---
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.