From ce9aef392998e471d41f88beb54d07e58dbf57d3 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 20 Feb 2021 15:56:36 +0100 Subject: Add "Probability current" --- content/know/category/quantum-mechanics.md | 3 + content/know/concept/index.md | 3 + latex/know/concept/probability-current/source.md | 85 ++++++++++++++++ static/know/concept/probability-current/index.html | 108 +++++++++++++++++++++ 4 files changed, 199 insertions(+) create mode 100644 latex/know/concept/probability-current/source.md create mode 100644 static/know/concept/probability-current/index.html diff --git a/content/know/category/quantum-mechanics.md b/content/know/category/quantum-mechanics.md index dcb6eb6..d302dfe 100644 --- a/content/know/category/quantum-mechanics.md +++ b/content/know/category/quantum-mechanics.md @@ -11,3 +11,6 @@ Alphabetical list of concepts in this category. ## D * [Dirac notation](/know/concept/dirac-notation/) + +## P +* [Probability current](/know/concept/probability-current/) diff --git a/content/know/concept/index.md b/content/know/concept/index.md index 6604aba..19f2027 100644 --- a/content/know/concept/index.md +++ b/content/know/concept/index.md @@ -11,3 +11,6 @@ Alphabetical list of concepts in this knowledge base. ## D * [Dirac notation](/know/concept/dirac-notation/) + +## P +* [Probability current](/know/concept/probability-current/) diff --git a/latex/know/concept/probability-current/source.md b/latex/know/concept/probability-current/source.md new file mode 100644 index 0000000..a6780f7 --- /dev/null +++ b/latex/know/concept/probability-current/source.md @@ -0,0 +1,85 @@ +# Probability current + +In quantum mechanics, the *probability current* expresses the movement +of the probability of finding a particle. Or 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 given by: + +$$\begin{aligned} + P = \int_{V} | \psi |^2 \dd[3]{\vec{r}} +\end{aligned}$$ + +As the system evolves in time, this probability may change, so we take +its derivative with respect to time $t$, and when necessary substitute +in the other side of the Schrödinger equation to get: + +$$\begin{aligned} + \pd{P}{t} + &= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}} + = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}} + \\ + &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big) + - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vec{r}) \psi \Big) \dd[3]{\vec{r}} + \\ + &= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \dd[3]{\vec{r}} + = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} +\end{aligned}$$ + +Where we have defined the probability current $\vec{J}$ as follows in +the $\vec{r}$-basis: + +$$\begin{aligned} + \vec{J} + = \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi) + = \mathrm{Re} \Big\{ \psi \frac{i \hbar}{m} \psi^* \Big\} +\end{aligned}$$ + +Let us rewrite this using the momentum operator +$\hat{p} = -i \hbar \nabla$ as follows, noting that $\hat{p} / m$ is +simply the velocity operator $\hat{v}$: + +$$\begin{aligned} + \boxed{ + \vec{J} + = \frac{1}{2 m} ( \psi^* \hat{p} \psi - \psi \hat{p} \psi^*) + = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p}}{m} \psi \Big\} + = \mathrm{Re} \{ \psi^* \hat{v} \psi \} + } +\end{aligned}$$ + +Returning to the derivation of $\vec{J}$, we now have the following +equation: + +$$\begin{aligned} + \pd{P}{t} + = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}} + = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} +\end{aligned}$$ + +By removing the integrals, we thus arrive at the *continuity equation* +for $\vec{J}$: + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vec{J} + = - \pd{|\psi|^2}{t} + } +\end{aligned}$$ + +This states that 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 +$\vec{J}$ 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 $\vec{A}$, 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: + +$$\begin{aligned} + \boxed{ + \vec{J} + = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p} - q \vec{A}}{m} \psi \Big\} + } +\end{aligned}$$ diff --git a/static/know/concept/probability-current/index.html b/static/know/concept/probability-current/index.html new file mode 100644 index 0000000..d736e49 --- /dev/null +++ b/static/know/concept/probability-current/index.html @@ -0,0 +1,108 @@ + + + + + + + Prefetch | source + + + + + + +
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Probability current

+

In quantum mechanics, the probability current expresses the movement of the probability of finding a particle. Or 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 given by:

+

\[\begin{aligned} + P = \int_{V} | \psi |^2 \dd[3]{\vec{r}} +\end{aligned}\]

+

As the system evolves in time, this probability may change, so we take its derivative with respect to time \(t\), and when necessary substitute in the other side of the Schrödinger equation to get:

+

\[\begin{aligned} + \pd{P}{t} + &= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}} + = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}} + \\ + &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big) + - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vec{r}) \psi \Big) \dd[3]{\vec{r}} + \\ + &= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \dd[3]{\vec{r}} + = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} +\end{aligned}\]

+

Where we have defined the probability current \(\vec{J}\) as follows in the \(\vec{r}\)-basis:

+

\[\begin{aligned} + \vec{J} + = \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi) + = \mathrm{Re} \Big\{ \psi \frac{i \hbar}{m} \psi^* \Big\} +\end{aligned}\]

+

Let us rewrite this using the momentum operator \(\hat{p} = -i \hbar \nabla\) as follows, noting that \(\hat{p} / m\) is simply the velocity operator \(\hat{v}\):

+

\[\begin{aligned} + \boxed{ + \vec{J} + = \frac{1}{2 m} ( \psi^* \hat{p} \psi - \psi \hat{p} \psi^*) + = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p}}{m} \psi \Big\} + = \mathrm{Re} \{ \psi^* \hat{v} \psi \} + } +\end{aligned}\]

+

Returning to the derivation of \(\vec{J}\), we now have the following equation:

+

\[\begin{aligned} + \pd{P}{t} + = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}} + = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} +\end{aligned}\]

+

By removing the integrals, we thus arrive at the continuity equation for \(\vec{J}\):

+

\[\begin{aligned} + \boxed{ + \nabla \cdot \vec{J} + = - \pd{|\psi|^2}{t} + } +\end{aligned}\]

+

This states that 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 \(\vec{J}\) 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 \(\vec{A}\), 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:

+

\[\begin{aligned} + \boxed{ + \vec{J} + = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p} - q \vec{A}}{m} \psi \Big\} + } +\end{aligned}\]

+
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