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| -rw-r--r-- | latex/know/concept/probability-current/source.md | 10 | ||||
| -rw-r--r-- | static/know/concept/probability-current/index.html | 10 |
2 files changed, 10 insertions, 10 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}$$ diff --git a/static/know/concept/probability-current/index.html b/static/know/concept/probability-current/index.html index d736e49..7b7ac32 100644 --- a/static/know/concept/probability-current/index.html +++ b/static/know/concept/probability-current/index.html @@ -56,8 +56,8 @@ \end{aligned}\]</span></p> <p>As the system evolves in time, this probability may change, so we take its derivative with respect to time <span class="math inline">\(t\)</span>, and when necessary substitute in the other side of the Schrödinger equation to get:</p> <p><span class="math display">\[\begin{aligned} - \pd{P}{t} - &= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}} + \pdv{P}{t} + &= \int_{V} \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \dd[3]{\vec{r}} = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}} \\ &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big) @@ -83,15 +83,15 @@ \end{aligned}\]</span></p> <p>Returning to the derivation of <span class="math inline">\(\vec{J}\)</span>, we now have the following equation:</p> <p><span class="math display">\[\begin{aligned} - \pd{P}{t} - = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}} + \pdv{P}{t} + = \int_{V} \pdv{|\psi|^2}{t} \dd[3]{\vec{r}} = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} \end{aligned}\]</span></p> <p>By removing the integrals, we thus arrive at the <em>continuity equation</em> for <span class="math inline">\(\vec{J}\)</span>:</p> <p><span class="math display">\[\begin{aligned} \boxed{ \nabla \cdot \vec{J} - = - \pd{|\psi|^2}{t} + = - \pdv{|\psi|^2}{t} } \end{aligned}\]</span></p> <p>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 <span class="math inline">\(\vec{J}\)</span> represents the flow of probability, which is analogous to the motion of a particle.</p> |