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Quantum Hamilton-Jacobi theory.

Marco Roncadelli1, L S Schulman

  • 1INFN, Sezione di Pavia, Via A. Bassi 6, I-27100 Pavia, Italy. marco.roncadelli@pv.infn.it

Physical Review Letters
|November 13, 2007
PubMed
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Researchers have solved the quantum Hamilton-Jacobi equation (QHJE) by using the Schrödinger equation

Area of Science:

  • Quantum mechanics
  • Theoretical physics
  • Mathematical physics

Background:

  • Quantum canonical transformations are fundamental in quantum theory.
  • Solving the nonlinear operator partial differential equation of the quantum Hamilton-Jacobi equation (QHJE) has been a significant challenge.
  • This difficulty has limited the practical application of quantum Hamilton-Jacobi theory.

Purpose of the Study:

  • To overcome the challenge of solving the quantum Hamilton-Jacobi equation (QHJE).
  • To enable the practical application of quantum Hamilton-Jacobi theory.
  • To explore the relationship between operator ordering and path density in semiclassical trajectories.

Main Methods:

  • Developing a novel prescription to construct solutions for the QHJE.

Related Experiment Videos

  • Utilizing the propagator of the associated Schrödinger equation as a starting point.
  • Applying the derived solutions to investigate operator ordering and path density.

    • Main Results:

    • A straightforward method for solving the QHJE has been established.
    • Solutions to the QHJE can be directly constructed from the Schrödinger equation's propagator.
    • A novel connection between operator ordering and the density of paths near semiclassical trajectories has been revealed.

    Conclusions:

    • The developed method simplifies the solution of the QHJE, paving the way for its practical use.
    • This breakthrough opens new avenues for research in quantum canonical transformations and quantum dynamics.
    • The findings offer new insights into the interpretation of quantum mechanics and semiclassical approximations.