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This study introduces a hydrodynamic model for quantum particles, revealing classical dynamics that explain quantum statistics and Born's rule. It offers a new perspective on quantum mechanics and particle behavior in potential wells.

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Area of Science:

  • Quantum Mechanics
  • Fluid Dynamics
  • Mathematical Physics

Background:

  • The Schrödinger equation describes quantum particle behavior.
  • Understanding quantum statistics and Born's rule remains a challenge.
  • Classical analogies can offer new insights into quantum phenomena.

Purpose of the Study:

  • To develop a hydrodynamic analogy for nonrelativistic quantum particles in potential wells.
  • To explore similarities between quantum mechanics and shallow water waves.
  • To provide a classical interpretation of quantum statistics and Born's rule.

Main Methods:

  • Analyzing similarities between a real variant of the Schrödinger equation and gravity-capillary shallow water waves.
  • Investigating particle trajectories guided by wave gradients.
  • Examining the particle probability distribution function.

Main Results:

  • Particles exhibit periodic or chaotic dynamics influenced by wave gradients.
  • The analogy reproduces quantum statistics found in standard Schrödinger equation solutions.
  • A classical interpretation of Born's rule is demonstrated.
  • A mechanism for transitions between quasi-stationary states is proposed.

Conclusions:

  • The hydrodynamic analogy offers a classical, deterministic framework for understanding quantum mechanics.
  • This model provides new insights into particle behavior and quantum statistics.
  • The proposed mechanism may explain transitions between quantum states.