Path Integrals in Quantum Mechanics
1st Edition
Published on 18. November 2004
335 pages
978-0-19-158142-7 (ISBN)
Description
Quantum field theory is hardly comprehensible without path integrals: 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.
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Jean Zinn-Justin
Path Integrals in Quantum Mechanics
Book
07/2010
Oxford University Press
€66.85
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Content
- Cover
- Contents
- 1 Gaussian integrals
- 1.1 Generating function
- 1.2 Gaussian expectation values. Wick's theorem
- 1.3 Perturbed gaussian measure. Connected contributions
- 1.4 Expectation values. Generating function. Cumulants
- 1.5 Steepest descent method
- 1.6 Steepest descent method: several variables, generating functions
- 1.7 Gaussian integrals: complex matrices
- Exercises
- 2 Path integrals in quantum mechanics
- 2.1 Local markovian processes
- 2.2 Solution of the evolution equation for short times
- 2.3 Path integral representation
- 2.4 Explicit calculation: gaussian path integrals
- 2.5 Correlation functions: generating functional
- 2.6 General gaussian path integral and correlation functions
- 2.7 Quantum harmonic oscillator: the partition function
- 2.8 Perturbed harmonic oscillator
- 2.9 Perturbative expansion in powers of h
- 2.10 Semi-classical expansion
- Exercises
- 3 Partition function and spectrum
- 3.1 Perturbative calculation
- 3.2 Semi-classical or WKB expansion
- 3.3 Path integral and variational principle
- 3.4 O(N) symmetric quartic potential for N 8
- 3.5 Operator determinants
- 3.6 Hamiltonian: structure of the ground state
- Exercises
- 4 Classical and quantum statistical physics
- 4.1 Classical partition function. Transfer matrix
- 4.2 Correlation functions
- 4.3 Classical model at low temperature: an example
- 4.4 Continuum limit and path integral
- 4.5 The two-point function: perturbative expansion, spectral representation
- 4.6 Operator formalism. Time-ordered products
- Exercises
- 5 Path integrals and quantization
- 5.1 Gauge transformations
- 5.2 Coupling to a static magnetic field: gauge symmetry
- 5.3 Quantization and path integrals
- 5.4 Static magnetic field: direct calculation
- 5.5 Diffusion, random walk, Fokker-Planck equation
- 5.6 The spectrum of the O(2) rigid rotator
- Exercises
- 6 Path integrals and holomorphic formalism
- 6.1 Complex integrals and Wick's theorem
- 6.2 Holomorphic representation
- 6.3 Kernel of operators
- 6.4 Path integral: the harmonic oscillator
- 6.5 Path integral: general hamiltonians
- 6.6 Bosons: second quantization
- 6.7 Quantum statistical physics: the partition function
- 6.8 Bose-Einstein condensation
- 6.9 Generalized path integrals: the quantum Bose gas
- 6.10 Partition function: the field integral representation
- Exercises
- 7 Path integrals: fermions
- 7.1 Grassmann algebras
- 7.2 Differentiation in Grassmann algebras
- 7.3 Integration in Grassmann algebras
- 7.4 Gaussian integrals and perturbative expansion
- 7.5 Fermion vector space and operators: one state
- 7.6 General Grassmann analytic functions
- 7.7 Many-fermion states. Hamiltonians
- 7.8 Second quantization representation
- 7.9 Grassmann path integral: one-state problem
- 7.10 Grassmann path integrals: generalization
- 7.11 Quantum Fermi gas
- 7.12 Real gaussian integrals. Wick's theorem
- 7.13 Mixed change of variables: berezinian and supertrace
- Exercises
- 8 Barrier penetration: semi-classical approximation
- 8.1 Quartic double-well potential and instantons
- 8.2 Degenerate minima: semi-classical approximation
- 8.3 Collective coordinates and gaussian integration
- 8.4 Instantons and metastable states
- 8.5 Collective coordinates: alternative method
- 8.6 The jacobian
- 8.7 Instantons: the quartic anharmonic oscillator
- Exercises
- 9 Quantum evolution and scattering matrix
- 9.1 Evolution of the free particle and S-matrix
- 9.2 Perturbative expansion of the S-matrix
- 9.3 S-matrix: bosons and fermions
- 9.4 S-matrix in the semi-classical limit
- 9.5 Semi-classical approximation: one dimension
- 9.6 Eikonal approximation
- 9.7 Perturbation theory and operators
- Exercises
- 10 Path integrals in phase space
- 10.1 A few elements of classical mechanics
- 10.2 The path integral in phase space
- 10.3 Harmonic oscillator. Perturbative calculations
- 10.4 Lagrangians quadratic in the velocities
- 10.5 Free motion on the sphere or rigid rotator
- Exercises
- Appendix: Quantum mechanics: minimal background
- A1 Hilbert space and operators
- A2 Quantum evolution, symmetries
- A3 Density matrix
- A4 Position and momentum operators
- A5 Hamiltonian and Schr¨odinger equation
- Bibliography
- Index
- A
- B
- C
- D
- E
- F
- G
- H
- I
- K
- L
- M
- N
- O
- P
- Q
- R
- S
- T
- U
- V
- W
- Z
