Path Integrals in Quantum Mechanics

The mathematical formalism based on path integrals, as introduced by Feynman, has changed our view about quantum mechanics. The modern theory of fundamental interactions at the microscopic level (quantum field theory) is hardly comprehensible without path integrals. Moreover, path integrals have allowed us to establish a direct mathematical relation between the theory of ordinary phase transitions and quantum field theory. The goal of this book is to introduce students to this topic within the context of ordinary quantum mechanics and non-relativistic many-body theory, before facing the problems associated with the more involved quantum field theory formalism.

Path integrals have allowed us to establish a direct mathematical relation between the theory of ordinary phase transitions and quantum field theory. This book introduces students to this topic within the context of ordinary quantum mechanics and non-relativistic many-body theory.

... very well written ... and not only pedagogically useful, but also useful to the experienced practitioner.

Autorentext
Jean Zinn-Justin is Head of DAPNIA/DSM/CEA-Saclay in France.

Klappentext
The main goal of this book is to familiarize the reader with a tool, the path integral, that not only offers an alternative point of view on quantum mechanics, but more importantly, under a generalized form, has also become the key to a deeper understanding of quantum field theory and its
applications, extending from particle physics to phase transitions or properties of quantum gases.
Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals. They share the algebraic properties of usual integrals, but have new properties from the viewpoint of analysis. They are powerful tools for
the study of quantum mechanics, since they emphasize very explicitly the correspondence between classical and quantum mechanics. Physical quantities are expressed as averages over all possible paths but, in the semi-classical limit, the leading contributions come from paths close to classical paths.
Thus, path integrals lead to an intuitive understanding of physical quantities in the semi-classical limit, as well as simple calculations of such quantities. This observation can be illustrated with scattering processes, spectral properties or barrier penetration effects. Even though the
formulation of quantum mechanics based on path integrals seems mathematically more complicated than the usual formulation based on partial differential equations, the path integral formulations well adapted to systems with many degrees of freedom, where a formalism of Schrodinger type is much less
useful. It allows simple construction of a many-body theory both for bosons and fermions.

Inhalt
1. Gaussian integrals; 2. Path integral in quantum mechanics; 3. Partition function and spectrum; 4. Classical and quantum statistical physics; 5. Path integrals and quantization; 6. Path integral and holomorphic formalism; 7. Path integrals: fermions; 8. Barrier penetration: semi-classical approximation; 9. Quantum evolution and scattering matrix; 10. Path integrals in phase space; QUANTUM MECHANICS: MINIMAL BACKGROUND; A1 Hilbert space and operators; A2 Quantum evolution, symmetries and density matrix; A3 Position and momentum. Scrodinger equation

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