In quantum mechanics, the propagator
gives the probability amplitude that a particle
starting at at ends up at position at .
It is defined as follows:
Where is the time-evolution operator.
The probability that a particle travels
from to is then given by:
Given a general (i.e. non-collapsed) initial state ,
we must integrate over :
And if the final state
is not a basis vector either, then we integrate twice:
Given a , the propagator can also be used
to find the full final wave function:
Sometimes the name “propagator” is also used to refer to
the fundamental solution
of the time-dependent Schrödinger equation,
which is related to by:
Where is the Heaviside step function.