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-rw-r--r--source/know/concept/electric-dipole-approximation/index.md89
1 files changed, 52 insertions, 37 deletions
diff --git a/source/know/concept/electric-dipole-approximation/index.md b/source/know/concept/electric-dipole-approximation/index.md
index 35cf00c..06f0f45 100644
--- a/source/know/concept/electric-dipole-approximation/index.md
+++ b/source/know/concept/electric-dipole-approximation/index.md
@@ -13,20 +13,22 @@ 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:
+The general Hamiltonian of an electron in an electromagnetic field is:
$$\begin{aligned}
\hat{H}
- &= \frac{(\vu{P} - q \vb{A})^2}{2 m} + q \varphi
+ &= \frac{(\vu{P} - q \vb{A})^2}{2 m} + q \Phi
\\
- &= \frac{\vu{P}{}^2}{2 m} - \frac{q}{2 m} (\vb{A} \cdot \vu{P} + \vu{P} \cdot \vb{A}) + \frac{q^2 \vb{A}^2}{2m} + q \varphi
+ &= \frac{\vu{P}{}^2}{2 m} - \frac{q}{2 m} (\vb{A} \cdot \vu{P} + \vu{P} \cdot \vb{A}) + \frac{q^2 \vb{A}^2}{2m} + q \Phi
\end{aligned}$$
-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 $$\vb{A} \cdot \vu{P} = \vu{P} \cdot \vb{A}$$:
+Where $$q < 0$$ is the electron's charge,
+$$\vu{P} = - i \hbar \nabla$$ is the canonical momentum operator,
+$$\vb{A}$$ is the magnetic vector potential,
+and $$\Phi$$ is the electric scalar potential.
+We start by fixing the Coulomb gauge $$\nabla \cdot \vb{A} = 0$$
+such that $$\vu{P}$$ and $$\vb{A}$$ commute;
+let $$\psi$$ be an arbitrary test function:
$$\begin{aligned}
\comm{\vb{A}}{\vu{P}} \psi
@@ -35,20 +37,29 @@ $$\begin{aligned}
&= -i \hbar \vb{A} \cdot (\nabla \psi) + i \hbar \nabla \cdot (\vb{A} \psi)
\\
&= i \hbar (\nabla \cdot \vb{A}) \psi
- = 0
+ \\
+ &= 0
+\end{aligned}$$
+
+Meaning $$\vb{A} \cdot \vu{P} = \vu{P} \cdot \vb{A}$$.
+Furthermore, we assume that $$\vb{A}$$ is so small that $$\vb{A}{}^2$$ is negligible,
+so the Hamiltonian is reduced to:
+
+$$\begin{aligned}
+ \hat{H}
+ &\approx \frac{\vu{P}{}^2}{2 m} - \frac{q}{m} \vu{P} \cdot \vb{A} + q \Phi
\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$$:
+We now split $$\hat{H}$$ like so,
+where $$\hat{H}_1$$ can be regarded as a perturbation to the "base" $$\hat{H}_0$$:
$$\begin{aligned}
\hat{H}
= \hat{H}_0 + \hat{H}_1
- \qquad \quad
+ \qquad\qquad
\hat{H}_0
- \equiv \frac{\vu{P}{}^2}{2 m} + q \varphi
- \qquad \quad
+ \equiv \frac{\vu{P}{}^2}{2 m} + q \Phi
+ \qquad\qquad
\hat{H}_1
\equiv - \frac{q}{m} \vu{P} \cdot \vb{A}
\end{aligned}$$
@@ -56,14 +67,16 @@ $$\begin{aligned}
In an electromagnetic wave, $$\vb{A}$$ is oscillating sinusoidally in time and space:
$$\begin{aligned}
- \vb{A}(\vb{x}, t) = \vb{A}_0 \sin(\vb{k} \cdot \vb{x} - \omega t)
+ \vb{A}(\vb{x}, t)
+ = \vb{A}_0 \sin(\vb{k} \cdot \vb{x} - \omega t)
\end{aligned}$$
Mathematically, it is more convenient to represent this with a complex exponential,
whose real part should be taken at the end of the calculation:
$$\begin{aligned}
- \vb{A}(\vb{x}, t) = - i \vb{A}_0 \exp(i \vb{k} \cdot \vb{x} - i \omega t)
+ \vb{A}(\vb{x}, t)
+ = - i \vb{A}_0 \exp(i \vb{k} \cdot \vb{x} - i \omega t)
\end{aligned}$$
The corresponding perturbative [electric field](/know/concept/electric-field/) $$\vb{E}$$ is then given by:
@@ -75,10 +88,10 @@ $$\begin{aligned}
\end{aligned}$$
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 radii $$\sim 10^{-10} \:\mathrm{m}$$,
-so $$\vb{k} \cdot \vb{x}$$ is negligible:
+Light in and around the visible spectrum
+has a wavelength $$2 \pi / |\vb{k}| \sim 10^{-7} \:\mathrm{m}$$,
+while an atomic orbital is several Bohr radii $$\sim 10^{-10} \:\mathrm{m}$$,
+so $$\vb{k} \cdot \vb{x}$$ is very small. Therefore:
$$\begin{aligned}
\boxed{
@@ -96,14 +109,13 @@ and the electron quantum-mechanically.
Next, we want to rewrite $$\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}$$
+To do so, we use that momentum $$\vu{P} \equiv 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 \dv{\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}$$
Taking the off-diagonal inner product with
@@ -111,34 +123,37 @@ the two-level system's states $$\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}
+ &= 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$$ is the resonance of the energy gap,
+Where $$\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$$.
-We thus get:
+Therefore, $$\vu{P} / m = i \omega_0 \vu{x}$$, so we get:
$$\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)
+ &= - (- i i) q \omega_0 \vu{x} \cdot \vb{A}_0 \exp(- i \omega t)
+ \\
+ &\approx - \vu{d} \cdot \vb{E}_0 \exp(- i \omega t)
\end{aligned}$$
-Where $$\vu{d} \equiv q \vu{x} = - e \vu{x}$$ is
+Where $$\vu{d} \equiv q \vu{x}$$ is
the **transition dipole moment operator** of the electron,
-hence the name **electric dipole approximation**.
+hence the name *electric dipole approximation*.
Finally, we take the real part, yielding:
$$\begin{aligned}
\boxed{
- \hat{H}_1(t)
- = - \vu{d} \cdot \vb{E}(t)
- = - q \vu{x} \cdot \vb{E}_0 \cos(\omega t)
+ \begin{aligned}
+ \hat{H}_1(t)
+ &= - \vu{d} \cdot \vb{E}(t)
+ \\
+ &= - q \vu{x} \cdot \vb{E}_0 \cos(\omega t)
+ \end{aligned}
}
\end{aligned}$$