Abstract
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 of the particle’s position over the complex plane is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution is verified by the solution of the complex Fokker–Planck equation. It is shown that quantum probability and classical probability can be integrated under the framework of complex probability , such that they can both be derived from by different statistical ways of collecting spatial points.
Highlights
Mechanics and stochastic mechanics are related to the imaginary part of the complex velocity in complex mechanics
Probability throughout quantum mechanics is the result of empirical observations, which is so different to our familiar classical theory and is counterintuitive
We introduced and compared three trajectory interpretations on the basis of Bohmian mechanics, stochastic mechanics, and complex mechanics
Summary
Probability is the most subtle setting in quantum mechanics which extracts information from the abstract complex wave function. In their experiment, that a particle guided by the pilot wave is non-local, even its initial position is a locally defined hidden variable in Bohmian mechanics [12]. The quantum trajectories observed through the weak measurements indicate that the quantum world is not purely probabilistic but is deterministic to a certain extent [13,14] These experimental observations motivated us to study how to connect the deterministic ensemble to the probability distribution on the basis of the statistical language. Some anomalous trajectories and complex probability expressions have been observed in some optical experiments It is shown how an optical weak measurement of diagonal polarization can be realized by path interference between the horizontal and vertical polarization components of the input beam [31]. A quantum particle with random motion in complex space is considered in this new trajectory interpretation.
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