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. Propagators 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).

The Green's function G for the Schrödinger equation is a function $${\displaystyle G(x,t;x',t')={\frac {1}{i\hbar }}\Theta (t-t')K(x,t;x',t')}$$ satisfying $${\displaystyle \left(i\hbar {\frac {\partial }{\partial t}}-H_{x}\right)G(x,t;x',t')=\delta (x-x')\delta (t-t'),}$$ where H denotes the Hamiltonian, δ(x) denotes the Dirac delta-function and Θ(t) is the Heaviside step function. The kernel of the above Schrödinger differential operator in the big parentheses is denoted by K(x, t ;x′, t′) and called the propagator.

This propagator may also be written as the transition amplitude $${\displaystyle K(x,t;x',t')={\big \langle }x{\big |}U(t,t'){\big |}x'{\big \rangle },}$$ where U(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 $${\displaystyle \lim _{t\to t'}K(x,t;x',t')=\delta (x-x').}$$ The propagator may also be found by using a path integral:

where L denotes the Lagrangian and the boundary conditions are given by q(t) = x, q(t′) = x′. The paths that are summed over move only forwards in time and are integrated with the differential ${\displaystyle D[q(t)]}$ following the path in time.

The propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is given by

If K(x, t; x′, t′) only depends on the difference x − x′, this is a convolution of the initial wave function and the propagator.

For a time-translationally invariant system, the propagator only depends on the time difference t − t′, so it may be rewritten as $${\displaystyle K(x,t;x',t')=K(x,x';t-t').}$$