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Hilbert space average of transition probabilities.

Nico Hahn1, Thomas Guhr1, Daniel Waltner1

  • 1Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, 47048 Duisburg, Germany.

Physical Review. E
|July 22, 2020
PubMed
Summary

We extend quantum statistical mechanics methods to calculate transition probabilities, linking them to spectral statistics. This approach handles non-orthogonal states and is demonstrated for a kicked spin chain.

Area of Science:

  • Quantum statistical mechanics
  • Quantum mechanics

Background:

  • The typicality approach and Hilbert space averaging are key in quantum statistical mechanics.
  • These methods are typically used for expectation values.

Purpose of the Study:

  • Extend typicality and Hilbert space averaging to transition probabilities.
  • Investigate the connection between transition probabilities and spectral statistics.
  • Develop a method to handle non-orthogonal states.

Main Methods:

  • Applied the typicality approach and Hilbert space averaging to transition probabilities.
  • Analyzed the relationship between transition probabilities of random states and spectral statistics.
  • Developed a generalized framework for non-orthogonal states.

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Main Results:

  • Successfully extended the typicality approach to transition probabilities.
  • Established a link between transition probabilities and spectral statistics for random states.
  • Demonstrated the capability to analyze distributions of matrix elements for non-orthogonal states.

Conclusions:

  • The extended approach provides a general method for calculating transition probabilities in quantum systems.
  • Numerical simulations on a kicked spin chain validate the findings in both integrable and chaotic regimes.