This appendix presents some basic results on Borel resummation and the theory of resurgence.

This chapter presents Feynman’s formulation of quantum mechanics, based on a path integral representation of the evolution operator. The chapter presents detailed examples which make it possible to understand clearly Feynman’s “sum over paths,” and it contains a complete discussion of how to calculate Gaussian path integrals. It also discusses the Euclidean version of the path integral, as well as Wick’s theorem and Feynman diagrams. Finally, it discusses instantons in quantum mechanics.

This chapter discusses metastable states or resonances in quantum mechanics. It develops different techniques to compute their complex energies, like complex dilatation techniques, and the uniform and the exact WKB method. The cubic and the inverted quartic oscillators are discussed in detail to exemplify these procedures. Finally, the chapter discusses the analytic continuation of eigenvalue problems, and the path integral formulation of metastable states.

This appendix lists some basic formulae on integrals of Gaussian functions which are used in the book.

It presents the unitary time evolution operator and its integral kernel in the space representation, also known as the quantum-mechanical propagator. It introduces the resolvent operator and its analytical properties, and it discusses in some detail scattering theory in one dimension.

This appendix lists some results on special functions used in the book, specially the Airy functions.

It gives a detailed and rigorous exposition of the WKB method in one dimension and its exact version, which includes nonperturbative effects in the Planck constant. This is illustrated in many examples, including the double-well potential. It also includes a description of the semiclassical quantization of higher-dimensional integrable systems, which are illustated by the Toda lattice.

This chapter presents Wigner’s approach to quantum mechanics, based on the Wigner function in phase space. It explains Wigner–Weyl quantization, which makes it possible to associate functions on phase space to wave functions and operators, and it develops the technology to do quantum mechanics in this formalism. This includes the star product, Moyal evolution,and star-eigenvalue equations. It also develops semiclassical methods in this formulation, and it has a section on Berry’s semiclassical formula for the Wigner function in one-dimensional systems.

We are now ready to study a generic class of three-dimensional physical systems. They are the systems that evolve in a central potential, i.e. a potential energy that depends only on the distance 𝑟 from the origin.

We shall now look at the solutions of Schrödinger’s equation for the quantum harmonic oscillator. In this chapter we will focus on the one-dimensional case, which can be seen as another example of a one-dimensional potential. Unlike the infinite potential well, the potential for the harmonic oscillator is finite for all finite values of 𝑥, and only diverges when 𝑥 → ±∞.