diff options
Diffstat (limited to 'source/know/concept/propagator')
| -rw-r--r-- | source/know/concept/propagator/index.md | 22 |
1 files changed, 11 insertions, 11 deletions
diff --git a/source/know/concept/propagator/index.md b/source/know/concept/propagator/index.md index 437c57e..54e9eb6 100644 --- a/source/know/concept/propagator/index.md +++ b/source/know/concept/propagator/index.md @@ -8,9 +8,9 @@ categories: layout: "concept" --- -In quantum mechanics, the **propagator** $K(x_f, t_f; x_i, t_i)$ +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 $t_f$. +starting at $$x_i$$ at $$t_i$$ ends up at position $$x_f$$ at $$t_f$$. It is defined as follows: $$\begin{aligned} @@ -20,24 +20,24 @@ $$\begin{aligned} } \end{aligned}$$ -Where $\hat{U} \equiv \exp(- i t \hat{H} / \hbar)$ is the time-evolution operator. +Where $$\hat{U} \equiv \exp(- i t \hat{H} / \hbar)$$ is the time-evolution operator. The probability that a particle travels -from $(x_i, t_i)$ to $(x_f, t_f)$ is then given by: +from $$(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}$$ -Given a general (i.e. non-collapsed) initial state $\psi_i(x) \equiv \psi(x, t_i)$, -we must integrate over $x_i$: +Given a general (i.e. non-collapsed) initial state $$\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}$$ -And if the final state $\psi_f(x) \equiv \psi(x, t_f)$ +And if the final state $$\psi_f(x) \equiv \psi(x, t_f)$$ is not a basis vector either, then we integrate twice: $$\begin{aligned} @@ -45,7 +45,7 @@ $$\begin{aligned} &= \bigg| \iint_{-\infty}^\infty \psi_f^*(x_f) \: K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \dd{x_f} \bigg|^2 \end{aligned}$$ -Given a $\psi_i(x)$, the propagator can also be used +Given a $$\psi_i(x)$$, the propagator can also be used to find the full final wave function: $$\begin{aligned} @@ -56,9 +56,9 @@ $$\begin{aligned} \end{aligned}$$ Sometimes the name "propagator" is also used to refer to -the [fundamental solution](/know/concept/fundamental-solution/) $G$ +the [fundamental solution](/know/concept/fundamental-solution/) $$G$$ of the time-dependent Schrödinger equation, -which is related to $K$ by: +which is related to $$K$$ by: $$\begin{aligned} \boxed{ @@ -67,4 +67,4 @@ $$\begin{aligned} } \end{aligned}$$ -Where $\Theta(t)$ is the [Heaviside step function](/know/concept/heaviside-step-function/). +Where $$\Theta(t)$$ is the [Heaviside step function](/know/concept/heaviside-step-function/). |