Use of mathematical methodologies in theoretical physics

Abstract

The present article is of great interest for young scientists-researchers and teachers, PhD students, master students, bachelor students and even for pupils of higher classes of schools willing to strengthen their knowledge in the field of mathematics and physics concerning it. In particular, there is a consideration of the methodology of calculation of a sum of series of natural numbers; knowledge of this methodology is of use for those who study quantum mechanics. For example, the methodology is used in quantum angular momentum theory when proving the quantization of angular momentum from considerations of probabilities theory in assumption that the possible projections of momentum for arbitrary axes are equal to m (m=l, l-1,..., -l)  and all these values of the momentum projections are equally probable and the axes are equivalent. Three methods for calculation of the sum of series of natural numbers squared are given: method of induction, method of differential calculus and method of finite differences. A solution of the problem by several techniques might be of use since when the results
obtained by different ways coincide, one should have no doubt in correctness of the result; some of the methods as one can see below can be generalized when solving similar or even more difficult problems. There is also a consideration of the well-known de Moivre-Euler’s formula which is often used by theoretical physicists when proving theorems and formulas, for example in the Born approximation and partial waves methods in quantum scattering theory. Euler solved complicated mathematical problems, results of which have practical applications in theoretical physics, but the surprising fact is that when solving this problems Euler only used ordinary mathematical knowledge and simple derivations with simplest mathematical functions. There is a consideration of obtaining of a sum of reverse squares of natural numbers in the article. A short description of how Fermat’s great theorem was proved by a group of mathematicians living in different time periods. But the main intrigue is that it is not known yet how Fermat himself proved it.