From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- .../concept/electric-dipole-approximation/index.md | 165 +++++++++++++++++++++ 1 file changed, 165 insertions(+) create mode 100644 source/know/concept/electric-dipole-approximation/index.md (limited to 'source/know/concept/electric-dipole-approximation') diff --git a/source/know/concept/electric-dipole-approximation/index.md b/source/know/concept/electric-dipole-approximation/index.md new file mode 100644 index 0000000..c3e6dc0 --- /dev/null +++ b/source/know/concept/electric-dipole-approximation/index.md @@ -0,0 +1,165 @@ +--- +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. -- cgit v1.2.3