Yang Ciann-Dong, Han Shiang-Yi
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan.
Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan.
Entropy (Basel). 2021 Feb 8;23(2):210. doi: 10.3390/e23020210.
Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle's random motion in the complex plane. The probability distribution ρc(t,x,y) of the particle's position over the complex plane z=x+iy is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution ρc(t,x,y) is verified by the solution of the complex Fokker-Planck equation. It is shown that quantum probability Ψ2 and classical probability can be integrated under the framework of complex probability ρc(t,x,y), such that they can both be derived from ρc(t,x,y) by different statistical ways of collecting spatial points.
概率是量子力学本体论诠释中的一个重要问题。在一些轨迹诠释中，如玻姆力学和随机力学，已经对其进行了讨论。当概率域扩展到复空间时，会出现新的问题，包括复轨迹的生成、复概率的定义以及复概率与量子概率的关系。本文提出的复处理方法应用最优量子制导律来推导控制粒子在复平面中随机运动的随机微分方程。粒子在复平面(z = x + iy)上位置的概率分布(\rho_c(t,x,y))由复量子随机轨迹的系综形成，这些轨迹是从复随机微分方程求解得到的。同时，概率分布(\rho_c(t,x,y))通过复福克 - 普朗克方程的解得到验证。结果表明，量子概率(\Psi^2)和经典概率可以在复概率(\rho_c(t,x,y))的框架下统一起来，使得它们都可以通过不同的收集空间点的统计方式从(\rho_c(t,x,y))导出。
