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Extending Quantum Probability from Real Axis to Complex Plane
oleh: Ciann-Dong Yang, Shiang-Yi Han
| Format: | Article |
|---|---|
| Diterbitkan: | MDPI AG 2021-02-01 |
Deskripsi
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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>c</mi></msub><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of the particle’s position over the complex plane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></mrow></semantics></math></inline-formula> is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>c</mi></msub><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is verified by the solution of the complex Fokker–Planck equation. It is shown that quantum probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mrow><mo>|</mo><mi>Ψ</mi><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula> and classical probability can be integrated under the framework of complex probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>c</mi></msub><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, such that they can both be derived from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>c</mi></msub><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> by different statistical ways of collecting spatial points.