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Title: Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling

Abstract

Picard–Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute real-time dynamics directly. After discussing basic properties of real-time path integrals on Lefschetz thimbles, we demonstrate its computational method in a concrete way by solving three simple examples of quantum mechanics. It is applied to quantum mechanics of a double-well potential, and quantum tunneling is discussed. We identify all of the complex saddle points of the classical action, and their properties are discussed in detail. However a big theoretical difficulty turns out to appear in rewriting the original path integral into a sum of path integrals on Lefschetz thimbles. We discuss generality of that problem and mention its importance. Real-time tunneling processes are shown to be described by those complex saddle points, and thus semi-classical description of real-time quantum tunneling becomes possible on solid ground if we could solve that problem. - Highlights: • Real-time path integral is studied based on Picard–Lefschetz theory. • Lucid demonstration is given through simple examples of quantum mechanics. • This technique is applied to quantum mechanics of the double-well potential. • Difficulty for practical applications is revealed, and we discuss its generality. • Quantum tunneling is shown to be closelymore » related to complex classical solutions.« less

Authors:
 [1];  [2]
  1. Department of Physics, The University of Tokyo, Tokyo 113-0033 (Japan)
  2. Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914 (Japan)
Publication Date:
OSTI Identifier:
22403469
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 351; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FEYNMAN PATH INTEGRAL; MATHEMATICAL SOLUTIONS; POTENTIALS; QUANTUM MECHANICS; QUANTUM WELLS; TUNNEL EFFECT

Citation Formats

Tanizaki, Yuya, Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, and Koike, Takayuki. Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling. United States: N. p., 2014. Web. doi:10.1016/J.AOP.2014.09.003.
Tanizaki, Yuya, Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, & Koike, Takayuki. Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling. United States. https://doi.org/10.1016/J.AOP.2014.09.003
Tanizaki, Yuya, Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, and Koike, Takayuki. 2014. "Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling". United States. https://doi.org/10.1016/J.AOP.2014.09.003.
@article{osti_22403469,
title = {Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling},
author = {Tanizaki, Yuya and Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198 and Koike, Takayuki},
abstractNote = {Picard–Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute real-time dynamics directly. After discussing basic properties of real-time path integrals on Lefschetz thimbles, we demonstrate its computational method in a concrete way by solving three simple examples of quantum mechanics. It is applied to quantum mechanics of a double-well potential, and quantum tunneling is discussed. We identify all of the complex saddle points of the classical action, and their properties are discussed in detail. However a big theoretical difficulty turns out to appear in rewriting the original path integral into a sum of path integrals on Lefschetz thimbles. We discuss generality of that problem and mention its importance. Real-time tunneling processes are shown to be described by those complex saddle points, and thus semi-classical description of real-time quantum tunneling becomes possible on solid ground if we could solve that problem. - Highlights: • Real-time path integral is studied based on Picard–Lefschetz theory. • Lucid demonstration is given through simple examples of quantum mechanics. • This technique is applied to quantum mechanics of the double-well potential. • Difficulty for practical applications is revealed, and we discuss its generality. • Quantum tunneling is shown to be closely related to complex classical solutions.},
doi = {10.1016/J.AOP.2014.09.003},
url = {https://www.osti.gov/biblio/22403469}, journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = ,
volume = 351,
place = {United States},
year = {Mon Dec 15 00:00:00 EST 2014},
month = {Mon Dec 15 00:00:00 EST 2014}
}