---
title: "Propagator"
sort_title: "Propagator"
date: 2021-07-04
categories:
- Physics
- Quantum mechanics
layout: "concept"
---

In quantum mechanics, the **propagator** $$K(x, t; x_0, t_0)$$
gives the probability amplitude that a (spinless) particle
starting at $$(x_0, t_0)$$ ends up at $$(x, t)$$.
It is defined as:

$$\begin{aligned}
    \boxed{
        K(x, t; x_0, t_0)
        \equiv \matrixel{x}{\hat{U}(t, t_0)}{x_0}
    }
\end{aligned}$$

With $$\hat{U}$$ the [time evolution operator](/know/concept/time-evolution-operator/),
given by $$\hat{U}(t, t_0) = e^{- i (t - t_0) \hat{H} / \hbar}$$
for a time-independent $$\hat{H}$$.
Practically, $$K$$ is often calculated using
[path integrals](/know/concept/path-integral-formulation/).

The principle here is straightforward:
evolve the initial state with $$\hat{U}$$,
and project the resulting superposition $$\ket{\psi}$$ onto the queried final state.
The probability density $$P$$ that the particle has travelled
from $$(x_0, t_0)$$ to $$(x, t)$$ is then:

$$\begin{aligned}
    P
    \propto \big| K(x, t; x_0, t_0) \big|^2
\end{aligned}$$

The propagator is also useful if the particle
starts in a general superposition $$\ket{\psi(t_0)}$$,
in which case the final wavefunction $$\psi(x, t)$$ is as follows:

$$\begin{aligned}
    \psi(x, t)
    &= \inprod{x}{\psi(t)}
    \\
    &= \matrixel{x}{\hat{U}(t, t_0)}{\psi(t_0)}
    \\
    &= \int_{-\infty}^\infty \bra{x} \hat{U}(t, t_0) \Big( \exprod{x_0}{x_0} \Big) \ket{\psi(t_0)} \dd{x_0}
\end{aligned}$$

Where we introduced an identity operator
and recognized $$\psi(x_0, t_0) = \inprod{x_0}{\psi(t_0)}$$, so:

$$\begin{aligned}
    \boxed{
        \psi(x, t)
        = \int_{-\infty}^\infty K(x, t; x_0, t_0) \: \psi(x_0, t_0) \dd{x_0}
    }
\end{aligned}$$

The probability density of finding
the particle at $$(x, t)$$ is then
$$P \propto \big| \psi(x, t) \big|^2 $$ as usual.

Sometimes the name *propagator* is also used to refer to
the [fundamental solution](/know/concept/fundamental-solution/) $$G$$
of the time-dependent Schrödinger equation,
which is related to $$K$$ by:

$$\begin{aligned}
    G(x, t; x_0, t_0)
    = - \frac{i}{\hbar} \: \Theta(t - t_0) \: K(x, t; x_0, t_0)
\end{aligned}$$

Where $$\Theta(t)$$ is the [Heaviside step function](/know/concept/heaviside-step-function/).
This $$G$$ is a particular example
of a [Green's function](/know/concept/greens-functions/),
but not all Green's functions are fundamental solutions
to the Schrödinger equation.
To add to the confusion, older literature tends to
call *all* fundamental solutions *Green's functions*,
even in classical contexts,
    so the term has a distinct (but related) meaning
inside and outside quantum mechanics.
The result is a mess where the terms *propagator*,
*fundamental solution* and *Green's function*
are used more or less interchangeably.