Propagator

In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function). In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a particle to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t). Consider a system with Hamiltonian H. The Green's function (fundamental solution) for the Schrödinger equation is a function satisfying where Hx denotes the Hamiltonian written in terms of the x coordinates, δ(x) denotes the Dirac delta-function, Θ(t) is the Heaviside step function and K(x, t ;x′, t′) is the kernel of the above Schrödinger differential operator in the big parentheses. The term propagator is sometimes used in this context to refer to G, and sometimes to K. This article will use the term to refer to K (see Duhamel's principle). This propagator may also be written as the transition amplitude where Û(t, t′) is the unitary time-evolution operator for the system taking states at time t′ to states at time t. Note the initial condition enforced by . The quantum-mechanical propagator may also be found by using a path integral: where the boundary conditions of the path integral include q(t) = x, q(t′) = x′. Here L denotes the Lagrangian of the system. The paths that are summed over move only forwards in time and are integrated with the differential following the path in time.