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Integral transforms of the quantum mechanical path integral: Hit function and path-averaged potential.

James P Edwards1, Urs Gerber1,2, Christian Schubert1

  • 1Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán, Mexico.

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We developed new integral transforms for the quantum mechanical transition kernel to analyze path integrals. These methods provide insights into particle trajectories and potential energy contributions.

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Area of Science:

  • Quantum mechanics
  • Path integral formulation
  • Theoretical physics

Background:

  • The path integral formulation is a powerful tool in quantum mechanics.
  • Understanding particle trajectories and potential interactions is crucial.
  • Existing methods may not fully capture trajectory-specific information.

Purpose of the Study:

  • Introduce novel integral transforms of the quantum mechanical transition kernel.
  • Develop tools to extract physical information from path integrals.
  • Interpret these transforms as probability distributions on particle trajectories.

Main Methods:

  • Define two specific integral transforms of the quantum mechanical transition kernel.
  • Interpret the first transform as a 'hit function' measuring path contributions through a spatial point.
  • Interpret the second transform as a 'path-averaged potential' quantifying likelihoods of potential line integrals.

Main Results:

  • The hit function quantifies the relative contribution of paths crossing a specific spatial point.
  • The path-averaged potential quantifies the likelihood of potential values along paths.
  • These transforms offer new perspectives on the physical information encoded in path integrals.

Conclusions:

  • The introduced integral transforms provide novel interpretations of the path integral.
  • These methods offer new ways to analyze particle trajectories and potential interactions in quantum mechanics.
  • The hit function and path-averaged potential serve as valuable tools for theoretical physics research.