polkuintegraalit

Polkuintegraalit, also known as Feynman path integrals, are a formulation of quantum mechanics developed by Richard Feynman. This approach offers a different perspective on how quantum systems evolve over time compared to the more traditional Hamiltonian or Schrödinger formulations. Instead of focusing on a single trajectory, the path integral method considers all possible paths a quantum particle can take between two points in spacetime.

The core idea is that the probability amplitude for a particle to travel from an initial state to a final state is obtained by summing up contributions from every conceivable path. Each path is assigned a complex number, or weight, which is determined by the classical action associated with that path. The action is a quantity that describes the dynamics of the system according to classical physics. Paths that are closer to the classical path tend to have larger contributions, while highly non-classical paths tend to cancel each other out due to their rapidly oscillating phases.

The mathematical expression for a path integral involves an integral over an infinite-dimensional space of functions, representing all possible paths. This formulation has proven to be incredibly powerful and has found significant applications in various fields of physics, including quantum field theory, statistical mechanics, and condensed matter physics. It provides a unifying framework and offers elegant solutions to complex problems. The concept is conceptually linked to the principle of least action in classical mechanics, extended to the quantum realm.