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The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics.It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
This expression actually defines the manner in which the path integrals are to be taken. The coefficient in front is needed to ensure that the expression has the correct dimensions, but it has no actual relevance in any physical application. This recovers the path integral formulation from Schrödinger's equation.
Line integral, the integral of a function along a curve; Contour integral, the integral of a complex function along a curve used in complex analysis; Functional integration, the integral of a functional over a space of curves; Path integral formulation, Richard Feynman's formulation of quantum mechanics using functional integration
Path integral Monte Carlo (PIMC) is a quantum Monte Carlo method used to solve quantum statistical mechanics problems numerically within the path integral formulation.The application of Monte Carlo methods to path integral simulations of condensed matter systems was first pursued in a key paper by John A. Barker.
Hamilton's principle has also been extended to quantum mechanics and quantum field theory—in particular the path integral formulation of quantum mechanics makes use of the concept—where a physical system explores all possible paths, with the phase of the probability amplitude for each path being determined by the action for the path; the ...
This is closely tied to the functional integral formulation of quantum mechanics, also invented by Feynman—see path integral formulation. The naïve application of such calculations often produces diagrams whose amplitudes are infinite , because the short-distance particle interactions require a careful limiting procedure, to include particle ...
Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases. Path integral formulation; Phase-space formulation of quantum mechanics & geometric quantization; quantum field theory in curved spacetime
The stochastic interpretation interprets the paths in the path integral formulation of quantum mechanics as the sample paths of a stochastic process. [9] It posits that quantum particles are localized on one of these paths, but observers cannot predict with certainty where the particle is localized.