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diff --git a/source/know/concept/probability-current/index.md b/source/know/concept/probability-current/index.md new file mode 100644 index 0000000..6f85149 --- /dev/null +++ b/source/know/concept/probability-current/index.md @@ -0,0 +1,99 @@ +--- +title: "Probability current" +date: 2021-02-22 +categories: +- Quantum mechanics +- Physics +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: + +$$\begin{aligned} + P = \int_{V} | \psi |^2 \ddn{3}{\vb{r}} +\end{aligned}$$ + +As the system evolves in time, this probability may change, so we take +its derivative with respect to time $t$, and when necessary substitute +in the other side of the Schrödinger equation to get: + +$$\begin{aligned} + \pdv{P}{t} + &= \int_{V} \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \ddn{3}{\vb{r}} + = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \ddn{3}{\vb{r}} + \\ + &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vb{r}) \psi^* \Big) + - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vb{r}) \psi \Big) \ddn{3}{\vb{r}} + \\ + &= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \ddn{3}{\vb{r}} + = - \int_{V} \nabla \cdot \vb{J} \ddn{3}{\vb{r}} +\end{aligned}$$ + +Where we have defined the probability current $\vb{J}$ as follows in +the $\vb{r}$-basis: + +$$\begin{aligned} + \vb{J} + = \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi) + = \mathrm{Re} \Big\{ \psi \frac{i \hbar}{m} \psi^* \Big\} +\end{aligned}$$ + +Let us rewrite this using the momentum operator +$\vu{p} = -i \hbar \nabla$ as follows, noting that $\vu{p} / m$ is +simply the velocity operator $\vu{v}$: + +$$\begin{aligned} + \boxed{ + \vb{J} + = \frac{1}{2 m} ( \psi^* \vu{p} \psi - \psi \vu{p} \psi^*) + = \mathrm{Re} \Big\{ \psi^* \frac{\vu{p}}{m} \psi \Big\} + = \mathrm{Re} \{ \psi^* \vu{v} \psi \} + } +\end{aligned}$$ + +Returning to the derivation of $\vb{J}$, we now have the following +equation: + +$$\begin{aligned} + \pdv{P}{t} + = \int_{V} \pdv{|\psi|^2}{t} \ddn{3}{\vb{r}} + = - \int_{V} \nabla \cdot \vb{J} \ddn{3}{\vb{r}} +\end{aligned}$$ + +By removing the integrals, we thus arrive at the **continuity equation** +for $\vb{J}$: + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vb{J} + = - \pdv{|\psi|^2}{t} + } +\end{aligned}$$ + +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 +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 +equation. We can thus extend the definition to a particle with charge +$q$ in an SI-unit field, neglecting spin: + +$$\begin{aligned} + \boxed{ + \vb{J} + = \mathrm{Re} \Big\{ \psi^* \frac{\vu{p} - q \vb{A}}{m} \psi \Big\} + } +\end{aligned}$$ + + + +## References +1. L.E. Ballentine, + *Quantum mechanics: a modern development*, 2nd edition, + World Scientific. |