Probability Density in the Complex Plane

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 $\rho (z)$ in the complex plane. It is shown that there exist infinitely many complex contours $C$ of infinite length on which $\rho (z)dz$ is real and positive. Furthermore, the probability integral ${\int }_C\rho (z)dz$ 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.

38 pages, 17figures

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• The ODE/IM Correspondence
• Mean-field dynamics of a non-Hermitian Bose-Hubbard dimer
• PT-Symmetric Wave Chaos
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• Complex Correspondence Principle
• Classical Trajectories for Complex Hamiltonians
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• Spontaneous Breaking of Classical PT Symmetry
• Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve ${\mu }^2={\nu }^n-1,n\in \mathbb{Z}$: ergodicity, isochrony, periodicity and fractals
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• Euler Incognito

Cited by in corpus (13)

• Probability-preserving evolution in a non-Hermitian two-band model
• Quantum tunneling as a classical anomaly
• Resonances in Extreme Mass-Ratio Inspirals: Asymptotic and Hyperasymptotic Analysis
• Understanding complex dynamics by means of an associated Riemann surface
• Periodic orbits for classical particles having complex energy
• CPT-Frames for PT-symmetric Hamiltonians
• Nonlinear eigenvalue problems
• Extending Quantum Probability from Real Axis to Complex Plane
• Hypercomplex Fock States for Discrete Electromagnetic Schrödinger Operators: A Bayesian Probability Perspective
• Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue
• Time evolution and adiabatic approximation in $PT$-symmetric quantum mechanics
• First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior
• PT-Rotations, PT-Spherical Harmonics and the PT-Hydrogen Atom