Categories: Physics, Quantum mechanics.

Propagator

In quantum mechanics, the propagator K(x,t;x0,t0)K(x, t; x_0, t_0) gives the probability amplitude that a (spinless) particle starting at (x0,t0)(x_0, t_0) ends up at (x,t)(x, t). It is defined as:

K(x,t;x0,t0)≡⟨x∣U^(t,t0)∣x0⟩\begin{aligned} \boxed{ K(x, t; x_0, t_0) \equiv \matrixel{x}{\hat{U}(t, t_0)}{x_0} } \end{aligned}

With U^\hat{U} the time evolution operator, given by U^(t,t0)=e−i(t−t0)H^/ℏ\hat{U}(t, t_0) = e^{- i (t - t_0) \hat{H} / \hbar} for a time-independent H^\hat{H}. Practically, KK is often calculated using path integrals.

The principle here is straightforward: evolve the initial state with U^\hat{U}, and project the resulting superposition ∣ψ⟩\ket{\psi} onto the queried final state. The probability density PP that the particle has travelled from (x0,t0)(x_0, t_0) to (x,t)(x, t) is then:

P∝∣K(x,t;x0,t0)∣2\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 ∣ψ(t0)⟩\ket{\psi(t_0)}, in which case the final wavefunction ψ(x,t)\psi(x, t) is as follows:

ψ(x,t)=⟨x∣ψ(t)⟩=⟨x∣U^(t,t0)∣ψ(t0)⟩=∫−∞∞⟨x∣U^(t,t0)(∣x0⟩⟨x0∣)∣ψ(t0)⟩dx0\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 ψ(x0,t0)=⟨x0∣ψ(t0)⟩\psi(x_0, t_0) = \inprod{x_0}{\psi(t_0)}, so:

ψ(x,t)=∫−∞∞K(x,t;x0,t0) ψ(x0,t0)dx0\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)(x, t) is then P∝∣ψ(x,t)∣2P \propto \big| \psi(x, t) \big|^2 as usual.

Sometimes the name propagator is also used to refer to the fundamental solution GG of the time-dependent Schrödinger equation, which is related to KK by:

G(x,t;x0,t0)=−iℏ Θ(t−t0) K(x,t;x0,t0)\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 Θ(t)\Theta(t) is the Heaviside step function. This GG is a particular example of a Green’s function, 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.