-
+
+{% include proof/start.html id="proof-relation" -%}
First, observe that we can rewrite the fraction using an integral:
$$\begin{aligned}
@@ -119,9 +116,8 @@ $$\begin{aligned}
= 4 \pi^2 \delta^2(x)
= 2 \pi \: \delta(x) \: t
\end{aligned}$$
+{% include proof/end.html id="proof-relation" %}
-
-
## One-photon absorption
@@ -187,6 +183,7 @@ Note that this transition is only possible when $$\matrixel{u}{\vu{p}}{0} \neq 0
i.e. for any odd-numbered final state $$\Ket{u}$$.
+
## Two-photon absorption
Next, we go to second-order perturbation theory.
@@ -255,6 +252,7 @@ Notice that the rate is proportional to $$|\vb{E}|^4$$,
so this effect is only noticeable at high light intensities.
+
## Three-photon absorption
For third-order perturbation theory,
@@ -327,6 +325,7 @@ The rate is proportional to $$|\vb{E}|^6$$,
so this effect only appears at extremely high light intensities.
+
## N-photon absorption
A pattern has appeared in these calculations:
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