Path Integrals in Quantum Mechanics
Path integrals provide in many instances an elegant complementary description of quantum mechanics and also for the quantization of fields, which we will study from a canonical point of view in Chap. 17 and following chapters. Path integrals are particul
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Path Integrals in Quantum Mechanics
Path integrals provide in many instances an elegant complementary description of quantum mechanics and also for the quantization of fields, which we will study from a canonical point of view in Chap. 17 and following chapters. Path integrals are particularly popular in scattering theory, because the techniques of path integration were originally developed in the study of time evolution operators. Other areas where path integrals are used include statistical physics and the description of dissipative systems. Path integration is based on a beautiful intuitive description of the quantum mechanical time evolution of particles or wave functions from initial to final states. The prize for the intuitive elegance in the description of time evolution is that the description of bound systems and the identification of the corresponding states is often cumbersome with path integral methods. On the other hand, path integration and canonical quantization complement each other particularly well in relativistic scattering theory, where canonical methods are needed for unitarity of the scattering matrix, for the normalization of the scattering states, and also for the correct choice of propagators in perturbation theory, while the path integral formulation provides an elegant tool for the development of rules for covariant perturbation theory. Path integrals had been developed by Richard Feynman as a tool for understanding the role of the classical action in quantum mechanics, and had then evolved into a basis for covariant perturbation theory in relativistic field theories [51]. Our introductory exposition will focus on the use of path integrals in scattering theory. The first authoritative textbook on path integrals was co-authored by Feynman himself [53]. Extensive discussions and many applications of path integrals can be found in [68] and [99]. The use of path integrals in perturbative relativistic quantum field theory from a particle physics perspective is discussed e.g. in [87, 133, 175].
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_14
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14 Path Integrals in Quantum Mechanics
14.1 Correlation and Green’s Functions for Free Particles Before we enter the discussion of free particle motion and potential scattering in terms of path integrals, it is useful to discuss Green’s functions for the Newton equation and canonical correlation functions for free particles. The equation of motion of a classical non-relativistic particle under the influence of a force F (t) is directly integrable, x(t) = x i + v i (t − ti ) +
1 m
t
dt
t
dt F (t ).
(14.1)
ti
ti
Partial integration of the acceleration term yields a Green’s function representation x(t) = x i + v i (t − ti ) +
1 m
1 = x i + v i (t − ti ) + m
t
dt (t − t )F (t )
ti
∞
−∞
dt Gi (t, t )F (t ),
(14.2)
with a Gre
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