1 files changed, 49 insertions, 30 deletions

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.