Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue

Yang, Ciann Dong; Han, Shiang-Yi

doi:10.1007/s10701-020-00363-3

Cited by 5 publications

(5 citation statements)

References 44 publications

(38 reference statements)

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“…As is well-known, there are nodes with

in the quantum harmonic oscillator. In our previous work [ 42 ], we solved this so-called nodal issue in the framework of complex mechanics. The statistical method we used was to collect all points of the CQRTs with the same real part

…”

Section: Extending Probability To the Complex Planementioning

confidence: 99%

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Extending Quantum Probability from Real Axis to Complex Plane

Yang

Han

2021

Entropy

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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.

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“…As is well-known, there are nodes with

…”

Section: Extending Probability To the Complex Planementioning

confidence: 99%

“… The statistical distribution of an ensemble of intersection points and projection points on the real axis of the CQRTs for

state. The red solid line is the quantum probability distribution and the black dotted line is the statistical distribution of point set B [ 42 ]. …”

Section: Figurementioning

confidence: 99%

Extending Quantum Probability from Real Axis to Complex Plane

Yang

Han

2021

Entropy

Self Cite

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show abstract

“…As is well-known, there are nodes with 𝛹(𝑥) = 0 in the quantum harmonic oscillator. In our previous work [42], we solved this so-called nodal issue in the framework of complex mechanics. The statistical method we used was to collect all points of the CQRTs with the same real part 𝑥:…”

Section: Extending Probability To the Complex Planementioning

confidence: 99%

Extending Quantum Probability from Real Axis to Complex Plane

Yang¹,

Han²

2020

Preprint

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Probability is an open 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 domain of probability extends to the complex space, including the generation of complex trajectories, the definition of the complex probability, the relation of the complex probability to the real quantum probability, and so on. The complex treatment proposed here applies the optimal quantum guidance law to derive the stochastic differential (SD) equation governing the particle’s random motions in the complex plane. The ensemble of the complex quantum random trajectories (CQRTs) solved from the complex SD equation is used to construct the probability distribution ρc(t,x,y) of the particle’s position over the complex plane z=x+iy. The correctness of the obtained complex probability is confirmed by the solution of the complex Fokker-Planck equation. The significant contribution of the complex probability is that it can be used to reconstruct both quantum probability and classical probability, and to clarify their relationship. Although quantum probability and classical probability are both defined on the real axis, they are obtained by projecting complex probability onto the real axis in different ways. This difference explains why the quantum probability cannot exactly converge to the classical probability when the quantum number is large.

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“…It is established on the foundation of the complex-space structure within which all physics quantities are complex-valued [51]. The quantum operators, quantization rules, uncertainty principle, correspondence principle, quantum probability, spin, and all other fundamental properties of quantum mechanics can have their corresponding trajectory-based descriptions underlying the framework of QHM in the complex domain [52][53][54][55]. It was found that the spin motion of the electron in the hydrogen atom is independent of the wave function, and is determined solely by the geometrical property of the complex-space structure [56,57].…”

Section: Introductionmentioning

confidence: 99%

Orbital and Spin Dynamics of Electron’s States Transition in Hydrogen Atom Driven by Electric Field

Yang¹,

Han²

2022

Preprint

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State transition in the multiple-levels system has the great potential applications in the quantum technology. In this article we employ a deterministic approach in complex space to analyze the dynamics of the 1s-2p electron transition in the hydrogen atom. The electron’s spin motion is embodied in the framework of quantum Hamilton mechanics that allows us to examine the transition dynamics more precisely. The transition is driven by an oscillating electric field in the z-direction. The electron’s transition process can be visualized by monitoring its motion in the complex space. The quantum potential and the total energy proposed in this paper provide new indices to observe the dynamic changes of electrons in the transition process.

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Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue

Cited by 5 publications

References 44 publications

Extending Quantum Probability from Real Axis to Complex Plane

Extending Quantum Probability from Real Axis to Complex Plane

Extending Quantum Probability from Real Axis to Complex Plane

Orbital and Spin Dynamics of Electron’s States Transition in Hydrogen Atom Driven by Electric Field

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