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diff --git a/source/know/concept/propagator/index.md b/source/know/concept/propagator/index.md index 54e9eb6..50228e2 100644 --- a/source/know/concept/propagator/index.md +++ b/source/know/concept/propagator/index.md @@ -8,63 +8,82 @@ categories: 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: +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: $$\begin{aligned} \boxed{ - K(x_f, t_f; x_i, t_i) - \equiv \matrixel{x_f}{\hat{U}(t_f, t_i)}{x_i} + K(x, t; x_0, t_0) + \equiv \matrixel{x}{\hat{U}(t, t_0)}{x_0} } \end{aligned}$$ -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: +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/). -$$\begin{aligned} - P - &= \big| K(x_f, t_f; x_i, t_i) \big|^2 -\end{aligned}$$ - -Given a general (i.e. non-collapsed) initial state $$\psi_i(x) \equiv \psi(x, t_i)$$, -we must integrate over $$x_i$$: +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: $$\begin{aligned} P - &= \bigg| \int_{-\infty}^\infty K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \bigg|^2 + \propto \big| K(x, t; x_0, t_0) \big|^2 \end{aligned}$$ -And if the final state $$\psi_f(x) \equiv \psi(x, t_f)$$ -is not a basis vector either, then we integrate twice: +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: $$\begin{aligned} - P - &= \bigg| \iint_{-\infty}^\infty \psi_f^*(x_f) \: K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \dd{x_f} \bigg|^2 + \psi(x, t) + &= \inprod{x}{\psi(t)} + \\ + &= \matrixel{x}{\hat{U}(t, t_0)}{\psi(t_0)} + \\ + &= \int_{-\infty}^\infty \bra{x} \hat{U}(t, t_0) \Big( \exprod{x_0}{x_0} \Big) \ket{\psi(t_0)} \dd{x_0} \end{aligned}$$ -Given a $$\psi_i(x)$$, the propagator can also be used -to find the full final wave function: +Where we introduced an identity operator +and recognized $$\psi(x_0, t_0) = \inprod{x_0}{\psi(t_0)}$$, so: $$\begin{aligned} \boxed{ - \psi(x_f, t_f) - = \int_{-\infty}^\infty \psi_i(x_i) K(x_f, t_f; x_i, t_i) \:dx_i + \psi(x, t) + = \int_{-\infty}^\infty K(x, t; x_0, t_0) \: \psi(x_0, t_0) \dd{x_0} } \end{aligned}$$ -Sometimes the name "propagator" is also used to refer to +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: $$\begin{aligned} - \boxed{ - G(x_f, t_f; x_i, t_i) - = - \frac{i}{\hbar} \: \Theta(t_f - t_i) \: K(x_f, t_f; x_i, t_i) - } + G(x, t; x_0, t_0) + = - \frac{i}{\hbar} \: \Theta(t - t_0) \: K(x, t; x_0, t_0) \end{aligned}$$ 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. |