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Title: Probability density in the complex plane
Authors: Bender, C.M.
Hook, D.W.
Meisinger, P.N.
Wang, Q.-H. 
Keywords: Correspondence principle
PT Symmetry
Issue Date: Nov-2010
Citation: Bender, C.M., Hook, D.W., Meisinger, P.N., Wang, Q.-H. (2010-11). Probability density in the complex plane. Annals of Physics 325 (11) : 2332-2362. ScholarBank@NUS Repository.
Abstract: The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years, many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density ρ(z) in the complex plane. It is shown that there exist infinitely many complex contours C of infinite length on which ρ(z). dz is real and positive. Furthermore, the probability integral ∫Cρ(z)dz is finite. Demonstrating the existence of such contours is the essential element in establishing the correspondence between complex quantum and classical mechanics. The mathematics needed to analyze these contours is subtle and involves the use of asymptotics beyond all orders. © 2010 Elsevier Inc.
Source Title: Annals of Physics
ISSN: 00034916
DOI: 10.1016/j.aop.2010.02.011
Appears in Collections:Staff Publications

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