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+---
+title: "Electric dipole approximation"
+firstLetter: "E"
+publishDate: 2021-09-14
+categories:
+- Physics
+- Quantum mechanics
+- Optics
+
+date: 2021-09-14T13:11:54+02:00
+draft: false
+markup: pandoc
+---
+
+# Electric dipole approximation
+
+Suppose that an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
+is travelling through an atom, and affecting the electrons.
+The general Hamiltonian of an electron in such a wave is given by:
+
+$$\begin{aligned}
+    \hat{H}
+    &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{2 m} (\vec{A} \cdot \vec{P} + \vec{P} \cdot \vec{A}) + \frac{q^2 \vec{A}{}^2}{2m} + V
+\end{aligned}$$
+
+With charge $q = - e$
+and electromagnetic vector potential $\vec{A}(\vec{r}, t)$.
+We reduce this by fixing the Coulomb gauge $\nabla \cdot \vec{A} = 0$,
+so that $\vec{A} \cdot \vec{P} = \vec{P} \cdot \vec{A}$,
+and assume that $\vec{A}{}^2$ is negligible:
+
+$$\begin{aligned}
+    \hat{H}
+    = \hat{H}_0 + \hat{H}_1
+    \qquad \quad
+    \hat{H}_0
+    \equiv \frac{\vec{P}{}^2}{2 m} + V
+    \qquad \quad
+    \hat{H}_1
+    \equiv - \frac{q}{m} \vec{P} \cdot \vec{A}
+\end{aligned}$$
+
+We have split $\hat{H}$ into $\hat{H}_0$
+and a perturbation $\hat{H}_1$, since $\vec{A}$ is small.
+In an electromagnetic wave,
+$\vec{A}$ is oscillating sinusoidally in time and space as follows:
+
+$$\begin{aligned}
+    \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
+\end{aligned}$$
+
+The corresponding perturbative
+[electric field](/know/concept/electric-field/) $\vec{E}$
+points in the same direction:
+
+$$\begin{aligned}
+    \vec{E}(\vec{r}, t)
+    = - \pdv{\vec{A}}{t}
+    = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
+\end{aligned}$$
+
+Where $\vec{E}_0 = i \omega \vec{A}_0$.
+Let us restrict ourselves to visible light,
+whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$.
+Meanwhile, an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$,
+so $\vec{k} \cdot \vec{r}$ is negligible:
+
+$$\begin{aligned}
+    \boxed{
+        \vec{E}(\vec{r}, t)
+        \approx \vec{E}_0 \exp\!(- i \omega t)
+    }
+\end{aligned}$$
+
+This is the **electric dipole approximation**:
+we ignore all spatial variation of $\vec{E}$,
+and only consider its temporal oscillation.
+Also, since we have not used the word "photon",
+we are implicitly treating the radiation classically,
+and the electron quantum-mechanically.
+
+Next, we want to convert $\hat{H}_1$
+to use the electric field $\vec{E}$ instead of the potential $\vec{A}$.
+To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$
+and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/):
+
+$$\begin{aligned}
+    \matrixel{2}{\dv*{\vec{r}}{t}}{1}
+    &= \frac{i}{\hbar} \matrixel{2}{[\hat{H}_0, \vec{r}]}{1}
+    = \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vec{r} - \vec{r} \hat{H}_0}{1}
+    \\
+    &= \frac{i}{\hbar} (E_2 - E_1) \matrixel{2}{\vec{r}}{1}
+    = i \omega_0 \matrixel{2}{\vec{r}}{1}
+\end{aligned}$$
+
+Therefore, $\vec{P} / m = i \omega_0 \vec{r}$,
+where $\omega_0 \equiv (E_2 - E_1) / \hbar$ is the resonance frequency of the transition,
+close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating.
+We thus get:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    &= - \frac{q}{m} \vec{P} \cdot \vec{A}
+    = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t)
+    \\
+    &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t)
+    = - \vec{d} \cdot \vec{E}_0 \exp\!(- i \omega t)
+\end{aligned}$$
+
+Where $\vec{d} \equiv q \vec{r} = - e \vec{r}$ is
+the **transition dipole moment operator** of the electron,
+hence the name **electric dipole approximation**.
+Finally, since electric fields are actually real
+(we let it be complex for mathematical convenience),
+we take the real part, yielding:
+
+$$\begin{aligned}
+    \boxed{
+        \hat{H}_1(t)
+        = - q \vec{r} \cdot \vec{E}_0 \cos\!(\omega t)
+    }
+\end{aligned}$$
+
+If this approximation is too rough,
+$\vec{E}$ can always be Taylor-expanded in $(i \vec{k} \cdot \vec{r})$:
+
+$$\begin{aligned}
+    \vec{E}(\vec{r}, t)
+    = \vec{E}_0 \Big( 1 + (i \vec{k} \cdot \vec{r}) + \frac{1}{2} (i \vec{k} \cdot \vec{r})^2 + \: ... \Big) \exp\!(- i \omega t)
+\end{aligned}$$
+
+Taking the real part then yields the following series of higher-order correction terms:
+
+$$\begin{aligned}
+    \vec{E}(\vec{r}, t)
+    = \vec{E}_0 \Big( \cos\!(\omega t) + (\vec{k} \cdot \vec{r}) \sin\!(\omega t) - \frac{1}{2} (\vec{k} \cdot \vec{r})^2 \cos\!(\omega t) + \: ... \Big)
+\end{aligned}$$
+
+
+
+## References
+1.  M. Fox,
+    *Optical properties of solids*, 2nd edition,
+    Oxford.
+2.  D.J. Griffiths, D.F. Schroeter,
+    *Introduction to quantum mechanics*, 3rd edition,
+    Cambridge.