Categories: Physics, Quantum mechanics.

# Propagator

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 so-called fundamental solution or Green’s function $$G$$ 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. The definition of $$G$$ is that it satisfies the following equation, where $$\delta$$ is the Dirac delta function:

\begin{aligned} \Big( i \hbar \pdv{t_f} - \hat{H} \Big) G = \delta(x_f - x_i) \: \delta(t_f - t_i) \end{aligned}