The chapter discusses the Path integral formulation of quantum mechanics, pioneered by Feynman, contrasting it with traditional Hamiltonian methods. It explores how quantum systems’ evolution can be understood through a sum over all possible paths, akin to classical trajectories but with quantum amplitudes replacing classical probabilities. Starting from the concept of probability amplitudes and propagators, it derives the Path integral expression using discretization of spacetime, defining the propagator as a sum over infinitesimal paths weighted by the action. This approach encapsulates quantum behavior beyond classical trajectories, highlighting its flexibility and foundational role in quantum field theory. The chapter concludes by connecting back to the Schrödinger equation, demonstrating how the Path integral framework can derive quantum dynamics from classical principles, underscoring its broad applicability in theoretical physics.

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Beyond the Least Action, Path Integral Formalism to Quantum Mechanics

  • Bertrand Berche,
  • Ernesto Medina

摘要

The chapter discusses the Path integral formulation of quantum mechanics, pioneered by Feynman, contrasting it with traditional Hamiltonian methods. It explores how quantum systems’ evolution can be understood through a sum over all possible paths, akin to classical trajectories but with quantum amplitudes replacing classical probabilities. Starting from the concept of probability amplitudes and propagators, it derives the Path integral expression using discretization of spacetime, defining the propagator as a sum over infinitesimal paths weighted by the action. This approach encapsulates quantum behavior beyond classical trajectories, highlighting its flexibility and foundational role in quantum field theory. The chapter concludes by connecting back to the Schrödinger equation, demonstrating how the Path integral framework can derive quantum dynamics from classical principles, underscoring its broad applicability in theoretical physics.