density. One problem shows that normal-ordering the Dirac current simply subtracts the number of negative-energy states. Additional problems expand on the discussion in the text of the interacting Dirac-scalar theory with a Yukawa coupling. One analyzes the processes described by the corresponding interaction-picture Hamiltonian. Another demonstrates that the coupling constant \(g^2/(4\pi hc^3)\) is dimensionless. Still another analyzes the field equations in the Schrödinger picture, and uses Ehrenfest’s theorem to establish classical correspondence. Finally, a problem establishes the direct analogy between the Poisson bracket formulation of classical mechanics and quantum theory.
Symmetries are the subject of Chapter 6. There are problems concerned with the construction of the generators for various internal symmetries and the evaluation of their commutation relations. Other problems exted the analysis to Dirac fields obeying canonical anticommutation rules. The analysis of the massive scalar field is extended in a problem to the isovector case, where there are three equal-mass components to the field. Another problem establishes the connection to the analysis of the complex scalar field. An isospin-invariant Yukawa coupling between isovector-scalar and isospinor-Dirac fields is constructed in a problem, and the consequences of this isospin invariance are examined. An extended problem on the \(\sigma\)-model uses Noether’s theorem to construct the conserved vector current and axial-vector currents in the model arising from isospin (CVC) and chiral symmetry. The partially conserved axial-vector current (PCAC), obtained when a small explicit chiral-symmetry-breaking term is included, is then derived. Finally, the Lagrangian density obtained through an expansion around the new minimum in the chiral-symmetric potential of the \(\sigma\)-model, which leads to the spontaneous breaking of chiral symmetry and a nucleon mass, is found. Still another problem explicitly exhibits the \(\mathrm{SU}(2)_L\otimes \mathrm{SU}(2)_R\)-symmetry of the \(\sigma\)-model. The generators of the \(\mathrm{SU}(3)\) internal symmetry of the Sakata model are constructed in still another problem, and manipulated to derive the Gell-Mann-Nishijima relation between charge, hypercharge, and third component of isospin. Finally, a problem explicitly verifies the fundamental \(\mathrm{SU}(3)\) matrix relations in the case of infinitesimal transformations.
Chapter 7 is entitled “Feynmann rules”. Here, the discussion is based on the use of Wick’s theorem to derive \(S\)-matrix elements for various theories and processes from the scattering operator, and thereby to construct the Feynmann diagrams and deduce the relevant Feynman rules. The Dirac-scalar theory serves as a prototype, and several problems deduce the \(S\)-matrix elements for various processes in this theory, extending the Feynman rules in the text to include antiparticles. One interesting problem displays all the relevant fourth-order Feynman diagrams for this theory. The decay rate for the scalar field \(\phi\rightarrow N+\tilde N\), if kinematically allowed, is calculated as an informative exercise in another. The mass renormalization of \(\phi\) is studied in still another. For illustration, the theory is enlarged to include two independent scalars in several additional problems. The analysis is extended in the problems to a Dirac-scalar-vector theory, where a neutral, massive vector field is included along with the neutral, massive scalar, \(S\)-matrix elements for various processes, Feynman diagrams, and Feynman rules are again obtained. One problem relaxes the restriction to a normal-ordered scalar density, and demostrates that the result is an additional fermion mass renormalization. Finally, there is a useful problem concerned with calculating various matrix elements of the creation and destruction operators.
Chapter 8 is on quantum electrodynamics (QED). First, there are problems on current conservation and commutation relations, the form of the Coulomb interaction, and basic relations satisfied by the Dirac matrices. The density of states required in the calculation of the Compton cross section is evaluated in one problem, while the contribution of the square of the direct term to the Klein-Nishina formula for this process is evaluated in another. While the details may become more complicated, these exercises provide the reader with the tools required to calculate any cross section. Limiting forms of various cross sections are examined. The amplitudes for pair production and bremsstrahlung are derived in one problem. The covariant photon polarization sum is derived in another. The amplitude and Feynman diagrams for Bhabha scattering, \(e^-+e^+\rightarrow e^-+e^+\) are derived in a problem. As an exercise, the Dirac equation for the external legs is used to remove the \(q_\mu\)-terms in the photon propagator for some specific processes. A pair of problems derive Furry’s theorem, which states that loops with an odd number of photon interactions are absent in QED. Finally, a pair of problems demonstrate that neither one possible modification of the current, nor the addition of a Schwinger term to the current commutator, alter the lowest-order \(S\)-matrix in QED.
Chapter 9, entitled “Higher-Order-Processes”, concerns the higher-order contributions in QED. One problem examines the role of the mass counter-term. A second derives the important Feynman parametrization relations, which allow one to complete the square and do momentum integrals. Still others examine the \(\epsilon\rightarrow0\) limit in the dimensional regulation of these momentum integrals in \(n=4-\epsilon\) dimensions. Another important problem shows how, by letting a photon end up everywhere else on a charged electron line running through a diagram, one can explicitly eliminate those terms in the photon propagator proportional to \(q_\mu\). The use of current conservation and equal-time commutation relations to eliminate such terms in a Dirac-vector theory is demonstrated in another. The amplitude and Feynman diagrams for the scattering of light-by-light through an electron loop are investigated in an problem. The properties of the adiabatic damping term are studied in another problem. The properties of the cut-off charge renormalization constant are examined in still another. One problem varifies Ward’s identity in fourth order. An important problem performs a counting of the powers of momentum in an arbitrary Feynman diagram, which, in a second problem, leads to an enumeration of the four classes of primitively-divergent diagrams. Additional problems demonstrate that one obtains the proper lowest-order results upon iteration of Dyson’s equation for the vertex and Ward’s equation for his vertex-construct. It is demonstrated in a problem that one obtains the correct result when the renormalized charge is used in skeleton diagrams into which the finite, renormalized photon and electron propagators have been inserted. It is shown in an additional problem that the Schwinger term makes no contribution to the third-order \(S\)-matrix in an external field. Finally, the origin of the concept of a running coupling constant is demonstrated in a QED theory which contains both muons and electrons.
Chapter 10 focuses on path integrals which provide an essential alternative for quantizing classical systems. Two problems deal with indices and left-variational derivatives for Dirac fields. An important exercise details the calculation of the generating functional for a scalar field with a \(\lambda\phi^4\)-interaction, from which the propagator and Feynman rules follow. One problem shows how the sum over the diagonal matrix elements with a complete set of states, the “Trace”, of the thermal operator \(\exp(-\beta\tilde H)\) is independent of basis. Then, with an analysis that parallels that in the text for the propagator, three additional problems consider a particle in a potential and demonstrate how that Trace can be expressed as a path integral over an exponential of the action evaluated for imaginary time. An additional problem shows how the thermal average of a quantity is obtained from that path integral. The relation expressing a determinant as an integral over Grassmann variables is verified in fourth order in another problem. A final problem studies an effective field theory involving nucleon, pion, and scalar fields, and shows how one limit of this effective theory reproduces the \(\sigma\)-model.
Chapter 11 studies canonical transformations for quantum systems. In this solutions manual, the methods developed in the text to describe quantum fluids are used to solve two basic problems in quantum field theory. The first problem involves a real, massive scalar field interacting with a time-independent, localized \(c\)-number source. It is demonstrated that the exact interaction of two point sources is given by the Yukawa potential. The overlap between the new and old vacua is then evaluated analytically. The second Bloch-Nordsieck problem concerns the quantized radiation field interacting with a time-independent, \(c\)-number, external current. It is shown that the overlap of the new and old vacua is given by \(\exp(-N)\) where \(N\) is the number of photons, and there is an infinite number of long-wavelength photons present due to the infrared divergence of QED. One implication is that the long-wavelength limit cannot be properly treated with low-order perturbation theory. There are three problems on boson interactions, one on the symmetry of the scattering amplitude and two on the low-energy form of that amplitude. The integral for the depletion of the boson ground state is verified in a problem. General properties of the two-body matrix element are found in another. One problem provides the details of the transformation of the fermion Hamiltonian, and another then examines the properties of a specific excited state.
The book contains eight appendices which supply essential details. Appendix (A) is on the multipole analysis of the radiation field, (B) considers functions of a complex variable, (C) the electromagnetic field, (D) irreducible representations of \(\mathrm{SU}(n)\), (E) Lorentz transformations in quantum field theory, (F) Green’s functions and other singular functions, (G) dimensional regularization, and (H) path integrals.
The only remark which may be made to improve the new book is that, repeating the problems formulated in “Advanced Modern Physics: Theoretical foundations” [loc. cit.], instead of the cited numbers of equations, the equations themselves should have been given, including the definitions of all used parameters.
The present work helps dedicated students to get familiar with the basics of graduated quantum mechanics and to make solving of problems less challenging, but even enjoyable. (Review constructed using the contents information given by the authors in the “Introduction”.)

MSC:

81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81P05 General and philosophical questions in quantum theory
00A07 Problem books
00A79 Physics
81Qxx General mathematical topics and methods in quantum theory
81Txx Quantum field theory; related classical field theories
81Vxx Applications of quantum theory to specific physical systems

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