In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
event described by the respective diagram. These may also be viewed as the
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
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).
Non-relativistic propagators
In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
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 . The
Green's function (
fundamental solution) for the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
is a function
:
satisfying
:
where denotes the Hamiltonian written in terms of the coordinates, denotes the
Dirac delta-function, is the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
and 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 , and sometimes to . This article will use the term to refer to (see
Duhamel's principle).
This propagator may also be written as the transition amplitude
:
where is the
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
time-evolution operator for the system taking states at time to states at time . 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 . Here denotes the
Lagrangian of the system. The paths that are summed over move only forwards in time and are integrated with the differential