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Ettore Majorana: Notes on theoretical physics. (English) Zbl 1041.81001

Fundamental Theories of Physics 133. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1649-2/hbk). xxv, 484 p. (2003).
In the book are published, for the first time, Majorana’s so called Volumetti; they comprise study notes he wrote in Roma between 1927 and 1931, possibly having in mind their inclusion in a future book. The present volume may thus be viewed as an excellent “modern” text of theoretical physics. Beyond that, it is also veritable goldmine of seminal new physical and mathematical ideas and suggestions, all still quite stimulating and applicable to modern research. A Preface dwells on the human and scientific personality of E. Majorana (Annotation).
In many etudes (\(\sim\)50) the wide series of problems and questions related to the mathematics, especially, to the mathematical physics are solved or described:
Volumetto I: Evaluation of some series. Table for the computation of \(x!\). Combinations. Definite integrals. Conformal transformations. Legendre polynomials. \(\nabla^2\) in spherical coordinates.
Volumetto II: \(\nabla^2\) in cylindrical coordinates. Expansion of the harmonic function in a plane. Diagonalization of a matrix. Numbers to the fifth, sixth and seventh power. Fourier expansions and integrals. About matrices. Series expansions. The equation \(y''+ Py= 0\). The equation \(y''=xy\). Various formulae. (Schwarz formula. Maximum value of random variables. Binomial coefficients. Expansion of \(1/(1- x)^n\). Relations between binomial coefficients. Mean values of \(r^n\) between concentric spherical surfaces).
Volumetto III.: Evaluation of some series. The equation \(\square H=r\). Evaluation of the integral \(\int^{\pi/2}_0 {\sin kx\over\sin x}\,dx\). Infinite products. Bernoulli numbers and polynomials. The group of proper unitary transformations in two variables. Exchange relations for infinitesimal transformations in the representations of continuous groups. The group of rotations \(O(3)\). The Lorentz group. Dirac matrices and the Lorentz group. Characters of \({\mathcal D}_j\) and reduction of \({\mathcal D}_j\times{\mathcal D}_j'\). Complete sets of first-order differential equations.
Volumetto IV: Expansions of Legendre polynomials in the interval \(-1\leq x\leq 1\). Multiplication rules for Legendre polynomials. Green functions for the differential equation \(y''+ (2/x-1)y+ \phi(x)= 0\). On the series expansion of the integral logarithm function. Fundamental characters of the group of permutations of \(f\) objects. Expansion of a plane wave in spherical harmonics. The Laplace equation. Integral representation of the Bessel functions. Cubic symmetry. Improper operators. The set of orthogonal functions defined by the equation \(y^{\prime\prime}_a= (x-a)y_a\). Fourier integral expansions. Circular integrals.
Volumetto V: Representations of the Lorentz group. Zeros of the half-order Bessel functions. Frequently used polynomials. Legendre polynomials. Spinor transformations. Infinite-dimensional unitary representations of the Lorentz group. The equation \((\square H+\lambda) A= p\).
The main contents of the present book defined the Majorana’s notes (\(\sim\)90), which are related to the basic areas of the physical sciences, first of all, of the theoretical physics in 20-th century. For illustration, the most of these etudes may be specified and classified, according to their physical meaning, as follows:
1. Classical and wave mechanics; Gravity: Perturbed keplerian motion in a plane. Equilibrium of a rotating heterogeneous liquid body (Clairaut problem). Wave mechanics of a mass point in conservative field; variational approach. Quantization of the linear harmonic oscillator. Wave quantization of a point particle that is attracted by a constant force towards a perfectly elastic wall. De Broghi waves. Poisson brackets. Intensity and selection rules for a central field. Rutherford formula deduced from classical mechanics and as a first approximation to the Born method. Gravitational attraction of an ellipsoid; special cases: prolate ellipsoid and spheroid. Inertial moment of the Earth.
2. Statistics and thermodynamics: Specific heat of an oscillator. Energy of specific heat of rotator. Thermodynamics in the thermoelectric cell. Imbalance of a pure three-phase system. Heat propagation from a certain cross section along an infinite-length bar endowed with another cross section acting as a heat well; a similarity with the crickets. Heat propagation in an isotropic and homogeneous medium; one-dimensional propagation. Entropy of the system in equilibrium. Perfect gases. Monatomic and diatomic gases. Numerical expansion for the entropy of gas. The energy of diatomic gases.
3. Classical and quantum electrodynamics; Radiation theory: Electrical and retarded potentials. Interaction energy of two electric or magnetic charge distributions. Energy of uniform circular distribution of electric or magnetic charges. Fields produced by circular and homogeneous distributions of charge in its own plane and by circular charge current in plane. Influence of a magnetic field on the melting point. Electric lines. Radiation theory (1–5). Classical theory of multipole radiation.
4. Theory of atoms and molecules: Statistical behavior of the fundamental terms in neutral atoms. Atomic polarizability. Resonance degeneracy for many-electron atoms. Second approximation for the potential inside atom. Empirical relations for a two-electron atom. On the spontaneous ionization of a hydrogen atom placed in a high potential region. Polarization forces between hydrogen atoms. Connection between the susceptibility and the electric moment of an atom in its ground state. Ionization probability for a hydrogen atom in an electric field. Relevant formulas for the atomic eigenfunctions. Quasi-stationary states. Integral representations of hydrogen eigenfunctions. The anomalous Zeeman effect (according to the Dirac theory). The interatomic potential without statistics. Diatomic molecule with identical nuclei. Statistical potential in molecules.
5. Theory of electron, nuclear and other particles. Electromagnetic mass of electron. Relativistic Hamiltonian for the motion of an electron. The electron scattering. Scattering of fast electrons: relativistic Born method. The spinning electron. Plane waves in the Dirac theory. Spherical functions with spin, spin 1 and spin 1/2. Scattering of alpha particle by a radioactive nucleus. Deflection of an alpha ray induced by a heavy nucleus (classical mechanics). Proton-neutron scattering.
Some other topics also are presented in the book.

MSC:

81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81-03 History of quantum theory
01A75 Collected or selected works; reprintings or translations of classics
81Qxx General mathematical topics and methods in quantum theory
81Rxx Groups and algebras in quantum theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
70Hxx Hamiltonian and Lagrangian mechanics
80Axx Thermodynamics and heat transfer
82B30 Statistical thermodynamics
81Vxx Applications of quantum theory to specific physical systems

Biographic References:

Majorana, Ettore
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