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+---
+title: "Electric dipole approximation"
+date: 2021-09-14
+categories:
+- Physics
+- Quantum mechanics
+- Optics
+- Electromagnetism
+- Perturbation
+layout: "concept"
+---
+
+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:
+
+$$\begin{aligned}
+    \hat{H}
+    &= \frac{(\vu{P} - q \vb{A})^2}{2 m} + q \varphi
+    \\
+    &= \frac{\vu{P}{}^2}{2 m} - \frac{q}{2 m} (\vb{A} \cdot \vu{P} + \vu{P} \cdot \vb{A}) + \frac{q^2 \vb{A}^2}{2m} + q \varphi
+\end{aligned}$$
+
+With charge $q = - e$,
+canonical momentum operator $\vu{P} = - i \hbar \nabla$,
+and magnetic vector potential $\vb{A}(\vb{x}, t)$.
+We reduce this by fixing the Coulomb gauge $\nabla \cdot \vb{A} = 0$,
+so that $\vb{A} \cdot \vu{P} = \vu{P} \cdot \vb{A}$:
+
+$$\begin{aligned}
+    \comm{\vb{A}}{\vu{P}} \psi
+    &= -i \hbar \vb{A} \cdot (\nabla \psi) + i \hbar \nabla \cdot (\vb{A} \psi)
+    \\
+    &= i \hbar (\nabla \cdot \vb{A}) \psi
+    = 0
+\end{aligned}$$
+
+Where $\psi$ is an arbitrary test function.
+Assuming $\vb{A}$ is so small that $\vb{A}{}^2$ is negligible, we split $\hat{H}$ as follows,
+where $\hat{H}_1$ can be regarded as a perturbation to $\hat{H}_0$:
+
+$$\begin{aligned}
+    \hat{H}
+    = \hat{H}_0 + \hat{H}_1
+    \qquad \quad
+    \hat{H}_0
+    \equiv \frac{\vu{P}{}^2}{2 m} + q \varphi
+    \qquad \quad
+    \hat{H}_1
+    \equiv - \frac{q}{m} \vu{P} \cdot \vb{A}
+\end{aligned}$$
+
+In an electromagnetic wave, $\vb{A}$ is oscillating sinusoidally in time and space:
+
+$$\begin{aligned}
+    \vb{A}(\vb{x}, t) = \vb{A}_0 \sin(\vb{k} \cdot \vb{x} - \omega t)
+\end{aligned}$$
+
+Mathematically, it is more convenient to represent this with a complex exponential,
+whose real part should be taken at the end of the calculation:
+
+$$\begin{aligned}
+    \vb{A}(\vb{x}, t) = - i \vb{A}_0 \exp(i \vb{k} \cdot \vb{x} - i \omega t)
+\end{aligned}$$
+
+The corresponding perturbative [electric field](/know/concept/electric-field/) $\vb{E}$ is then given by:
+
+$$\begin{aligned}
+    \vb{E}(\vb{x}, t)
+    = - \pdv{\vb{A}}{t}
+    = \vb{E}_0 \exp(i \vb{k} \cdot \vb{x} - i \omega t)
+\end{aligned}$$
+
+Where $\vb{E}_0 = \omega \vb{A}_0$.
+Let us restrict ourselves to visible light,
+whose wavelength $2 \pi / |\vb{k}| \sim 10^{-6} \:\mathrm{m}$.
+Meanwhile, an atomic orbital is several Bohr $\sim 10^{-10} \:\mathrm{m}$,
+so $\vb{k} \cdot \vb{x}$ is negligible:
+
+$$\begin{aligned}
+    \boxed{
+        \vb{E}(\vb{x}, t)
+        \approx \vb{E}_0 \exp(- i \omega t)
+    }
+\end{aligned}$$
+
+This is the **electric dipole approximation**:
+we ignore all spatial variation of $\vb{E}$,
+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 rewrite $\hat{H}_1$
+to use the electric field $\vb{E}$ instead of the potential $\vb{A}$.
+To do so, we use that $\vu{P} = m \: \idv{\vu{x}}{t}$
+and evaluate this in the [interaction picture](/know/concept/interaction-picture/):
+
+$$\begin{aligned}
+    \vu{P}
+    = m \idv{\vu{x}}{t}
+    = m \frac{i}{\hbar} \comm{\hat{H}_0}{\vu{x}}
+    = m \frac{i}{\hbar} (\hat{H}_0 \vu{x} - \vu{x} \hat{H}_0)
+\end{aligned}$$
+
+Taking the off-diagonal inner product with
+the two-level system's states $\Ket{1}$ and $\Ket{2}$ gives:
+
+$$\begin{aligned}
+    \matrixel{2}{\vu{P}}{1}
+    = m \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vu{x} - \vu{x} \hat{H}_0}{1}
+    = m i \omega_0 \matrixel{2}{\vu{x}}{1}
+\end{aligned}$$
+
+Therefore, $\vu{P} / m = i \omega_0 \vu{x}$,
+where $\omega_0 \equiv (E_2 \!-\! E_1) / \hbar$ is the resonance of the energy gap,
+close to which we assume that $\vb{A}$ and $\vb{E}$ are oscillating, i.e. $\omega \approx \omega_0$.
+We thus get:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    &= - \frac{q}{m} \vu{P} \cdot \vb{A}
+    = - (- i i) q \omega_0 \vu{x} \cdot \vb{A}_0 \exp(- i \omega t)
+    \\
+    &\approx - q \vu{x} \cdot \vb{E}_0 \exp(- i \omega t)
+    = - \vu{d} \cdot \vb{E}_0 \exp(- i \omega t)
+\end{aligned}$$
+
+Where $\vu{d} \equiv q \vu{x} = - e \vu{x}$ is
+the **transition dipole moment operator** of the electron,
+hence the name **electric dipole approximation**.
+Finally, we take the real part, yielding:
+
+$$\begin{aligned}
+    \boxed{
+        \hat{H}_1(t)
+        = - \vu{d} \cdot \vb{E}(t)
+        = - q \vu{x} \cdot \vb{E}_0 \cos(\omega t)
+    }
+\end{aligned}$$
+
+If this approximation is too rough,
+$\vb{E}$ can always be Taylor-expanded in $(i \vb{k} \cdot \vb{x})$:
+
+$$\begin{aligned}
+    \vb{E}(\vb{x}, t)
+    = \vb{E}_0 \Big( 1 + (i \vb{k} \cdot \vb{x}) + \frac{1}{2} (i \vb{k} \cdot \vb{x})^2 + \: ... \Big) \exp(- i \omega t)
+\end{aligned}$$
+
+Taking the real part then yields the following series of higher-order correction terms:
+
+$$\begin{aligned}
+    \vb{E}(\vb{x}, t)
+    = \vb{E}_0 \Big( \cos(\omega t) + (\vb{k} \cdot \vb{x}) \sin(\omega t) - \frac{1}{2} (\vb{k} \cdot \vb{x})^2 \cos(\omega t) + \: ... \Big)
+\end{aligned}$$
+
+
+
+## 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.