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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 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.

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