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This study introduces a new propagator for non-spin-relativistic quantum particles, enabling wave propagation analysis. The Bäumer propagator describes relativistic diffusion, interpolating between Cauchy and Gaussian processes for time-dependent anomalous diffusion.

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Area of Science:

  • Quantum mechanics
  • Relativistic quantum mechanics
  • Stochastic processes

Background:

  • The Salpeter Hamiltonian describes relativistic quantum particles.
  • Understanding wave propagation and diffusion is crucial in quantum mechanics.
  • Anomalous diffusion presents challenges in modeling stochastic processes.

Purpose of the Study:

  • To derive an analytical expression for the propagator of a (1+1)-dimensional Salpeter Hamiltonian.
  • To describe the wave propagation of non-spin-relativistic quantum particles.
  • To formulate and analyze a relativistic stochastic process with time-dependent anomalous diffusion.

Main Methods:

  • Utilizing integral representations and analytical solutions for the Salpeter Hamiltonian propagator.
  • Deriving the Green function in terms of special functions.
  • Employing analytical extension of the Hamiltonian in the complex plane to formulate the Bäumer equation.
  • Calculating the propagator to interpolate between Cauchy and Gaussian diffusion regimes.

Main Results:

  • An explicit expression for the propagator (Green function) was derived.
  • The wave propagation for massive and massless particles was determined by convolving the Green function with initial wave functions.
  • The Bäumer equation was formulated, describing relativistic stochastic processes with time-changing anomalous diffusion.
  • A Bäumer propagator was analytically calculated, interpolating between Cauchy and Gaussian diffusion.

Conclusions:

  • The derived Green function provides a framework for analyzing wave propagation of quantum particles.
  • The Bäumer propagator offers a novel approach to modeling relativistic diffusion with time-dependent anomalous diffusion.
  • This work establishes a connection between quantum mechanics and stochastic processes with time-dependent anomalous diffusion.