MATHEMATICAL PREPARATION FOR A STUDENT OF MODERN PHYSICAL SCIENCE

By Linus Pauling

In formulating my opinion of the mathematical training which a student of modern physics and physical chemistry should receive, I have drawn almost entirely on my own perhaps rather unorthodox experiences during the last ten years. These have not included the teaching of mathematics; in consequence I am not in a position to judge concerning the practicability of the suggestions which I make, but only of their desirability, and I am interested to know your feelings regarding them.

The amount of mathematical training needed by the student of physics and of physical chemistry increases appreciably every decade. The developments of the last thirty years, especially the theory of relativity and the quantum theory, have involved the application of many new mathematical methods. The importance of mathematical training to the theoretical physicist is illustrated by the example of the great theoretical physicist and teacher of theoretical physicists, Professor Arnold Sommerfeld of Munich, who was trained as a mathematician, having as a young man collaborated with Felix Klein in writing the five-volume book, “Über die Theorie des Kreisels". He has never forgotten this early training; one of his amusements is to play with a peculiarly-shaped spinning top, which is able to convert all of its kinetic energy into potential energy and then reverse its motion. His work on atomic structure has its feet in the spinning top; very often a student struggling with a difficult problem in atomic structure or theoretical spectroscopy is referred by Professor Sommerfeld to this book for aid. To his students he gives a similar sound training in mathematics as the tool for theoretical physics; and the men in the impressive group who have taken their doctorates with him, including Debye, Heisenberg, Pauli, Landé, Ewald, Herzfeld, Lenz, Fues, Kossel, London, Rogowski, Kratzer, Unsöld, Wentzel, and many others, owe a great deal to this.

The capabilities of students of physics and chemistry are at all stages in their education determined to a large extent by their mathematical preparation; even for the most elementary work in physics some mathematical knowledge is required. This was perhaps overlooked by Dr. H. E. Timerding of Braunschweig when he wrote the section, "Forschung und Unterricht," for Volume I of the monumental and authoritative 11 "Handouch der Physik," which has appeared in twenty-four volumes during the last six years. On page 208 he wrote "Instead of restricting the teaching of physics to the last school years, the noteworthy attempt is being made in England to begin instruction in physical observation in its simplest form in the very early years of childhood," the reference for this statement being "Physical Exercises for children under seven years of age, in the Reports of the Board of Education, 1920."

In the American institutions with which I have been connected, the California Institute of Technology, the University of California, and the Massachusetts Institute of Technology, and I believe in other universities also, the undergraduate curriculum of the student preparing for advanced work in physics and physical chemistry usually contains two courses beyond first and second year mathematics, advanced calculus and differential equations, including some discussion of partial differential equations. These courses, together preferably with a course in vector analysis, provided satisfactory preparation for the advanced work in physics of ten years ago, such as electromagnetic theory, advanced mechanics, geometrical and undulatory optics, and the Bohr theory of atomic structure. The gradual development of the student's in knowledge of Newtonian mechanics is striking. In freshman physics he works simple mechanical problems and learns simple non-differential laws; later he applies the differential and integral calculus to mechanics, and still later he learns the elegant methods of Lagrange and Hamilton. The Bohr theory of atomic structure, with its superposition of simple rules of quantization on the solutions of problems in classical mechanics, involves no additional prerequisite knowledge, and can be introduced to the student at any stage in his study of mechanics.

A similar gradual introduction of quantum mechanics to the undergraduate student cannot be made at present. Quantum mechanical problems can be handled by any one of a number of essentially equivalent methods - solution of a characteristic-value differential equation, application of matrix algebra, of operator calculus, etc.; - but none of these is familiar to the undergraduate or usual graduate student. But the quantum mechanics is here to stay, and it is essential for the physicist or chemist, at least for those in academic work, to become acquainted with the modern theory of atomic and molecular structure, even though he be not mainly a theoretical man. I believe that the mathematical preparation of the student should be such as to facilitate the attainment of this acquaintance.

In my opinion, the best approach to quantum mechanics is by way of the Schrödinger wave equation. The thorough mathematical preparation for this for students of theoretical physics and chemistry could be given in a year’s course in partial differential equations. This should include, after the development of the fundamental theory, the extensive discussion of complete sets of orthogonal functions, with the explicit consideration of, in addition to Fourier series and spherical harmonics, special functions such as those of Hermite, Laguerre, Sonine, Mathieu, etc., and with extensive practise in the solution of boundary - value problems, especially physical problems in heat conduction, elasticity, the Schrödinger wave equation, etc., to emphasize the physical interpretation of the equations.

In addition, provision must be made for students not specializing in theoretical work, by devoting to wave mechanics part of the work in the year's course "Introduction to Theoretical Physics" given senior and first-year graduate students at many universities, and by preparing the student for this. Mathematical preparation is needed; for example, Leigh Page, in his book "Introduction to Mathematical Physics," published in 1928, omits all discussion of matrix and wave mechanics, writing (P. 581) "On account of their mathematical complexity any detailed discussion of them is beyond the scope of this book." Wave mechanics is no more difficult than Maxwell's theory, relativity theory, the theory of optical wave motion, etc., which he discusses in detail. The undergraduate mathematics courses should include such topics as to permit an introduction to wave mechanics to be given the senior student. This could be achieved by devoting a part of second-year mathematics or of advanced calculus to the general properties of orthogonal functions, with Fourier series and surface harmonics as examples, the definition and properties of Hermite functions and other special functions, preferably with mention of their significance and interpretation in atomic physics, the use of recursion formulas and generating functions, etc. An appreciable fraction of the usual course in differential equations should be used in treating characteristic-value equations, emphasizing those occurring in physics. An introduction to matrix algebra might also be a part of some undergraduate mathematics course.

The student often desires advice as to which of the more advanced mathematical courses, such as theory of numbers, advanced analytic geometry, differential analytic functions of a real variable, elliptic functions, group theory, topology, etc., will be useful in his physical and chemical work. This is hard to give, for there is hardly a branch of mathematics which has not been used in physics, especially during the present century. I shall not draw upon quantum mechanics for illustrations, with which you are doubtless familiar, such as the extensive application of the theory of permutation groups in the discussion of the chemical bond and of ferromagnetism, but shall refer to a perhaps less familiar subject, the structure of crystals and its investigation by the diffraction of X-rays and electrons.

This summer marks the twentieth anniversary of Laue's discovery of the diffraction of X-rays by crystals - a discovery which may well be described as mathematical rather than physical. It was not the result of an accidental experimental observation, although interference phenomena were often overlooked. In the years between 1895, when X-rays were discovered, and 1912 Röntgen and other skillful investigators had passed X-rays through rock-salt crystals and crystal powder and had measured the absorption coefficients in metal foils and in various directions through single crystals. In all of these experiments diffraction phenomena occurred and might have been observed. P. P. Ewald as a student in Sommerfeld’s Institute for Theoretical Physics had for two years been working on his doctoral dissertation on electromagnetic potentials in crystal lattices, which contained within it the theory of X-ray diffraction. Max von Laue, Privatdozent in the Institute, was familiar with this work, with recent estimates of the wave-length of X-rays, and with the theory of optical diffraction by one and two dimensional gratings. He told in his Nobel lecture how in an inspired moment, as he walked home through the Englishergarten after talking with Ewald, the picture of a beam of X-rays diffracted by a three-dimensional grating occurred to him. The experiments immediately conducted by Friedrich and Knipping verified the existence of the phenomenon.

The theory of space groups is invaluable in the investigation of the structure of crystals. Forty years ago the 230 space groups, the discontinuous transformation groups in space involving three fundamental translational operations, were discovered by Federow, Schoenflies, and Barlow. Although discussed by crystallographers, the discussion was formal, for no sound method of determining the space-group symmetry was known. After Laue's discovery a number of simple structures were determined by W.H. and W. L. Bragg without the use of space-group theory; but as more and more complicated structures were investigated, especially on the Continent and in America, space-group theory was used more and more, until now no investigation is made without it. Extensions of the mathematical theory have also been made, such as the discussion of net and chain groups, three-dimensional groups involving two or one fundamental translational operations. In the years 1928-1930 some twenty papers on space-group theory and related topics appeared in the Zeitschrift für Kristallographie.

The theory of X-ray diffraction requires the use of the most powerful mathematical methods of treating diffraction problems, some of them being unusual in physical theory. Thus Darwin in 1914 in deriving his expression for the intensity of reflection of X-rays by a perfect crystal, with the aid of which he showed that many actual crystals have a mosaic structure, found it necessary to solve an interesting infinite set of simultaneous difference equations. Even in such a relatively simple problem as the calculation of the scattering power of atoms for X-rays, using hydrogen-like wave functions, Dr. Sherman and I were required to evaluate a rather difficult definite integral, this being accomplished with the use of generating functions and of an integral discussed sixty years ago by Hankel and Gegenbauer.

The Madelung-Born theory of ionic crystals in common with other branches of crystal physics, involves extensive application of tensor calculus. Madelung constants, which give the energy of the Coulomb interaction of ions arranged according to one of the space groups, can be calculated by potential-theory methods, such as by the use of solutions of Laplace's equation. A very elegant and powerful method, using three-dimensional theta functions, has been developed by Ewald. Because of the importance of Madelung constants to chemistry, it is to be hoped that some of the great amount of work remaining to be done in this field will be carried out. Some questions of fundamental importance remain to be answered, such as the question of the convergence of the series involved; a start on this has been made by Laue.

Recent developments in crystal chemistry have consisted mainly in the formulation of semi-empirical rules governing the structures of crystals of various types. These often could be made more exact and more powerful if related mathematical studies were made. Thus it has been found that many crystals behave as though they were composed of spherically-symmetrical atoms or ions piled together as closely as possible. It was shown by Lord Kelvin that there is only one way of piling spheres in homogeneous closest packing; that is, such that any sphere can be obtained from any other by a translational operation. This is the familiar face-centered cubic arrangement. Barlow pointed out that this condition is not of physical significance, and replaced it by the condition that the spheres be crystallographically equivalent. There are then two closest-packed arrangements, the second being hexagonal. It was found on studying the structure of topaz and of brookite that another type of closest packing of the large ions occurs in these crystals, the large ions being not crystallographically equivalent, but of two kinds. Four types of closest packing in which the spheres are of two crystallographic kinds are now known; it would be very valuable to know the results of a rigorous derivation of the possible arrangements of this nature, and also of the possible closest-packed arrangements of spheres of two or more sizes.

The tendency of small positive ions to co-ordinate large negative ions about them at the corners of an approximately regular polyhedron forms the basis of the polyhedron method of discussing and predicting structures of complex ionic crystals. In the three known forms of titanium dioxide, rutile, anatase, and brookite, each titanium ion is surrounded by an octahedron of six oxygen ions, each oxygen ion forming the common corner of three octahedral. Indeed, the structure of brookite was discovered with the aid of this conception, and verified by X-ray methods. It would be interesting to know all of the ways of arranging regular octahedra such that each corner is common to three octahedra and all of the octahedra are crystallographically equivalent; these would be possible structures for substances MX_{2}, with M having the coordination number 6. Several other such structures are known, two, the CdCl_{2} and CdI_{2} structures, occurring in nature. Similar arrangements of tetrahedra and of tetrahedra and octahedra, such as those known, in sodalite, Na_{4}Al_{3}Si_{3}O_{12}Cl, and garnet, Ca_{3}Al_{2}Si_{3}O_{12}, for example, should also be discussed. A start in this direction has been made by Niggli in his Topological Structure Analysis. I hope that you will find it possible to prepare an undergraduate student for an introductory course in wave mechanics; and I know that you agree with me that there is no branch of mathematics which may not be found useful in physics and chemistry.