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

In quantum mechanics, the **propagator** $K(x_f, t_f; x_i, t_i)$
gives the probability amplitude that a particle
starting at $x_i$ at $t_i$ ends up at position $x_f$ at $t_f$.
It is defined as follows:

$$\begin{aligned}
    \boxed{
        K(x_f, t_f; x_i, t_i)
        \equiv \matrixel{x_f}{\hat{U}(t_f, t_i)}{x_i}
    }
\end{aligned}$$

Where $\hat{U} \equiv \exp(- i t \hat{H} / \hbar)$ is the time-evolution operator.
The probability that a particle travels
from $(x_i, t_i)$ to $(x_f, t_f)$ is then given by:

$$\begin{aligned}
    P
    &= \big| K(x_f, t_f; x_i, t_i) \big|^2
\end{aligned}$$

Given a general (i.e. non-collapsed) initial state $\psi_i(x) \equiv \psi(x, t_i)$,
we must integrate over $x_i$:

$$\begin{aligned}
    P
    &= \bigg| \int_{-\infty}^\infty K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \bigg|^2
\end{aligned}$$

And if the final state $\psi_f(x) \equiv \psi(x, t_f)$
is not a basis vector either, then we integrate twice:

$$\begin{aligned}
    P
    &= \bigg| \iint_{-\infty}^\infty \psi_f^*(x_f) \: K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \dd{x_f} \bigg|^2
\end{aligned}$$

Given a $\psi_i(x)$, the propagator can also be used
to find the full final wave function:

$$\begin{aligned}
    \boxed{
        \psi(x_f, t_f)
        = \int_{-\infty}^\infty \psi_i(x_i) K(x_f, t_f; x_i, t_i) \:dx_i
    }
\end{aligned}$$

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:

$$\begin{aligned}
    \boxed{
        G(x_f, t_f; x_i, t_i)
        = - \frac{i}{\hbar} \: \Theta(t_f - t_i) \: K(x_f, t_f; x_i, t_i)
    }
\end{aligned}$$

Where $\Theta(t)$ is the [Heaviside step function](/know/concept/heaviside-step-function/).