Abstract
The path integral formulation of quantum mechanics is an alternative to Schrödinger’s wave mechanics or Heisenberg’s matrix mechanics. The path integral method allows for a uniform treatment of quantum mechanics, statistical mechanics and quantum field theory and can be regarded as a basic tool in modern theoretical physics. We introduce and discuss the path integral for quantum mechanics and quantum statistics. Thereby the time-evolution kernel, correlation functions, generating functionals of correlation functions and thermodynamic potentials are given by sums, or functional integrals, over an infinity of possible trajectories. The results are applied to the free particle and the harmonic oscillator. Both the semi-classical and high-temperature expansions of the partition function are considered in the end-of-chapter problems.
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Notes
- 1.
If we couple the system to a magnetic field, \(\hat{H}\) and \(\hat{K}(\tau)\) become complex quantities.
- 2.
To keep the notation simple, we use q as the final point.
References
P.A.M. Dirac, The Lagrangian in quantum mechanics. Phys. Z. Sowjetunion 3, 64 (1933)
P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, London, 1947)
N. Wiener, Differential space. J. Math. Phys. Sci. 2, 132 (1923)
R. Feynman, Spacetime approach to non-relativistic quantum mechanic. Rev. Mod. Phys. 20, 267 (1948)
R. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965)
J. Glimm, A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer, Berlin, 1981)
P.R. Chernoff, Note on product formulas for operator semigroups. J. Funct. Anal. 2, 238 (1968)
M. Reed, B. Simon, Methods of Modern Mathematical Physics I (Academic Press, New York, 1972)
M. Kac, Random walk and the theory of Brownian motion. Am. Math. Mon. 54, 369 (1947)
I.M. Gel’fand, A.M. Yaglom, Integration in functional spaces and its applications in quantum physics. J. Math. Phys. 1, 48 (1960)
G. Roepstorff, Path Integral Approach to Quantum Physics (Springer, Berlin, 1996)
L.S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)
E. Nelson, Feynman integrals and the Schrödinger equation. J. Math. Phys. 5, 332 (1964)
S.G. Brush, Functional integrals and statistical physics. Rev. Mod. Phys. 33, 79 (1961)
J. Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford University Press, London, 2004)
H. Kleinert, Path Integral, in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets (World Scientific, Singapore, 2006)
U. Mosel, Path Integrals in Field Theory: An Introduction (Springer, Berlin, 2004)
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Wipf, A. (2013). Path Integrals in Quantum and Statistical Mechanics. In: Statistical Approach to Quantum Field Theory. Lecture Notes in Physics, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33105-3_2
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DOI: https://doi.org/10.1007/978-3-642-33105-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33104-6
Online ISBN: 978-3-642-33105-3
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