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author | Prefetch | 2022-02-01 11:23:35 +0100 |
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committer | Prefetch | 2022-02-01 11:23:35 +0100 |
commit | 43c5b696aaf421dec7aee967002999d9145da35e (patch) | |
tree | 418dd75bb385a8922b9484883ae0d42124239c94 | |
parent | 88537b82784f71104c3a2771330c9e492f57fb03 (diff) |
Expand knowledge base
-rw-r--r-- | content/know/category/plasma-waves.md | 9 | ||||
-rw-r--r-- | content/know/category/two-level-system.md | 9 | ||||
-rw-r--r-- | content/know/concept/alfven-waves/index.pdc | 249 | ||||
-rw-r--r-- | content/know/concept/bloch-sphere/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/einstein-coefficients/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/fermis-golden-rule/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/interaction-picture/index.pdc | 3 | ||||
-rw-r--r-- | content/know/concept/ion-sound-wave/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/langmuir-waves/index.pdc | 1 | ||||
-rw-r--r-- | content/know/concept/maxwell-bloch-equations/index.pdc | 5 | ||||
-rw-r--r-- | content/know/concept/multi-photon-absorption/index.pdc | 3 | ||||
-rw-r--r-- | content/know/concept/rabi-oscillation/index.pdc | 15 | ||||
-rw-r--r-- | content/know/concept/rotating-wave-approximation/index.pdc | 126 |
13 files changed, 411 insertions, 13 deletions
diff --git a/content/know/category/plasma-waves.md b/content/know/category/plasma-waves.md new file mode 100644 index 0000000..6020a4c --- /dev/null +++ b/content/know/category/plasma-waves.md @@ -0,0 +1,9 @@ +--- +title: "Plasma waves" +firstLetter: "P" +date: 2022-01-31T22:45:52+01:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/category/two-level-system.md b/content/know/category/two-level-system.md new file mode 100644 index 0000000..aae4d0e --- /dev/null +++ b/content/know/category/two-level-system.md @@ -0,0 +1,9 @@ +--- +title: "Two-level system" +firstLetter: "T" +date: 2022-01-31T22:45:58+01:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/concept/alfven-waves/index.pdc b/content/know/concept/alfven-waves/index.pdc new file mode 100644 index 0000000..ba87bee --- /dev/null +++ b/content/know/concept/alfven-waves/index.pdc @@ -0,0 +1,249 @@ +--- +title: "Alfvén waves" +firstLetter: "A" +publishDate: 2022-01-31 +categories: +- Physics +- Plasma physics +- Plasma waves + +date: 2022-01-30T19:26:33+01:00 +draft: false +markup: pandoc +--- + +# Alfvén waves + +In the [magnetohydrodynamic](/know/concept/magnetohydrodynamics/) description of a plasma, +we split the velocity $\vb{u}$, electric current $\vb{J}$, +[magnetic field](/know/concept/magnetic-field/) $\vb{B}$ +and [electric field](/know/concept/electric-field/) $\vb{E}$ like so, +into a constant uniform equilibrium (subscript $0$) +and a small unknown perturbation (subscript $1$): + +$$\begin{aligned} + \vb{u} + = \vb{u}_0 + \vb{u}_1 + \qquad + \vb{J} + = \vb{J}_0 + \vb{J}_1 + \qquad + \vb{B} + = \vb{B}_0 + \vb{B}_1 + \qquad + \vb{E} + = \vb{E}_0 + \vb{E}_1 +\end{aligned}$$ + +Inserting this decomposition into the ideal form of the generalized Ohm's law +and keeping only terms that are first-order in the perturbation, we get: + +$$\begin{aligned} + 0 + &= (\vb{E}_0 + \vb{E}_1) + (\vb{u}_0 + \vb{u}_1) \cross (\vb{B}_0 + \vb{B}_1) + \\ + &= \vb{E}_1 + \vb{u}_1 \cross \vb{B}_0 +\end{aligned}$$ + +We do this for the momentum equation too, +assuming that $\vb{J}_0 \!=\! 0$ (to be justified later). +Note that the temperature is set to zero, such that the pressure vanishes: + +$$\begin{aligned} + \rho \pdv{\vb{u}_1}{t} + = \vb{J}_1 \cross \vb{B}_0 +\end{aligned}$$ + +Where $\rho$ is the uniform equilibrium density. +We would like an equation for $\vb{J}_1$, +which is provided by the magnetohydrodynamic form of Ampère's law: + +$$\begin{aligned} + \nabla \cross \vb{B}_1 + = \mu_0 \vb{J}_1 + \qquad \implies \quad + \vb{J}_1 + = \frac{1}{\mu_0} \nabla \cross \vb{B}_1 +\end{aligned}$$ + +Substituting this into the momentum equation, +and differentiating with respect to $t$: + +$$\begin{aligned} + \rho \pdv[2]{\vb{u}_1}{t} + = \frac{1}{\mu_0} \bigg( \Big( \nabla \cross \pdv{\vb{B}_1}{t} \Big) \cross \vb{B}_0 \bigg) +\end{aligned}$$ + +For which we can use Faraday's law to rewrite $\pdv*{\vb{B}_1}{t}$, +incorporating Ohm's law too: + +$$\begin{aligned} + \pdv{\vb{B}_1}{t} + = - \nabla \cross \vb{E}_1 + = \nabla \cross (\vb{u}_1 \cross \vb{B}_0) +\end{aligned}$$ + +Inserting this into the momentum equation for $\vb{u}_1$ +thus yields its final form: + +$$\begin{aligned} + \rho \pdv[2]{\vb{u}_1}{t} + = \frac{1}{\mu_0} \bigg( \Big( \nabla \cross \big( \nabla \cross (\vb{u}_1 \cross \vb{B}_0) \big) \Big) \cross \vb{B}_0 \bigg) +\end{aligned}$$ + +Suppose the magnetic field is pointing in $z$-direction, +i.e. $\vb{B}_0 = B_0 \vu{e}_z$. +Then Faraday's law justifies our earlier assumption that $\vb{J}_0 = 0$, +and the equation can be written as: + +$$\begin{aligned} + \pdv[2]{\vb{u}_1}{t} + = v_A^2 \bigg( \Big( \nabla \cross \big( \nabla \cross (\vb{u}_1 \cross \vu{e}_z) \big) \Big) \cross \vu{e}_z \bigg) +\end{aligned}$$ + +Where we have defined the so-called **Alfvén velocity** $v_A$ to be given by: + +$$\begin{aligned} + \boxed{ + v_A + \equiv \sqrt{\frac{B_0^2}{\mu_0 \rho}} + } +\end{aligned}$$ + +Now, consider the following plane-wave ansatz for $\vb{u}_1$, +with wavevector $\vb{k}$ and frequency $\omega$: + +$$\begin{aligned} + \vb{u}_1(\vb{r}, t) + &= \vb{u}_1 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) +\end{aligned}$$ + +Inserting this into the above differential equation for $\vb{u}_1$ leads to: + +$$\begin{aligned} + \omega^2 \vb{u}_1 + = v_A^2 \bigg( \Big( \vb{k} \cross \big( \vb{k} \cross (\vb{u}_1 \cross \vu{e}_z) \big) \Big) \cross \vu{e}_z \bigg) +\end{aligned}$$ + +To evaluate this, we rotate our coordinate system around the $z$-axis +such that $\vb{k} = (0, k_\perp, k_\parallel)$, +i.e. the wavevector's $x$-component is zero. +Calculating the cross products: + +$$\begin{aligned} + \omega^2 \vb{u}_1 + &= v_A^2 \bigg( \Big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix} + \cross \big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix} + \cross ( \begin{bmatrix} u_{1x} \\ u_{1y} \\ u_{1z} \end{bmatrix} + \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} ) \big) \Big) + \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \bigg) + \\ + &= v_A^2 \bigg( \Big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix} + \cross \big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix} + \cross \begin{bmatrix} u_{1y} \\ -u_{1x} \\ 0 \end{bmatrix} \big) \Big) + \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \bigg) + \\ + &= v_A^2 \bigg( \Big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix} + \cross \begin{bmatrix} k_\parallel u_{1x} \\ k_\parallel u_{1y} \\ -k_\perp u_{1y} \end{bmatrix} \Big) + \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \bigg) + \\ + &= v_A^2 \bigg( \begin{bmatrix} -(k_\perp^2 \!+ k_\parallel^2) u_{1y} \\ k_\parallel^2 u_{1x} \\ -k_\perp k_\parallel u_{1x} \end{bmatrix} + \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \bigg) + \\ + &= v_A^2 \begin{bmatrix} k_\parallel^2 u_{1x} \\ (k_\perp^2 \!+ k_\parallel^2) u_{1y} \\ 0 \end{bmatrix} +\end{aligned}$$ + +We rewrite this equation in matrix form, +using that $k_\perp^2 \!+ k_\parallel^2 = k^2 \equiv |\vb{k}|^2$: + +$$\begin{aligned} + \begin{bmatrix} + \omega^2 - v_A^2 k_\parallel^2 & 0 & 0 \\ + 0 & \omega^2 - v_A^2 k^2 & 0 \\ + 0 & 0 & \omega^2 + \end{bmatrix} + \vb{u}_1 + = 0 +\end{aligned}$$ + +This has the form of an eigenvalue problem for $\omega^2$, +meaning we must find non-trivial solutions, +where we cannot simply choose the components of $\vb{u}_1$ to satisfy the equation. +To achieve this, we demand that the matrix' determinant is zero: + +$$\begin{aligned} + \big(\omega^2 - v_A^2 k_\parallel^2\big) \: \big(\omega^2 - v_A^2 k^2\big) \: \omega^2 + = 0 +\end{aligned}$$ + +This equation has three solutions for $\omega^2$, +one for each of its three factors being zero. +The simplest case $\omega^2 = 0$ is of no interest to us, +because we are looking for waves. + +The first interesting case is $\omega^2 = v_A^2 k_\parallel^2$, +yielding the following dispersion relation: + +$$\begin{aligned} + \boxed{ + \omega + = \pm v_A k_\parallel + } +\end{aligned}$$ + +The resulting waves are called **shear Alfvén waves**. +From the eigenvalue problem, we see that in this case +$\vb{u}_1 = (u_{1x}, 0, 0)$, meaning $\vb{u}_1 \cdot \vb{k} = 0$: +these waves are **transverse**. +The phase velocity $v_p$ and group velocity $v_g$ are as follows, +where $\theta$ is the angle between $\vb{k}$ and $\vb{B}_0$: + +$$\begin{aligned} + v_p + = \frac{|\omega|}{k} + = v_A \frac{k_\parallel}{k} + = v_A \cos\!(\theta) + \qquad \qquad + v_g + = \pdv{|\omega|}{k} + = v_A +\end{aligned}$$ + +The other interesting case is $\omega^2 = v_A^2 k^2$, +which leads to so-called **compressional Alfvén waves**, +with the simple dispersion relation: + +$$\begin{aligned} + \boxed{ + \omega + = \pm v_A k + } +\end{aligned}$$ + +Looking at the eigenvalue problem reveals that $\vb{u}_1 = (0, u_{1y}, 0)$, +meaning $\vb{u}_1 \cdot \vb{k} = u_{1y} k_\perp$, +so these waves are not necessarily transverse, nor longitudinal (since $k_\parallel$ is free). +The phase velocity $v_p$ and group velocity $v_g$ are given by: + +$$\begin{aligned} + v_p + = \frac{|\omega|}{k} + = v_A + \qquad \qquad + v_g + = \pdv{|\omega|}{k} + = v_A +\end{aligned}$$ + +The mechanism behind both of these oscillations is magnetic tension: +the waves are "ripples" in the field lines, +which get straightened out by Faraday's law, +but the ions' inertia causes them to overshoot and form ripples again. + + + +## References +1. M. Salewski, A.H. Nielsen, + *Plasma physics: lecture notes*, + 2021, unpublished. + diff --git a/content/know/concept/bloch-sphere/index.pdc b/content/know/concept/bloch-sphere/index.pdc index f0c48a9..27abb54 100644 --- a/content/know/concept/bloch-sphere/index.pdc +++ b/content/know/concept/bloch-sphere/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-03-09 categories: - Quantum mechanics - Quantum information +- Two-level system date: 2021-03-09T15:35:33+01:00 draft: false diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc index f0f0f96..b56af77 100644 --- a/content/know/concept/einstein-coefficients/index.pdc +++ b/content/know/concept/einstein-coefficients/index.pdc @@ -7,6 +7,7 @@ categories: - Optics - Electromagnetism - Quantum mechanics +- Two-level system date: 2021-07-11T18:22:14+02:00 draft: false diff --git a/content/know/concept/fermis-golden-rule/index.pdc b/content/know/concept/fermis-golden-rule/index.pdc index 5ed273e..6fcc482 100644 --- a/content/know/concept/fermis-golden-rule/index.pdc +++ b/content/know/concept/fermis-golden-rule/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-07-10 categories: - Physics - Quantum mechanics +- Two-level system - Optics date: 2021-07-03T14:41:11+02:00 diff --git a/content/know/concept/interaction-picture/index.pdc b/content/know/concept/interaction-picture/index.pdc index 45950ff..89aff58 100644 --- a/content/know/concept/interaction-picture/index.pdc +++ b/content/know/concept/interaction-picture/index.pdc @@ -197,7 +197,8 @@ $$\begin{aligned} \mathcal{T} \bigg\{ \bigg( \int_{t_0}^{t} \hat{H}_{1,I}(t') \dd{t'} \bigg)^n \bigg\} \end{aligned}$$ -Here, we recognize the Taylor expansion of $\exp$, +This construction is occasionally called the **Dyson series**. +We recognize the well-known Taylor expansion of $\exp\!(x)$, leading us to a final expression for $\hat{K}_I$: $$\begin{aligned} diff --git a/content/know/concept/ion-sound-wave/index.pdc b/content/know/concept/ion-sound-wave/index.pdc index 657627d..38ab394 100644 --- a/content/know/concept/ion-sound-wave/index.pdc +++ b/content/know/concept/ion-sound-wave/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-10-31 categories: - Physics - Plasma physics +- Plasma waves - Perturbation date: 2021-10-31T09:38:14+01:00 diff --git a/content/know/concept/langmuir-waves/index.pdc b/content/know/concept/langmuir-waves/index.pdc index c5cd23e..caf2294 100644 --- a/content/know/concept/langmuir-waves/index.pdc +++ b/content/know/concept/langmuir-waves/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-10-30 categories: - Physics - Plasma physics +- Plasma waves - Perturbation date: 2021-10-15T20:31:46+02:00 diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc index 3f090a2..e3a3680 100644 --- a/content/know/concept/maxwell-bloch-equations/index.pdc +++ b/content/know/concept/maxwell-bloch-equations/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-10-02 categories: - Physics - Quantum mechanics +- Two-level system - Electromagnetism date: 2021-09-09T21:17:52+02:00 @@ -78,8 +79,8 @@ $$\begin{aligned} \end{aligned}$$ With these, the equations for $c_g$ and $c_e$ can be rewritten as shown below. -Note that $\vb{E}^{-}$ and $\vb{E}^{+}$ include the driving plane wave, -and the *rotating wave approximation* is still made: +Note that $\vb{E}^{-}$ and $\vb{E}^{+}$ include the driving plane wave, and the +[rotating wave approximation](/know/concept/rotating-wave-approximation/) is still made: $$\begin{aligned} \dv{c_g}{t} diff --git a/content/know/concept/multi-photon-absorption/index.pdc b/content/know/concept/multi-photon-absorption/index.pdc index cfdd234..b208cfe 100644 --- a/content/know/concept/multi-photon-absorption/index.pdc +++ b/content/know/concept/multi-photon-absorption/index.pdc @@ -29,7 +29,8 @@ $$\begin{aligned} Where $\vb{E}$ is the [electric field](/know/concept/electric-field/) amplitude, and $\vu{p} \equiv q \vu{x}$ is the transition dipole moment operator. -Here, we have made the *rotating wave approximation* +Here, we have made the +[rotating wave approximation](/know/concept/rotating-wave-approximation/) to neglect the $e^{i \omega t}$ term, because it turns out to be irrelevant in this discussion. diff --git a/content/know/concept/rabi-oscillation/index.pdc b/content/know/concept/rabi-oscillation/index.pdc index a488de0..c6a1227 100644 --- a/content/know/concept/rabi-oscillation/index.pdc +++ b/content/know/concept/rabi-oscillation/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-09-22 categories: - Physics - Quantum mechanics +- Two-level system - Optics date: 2021-09-18T00:41:43+02:00 @@ -74,17 +75,13 @@ $$\begin{aligned} &= - i \frac{V_{ab}}{2 \hbar} \Big( \exp\!\big(i (\omega \!+\! \omega_0) t\big) + \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t\big) \Big) \: c_a \end{aligned}$$ -Here, we make the *rotating wave approximation*: +Here, we make the +[rotating wave approximation](/know/concept/rotating-wave-approximation/): assuming we are close to resonance $\omega \approx \omega_0$, -we decide that $\exp\!(i (\omega \!+\! \omega_0) t)$ -oscillates so much faster than $\exp\!(i (\omega \!-\! \omega_0) t)$, -that its effect turns out negligible +we argue that $\exp\!(i (\omega \!+\! \omega_0) t)$ +oscillates so fast that its effect is negligible when the system is observed over a reasonable time interval. - -In other words, over this reasonably-sized time interval, -$\exp\!(i (\omega \!+\! \omega_0) t)$ averages to zero, -while $\exp\!(i (\omega \!-\! \omega_0) t)$ does not. -Dropping the respective terms thus leaves us with: +Dropping those terms leaves us with: $$\begin{aligned} \boxed{ diff --git a/content/know/concept/rotating-wave-approximation/index.pdc b/content/know/concept/rotating-wave-approximation/index.pdc new file mode 100644 index 0000000..874dc96 --- /dev/null +++ b/content/know/concept/rotating-wave-approximation/index.pdc @@ -0,0 +1,126 @@ +--- +title: "Rotating wave approximation" +firstLetter: "R" +publishDate: 2022-02-01 +categories: +- Physics +- Quantum mechanics +- Two-level system +- Optics + +date: 2022-01-31T19:29:43+01:00 +draft: false +markup: pandoc +--- + +# Rotating wave approximation + +Consider the following periodic perturbation $\hat{H}_1$ to a quantum system, +which represents e.g. an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) +in the [electric dipole approximation](/know/concept/electric-dipole-approximation/): + +$$\begin{aligned} + \hat{H}_1(t) + = \hat{V} \cos\!(\omega t) + = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) +\end{aligned}$$ + +Where $\hat{V}$ is some operator, and we assume that $\omega$ +is fairly close to a resonance frequency $\omega_0$ +of the system that is getting perturbed by $\hat{H}_1$. + +As an example, consider a two-level system +consisting of states $\ket{g}$ and $\ket{e}$, +with a resonance frequency $\omega_0 = (E_e \!-\! E_g) / \hbar$. +From the derivation of +[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/), +we know that the state $\ket{\Psi} = c_g \ket{g} + c_e \ket{e}$ evolves as: + +$$\begin{aligned} + i \hbar \dv{c_g}{t} + &= \matrixel{g}{\hat{H}_1(t)}{g} \: c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} \: c_e(t) \: e^{- i \omega_0 t} + \\ + i \hbar \dv{c_e}{t} + &= \matrixel{e}{\hat{H}_1(t)}{g} \: c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} \: c_e(t) +\end{aligned}$$ + +Typically, $\hat{V}$ has odd spatial parity, in which case +[Laporte's selection rule](/know/concept/selection-rules/) +reduces this to: + +$$\begin{aligned} + \dv{c_g}{t} + &= \frac{1}{i \hbar} \matrixel{g}{\hat{H}_1}{e} \: c_e \: e^{- i \omega_0 t} + \\ + \dv{c_e}{t} + &= \frac{1}{i \hbar} \matrixel{e}{\hat{H}_1}{g} \: c_g \: e^{i \omega_0 t} +\end{aligned}$$ + +We now insert the general $\hat{H}_1$ defined above, +and define $V_{eg} \equiv \matrixel{e}{\hat{V}}{g}$ to get: + +$$\begin{aligned} + \dv{c_g}{t} + &= \frac{V_{eg}^*}{i 2 \hbar} + \Big( e^{i (\omega - \omega_0) t} + e^{- i (\omega + \omega_0) t} \Big) \: c_e + \\ + \dv{c_e}{t} + &= \frac{V_{eg}}{i 2 \hbar} + \Big( e^{i (\omega + \omega_0) t} + e^{- i (\omega - \omega_0) t} \Big) \: c_g +\end{aligned}$$ + +At last, here we make the **rotating wave approximation**: +since $\omega$ is assumed to be close to $\omega_0$, +we argue that $\omega \!+\! \omega_0$ is so much larger than $\omega \!-\! \omega_0$ +that those oscillations turn out negligible +if the system is observed over a reasonable time interval. + +Specifically, since both exponentials have the same weight, +the fast ($\omega \!+\! \omega_0$) oscillations +have a tiny amplitude compared to the slow ($\omega \!-\! \omega_0$) ones. +Furthermore, since they average out to zero over most realistic time intervals, +the fast terms can be dropped, leaving: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + e^{i (\omega - \omega_0) t} + e^{- i (\omega + \omega_0) t} + &\approx e^{i (\omega - \omega_0) t} + \\ + e^{i (\omega + \omega_0) t} + e^{- i (\omega - \omega_0) t} + &\approx e^{- i (\omega - \omega_0) t} + \end{aligned} + } +\end{aligned}$$ + +Such that our example set of equations can be approximated as shown below, +and its analysis can continue; +see [Rabi oscillation](/know/concept/rabi-oscillation/) for more: + +$$\begin{aligned} + \dv{c_g}{t} + &= \frac{V_{eg}^*}{i 2 \hbar} c_e \: e^{i (\omega - \omega_0) t} + \\ + \dv{c_e}{t} + &= \frac{V_{eg}}{i 2 \hbar} c_g \: e^{- i (\omega - \omega_0) t} +\end{aligned}$$ + +This approximation's name is a bit confusing: +the idea is that going from the Schrödinger to +the [interaction picture](/know/concept/interaction-picture/) +has the effect of removing the exponentials of $\omega_0$ from the above equations, +i.e. multiplying them by $e^{i \omega_0 t}$ and $e^{- i \omega_0 t}$ +respectively, which can be regarded as a rotation. + +Relative to this rotation, when we split the wave $\cos\!(\omega t)$ +into two exponentials, one co-rotates, and the other counter-rotates. +We keep only the co-rotating waves, hence the name. + +The rotating wave approximation is usually used in the context +of the two-level quantum system for light-matter interactions, +as in the above example. +However, it is not specific to that case, +and it more generally refers to any approximation +where fast-oscillating terms are neglected. + + |