Abstract
There exist three apparently different formulations of quantum mechanics: Heisenberg’s matrix mechanics, Schrödinger’s wave mechanics, and Feynman’s path integral approach. In contrast to matrix and wave mechanics, which are based on the Hamiltonian approach, the latter is based on the Lagrangian approach.
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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 mechanics. Rev. Mod. Phys. 20, 267 (1948)
R. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals (Dover, New York, 2010)
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)
L.S. Schulman, Techniques and Applications of Path Integration (Dover, New York, 2005)
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, 2010)
J.R. Klauder, A Modern Approach to Functional Integration (Birkhäuser, Basel, 2011)
H. Kleinert, Path Integral, in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets (World Scientific, Singapore, 2009)
U. Mosel, Path Integrals in Field Theory: An Introduction (Springer, Berlin, 2013)
W. Dittrich, M. Reuter, Classical and Quantum Dynamics: From Classical Paths to Path Integrals, 6th edn. (Springer, Berlin, 2020)
P.R. Chernoff, Note on product formulas for operator semigroups. J. Funct. Anal. 2, 238 (1968)
P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications (Springer, Berlin, 2004)
J. Glimm, A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer, Berlin, 1981)
C. Grosche, F. Steiner, Handbook of Feynman Path Integrals. Springer Tracts in Modern Physics, vol. 145 (Springer, Berlin, 2013)
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Wipf, A. (2021). Path Integrals in Quantum and Statistical Mechanics. In: Statistical Approach to Quantum Field Theory. Lecture Notes in Physics, vol 992. Springer, Cham. https://doi.org/10.1007/978-3-030-83263-6_2
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DOI: https://doi.org/10.1007/978-3-030-83263-6_2
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