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This study shows how quantum energy level statistics change with transmission probability in a model system. The eigenvalue statistics transition from Poisson-like to Gaussian Orthogonal Ensemble and then to Gaussian as transmission increases.

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Area of Science:

  • Quantum mechanics
  • Statistical physics
  • Condensed matter physics

Background:

  • Energy level statistics in quantum systems correlate with classical dynamics.
  • Regular systems typically show Poisson statistics, while chaotic systems exhibit random matrix statistics.
  • System parameters can drive transitions between regular and chaotic classical dynamics, reflected in quantum eigenvalue statistics.

Purpose of the Study:

  • To investigate the change in quantum energy eigenvalue statistics in a model system with varying transmission probability.
  • To explore the relationship between transmission probability, eigenvalue statistics, and energy eigenstate structure.
  • To analyze the system as a quantum graph and compare results with theoretical expectations.

Main Methods:

  • A quantum system comprising an infinite square well with Dirac delta barriers was modeled.
  • Transmission probability (T) through the barriers was systematically varied.
  • Statistical properties of energy eigenvalue sequences, including level spacing and number variance, were analyzed.
  • The spatial distribution of energy eigenstates was examined.

Main Results:

  • At low transmission probability (T≈0), level spacing distribution was Poisson-like.
  • Increasing T shifted the distribution towards the Gaussian Orthogonal Ensemble (GOE) and then to a Gaussian distribution as T→1.
  • Number variance showed a similar transition.
  • Energy eigenstates evolved from localized to delocalized across the well as T increased.
  • Results align with quantum graph theory predictions for T→0.

Conclusions:

  • The model system demonstrates a clear transition in quantum eigenvalue statistics driven by transmission probability.
  • This transition reflects a change in underlying classical dynamics from regular to chaotic.
  • The study provides a concrete example of statistical mechanics principles in a tunable quantum system.