diff options
Diffstat (limited to 'latex/know/concept/probability-current')
| -rw-r--r-- | latex/know/concept/probability-current/source.md | 10 |
1 files changed, 5 insertions, 5 deletions
diff --git a/latex/know/concept/probability-current/source.md b/latex/know/concept/probability-current/source.md index a6780f7..69faf0c 100644 --- a/latex/know/concept/probability-current/source.md +++ b/latex/know/concept/probability-current/source.md @@ -14,8 +14,8 @@ its derivative with respect to time $t$, and when necessary substitute in the other side of the Schrödinger equation to get: $$\begin{aligned} - \pd{P}{t} - &= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}} + \pdv{P}{t} + &= \int_{V} \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \dd[3]{\vec{r}} = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}} \\ &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big) @@ -51,8 +51,8 @@ Returning to the derivation of $\vec{J}$, we now have the following equation: $$\begin{aligned} - \pd{P}{t} - = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}} + \pdv{P}{t} + = \int_{V} \pdv{|\psi|^2}{t} \dd[3]{\vec{r}} = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} \end{aligned}$$ @@ -62,7 +62,7 @@ for $\vec{J}$: $$\begin{aligned} \boxed{ \nabla \cdot \vec{J} - = - \pd{|\psi|^2}{t} + = - \pdv{|\psi|^2}{t} } \end{aligned}$$ |