Feynman's Path Integral Formulation

The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude.

The path integral formulation was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier, in the course of his doctoral thesis work with John Archibald Wheeler.

This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970s which unified quantum field theory with statistical mechanics (see the first three textbooks below). If we realize that the Schrödinger equation is essentially a diffusion equation with an imaginary diffusion constant, then the path integral is a method for summing up all possible of random walks.

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Caging Schrödinger's Cat - Quantum Nanotechnology3-Dec-2008

Weird new possibilities emerge as we explore the nanoworld, the universe at the length scale of a billionth of a metre. Here the theory of quantum mechanics bewilders our everyday common sense, as Erwin Schrödinger famously expressed when he imagined a cat that was both dead and alive at the same time! Now Dr Simon Benjamin shows us how experts in physics, chemistry and materials science are working together to harness this strange reality. Underlying their research is the promise of building what may be the most exotic and powerful technology ever conceived: the quantum computer.

Simon Benjamin, Oxford University
http://media.podcasts.ox.ac.uk/mat/nanotechnology/quantum4-medium-video.mp4?CAMEFROM=podcastsGET


Poisson Bracket

In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. In a more general setting, the Poisson bracket is used to define a Poisson algebra, of which the Poisson manifolds are a special case. These are all named in honour of Siméon-Denis Poisson.

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