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author | Prefetch | 2022-10-20 18:25:31 +0200 |
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committer | Prefetch | 2022-10-20 18:25:31 +0200 |
commit | 16555851b6514a736c5c9d8e73de7da7fc9b6288 (patch) | |
tree | 76b8bfd30f8941d0d85365990bcdbc5d0643cabc /source/know/concept/probability-current/index.md | |
parent | e5b9bce79b68a68ddd2e51daa16d2fea73b84fdb (diff) |
Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
Diffstat (limited to 'source/know/concept/probability-current/index.md')
-rw-r--r-- | source/know/concept/probability-current/index.md | 24 |
1 files changed, 12 insertions, 12 deletions
diff --git a/source/know/concept/probability-current/index.md b/source/know/concept/probability-current/index.md index 43683a3..bd41dab 100644 --- a/source/know/concept/probability-current/index.md +++ b/source/know/concept/probability-current/index.md @@ -10,15 +10,15 @@ layout: "concept" In quantum mechanics, the **probability current** describes the movement of the probability of finding a particle at given point in space. -In other words, it treats the particle as a heterogeneous fluid with density $|\psi|^2$. -Now, the probability of finding the particle within a volume $V$ is: +In other words, it treats the particle as a heterogeneous fluid with density $$|\psi|^2$$. +Now, the probability of finding the particle within a volume $$V$$ is: $$\begin{aligned} P = \int_{V} | \psi |^2 \ddn{3}{\vb{r}} \end{aligned}$$ As the system evolves in time, this probability may change, so we take -its derivative with respect to time $t$, and when necessary substitute +its derivative with respect to time $$t$$, and when necessary substitute in the other side of the Schrödinger equation to get: $$\begin{aligned} @@ -33,8 +33,8 @@ $$\begin{aligned} = - \int_{V} \nabla \cdot \vb{J} \ddn{3}{\vb{r}} \end{aligned}$$ -Where we have defined the probability current $\vb{J}$ as follows in -the $\vb{r}$-basis: +Where we have defined the probability current $$\vb{J}$$ as follows in +the $$\vb{r}$$-basis: $$\begin{aligned} \vb{J} @@ -43,8 +43,8 @@ $$\begin{aligned} \end{aligned}$$ Let us rewrite this using the momentum operator -$\vu{p} = -i \hbar \nabla$ as follows, noting that $\vu{p} / m$ is -simply the velocity operator $\vu{v}$: +$$\vu{p} = -i \hbar \nabla$$ as follows, noting that $$\vu{p} / m$$ is +simply the velocity operator $$\vu{v}$$: $$\begin{aligned} \boxed{ @@ -55,7 +55,7 @@ $$\begin{aligned} } \end{aligned}$$ -Returning to the derivation of $\vb{J}$, we now have the following +Returning to the derivation of $$\vb{J}$$, we now have the following equation: $$\begin{aligned} @@ -65,7 +65,7 @@ $$\begin{aligned} \end{aligned}$$ By removing the integrals, we thus arrive at the **continuity equation** -for $\vb{J}$: +for $$\vb{J}$$: $$\begin{aligned} \boxed{ @@ -77,13 +77,13 @@ $$\begin{aligned} This states that the total probability is conserved, and is reminiscent of charge conservation in electromagnetism. In other words, the probability at a point can only change by letting it "flow" towards or away from it. Thus -$\vb{J}$ represents the flow of probability, which is analogous to the +$$\vb{J}$$ represents the flow of probability, which is analogous to the motion of a particle. As a bonus, this still holds for a particle in an electromagnetic vector -potential $\vb{A}$, thanks to the gauge invariance of the Schrödinger +potential $$\vb{A}$$, thanks to the gauge invariance of the Schrödinger equation. We can thus extend the definition to a particle with charge -$q$ in an SI-unit field, neglecting spin: +$$q$$ in an SI-unit field, neglecting spin: $$\begin{aligned} \boxed{ |