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Diffstat (limited to 'source/know/concept/rotating-wave-approximation/index.md')
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1 files changed, 13 insertions, 19 deletions
diff --git a/source/know/concept/rotating-wave-approximation/index.md b/source/know/concept/rotating-wave-approximation/index.md index edb13e9..54e0675 100644 --- a/source/know/concept/rotating-wave-approximation/index.md +++ b/source/know/concept/rotating-wave-approximation/index.md @@ -16,7 +16,7 @@ in the [electric dipole approximation](/know/concept/electric-dipole-approximati $$\begin{aligned} \hat{H}_1(t) - = \hat{V} \cos(\omega t) + \equiv \hat{V} \cos(\omega t) = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) \end{aligned}$$ @@ -26,17 +26,17 @@ 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 $$\omega_0 = (E_e \!-\! E_g) / \hbar$$. +with a resonance frequency $$\omega_0 \equiv (E_e \!-\! E_g) / \hbar$$. From the [amplitude rate equations](/know/concept/amplitude-rate-equations/), we know that the general superposition state $$\ket{\Psi} = c_g \ket{g} + c_e \ket{e}$$ evolves as: $$\begin{aligned} i \hbar \dv{c_g}{t} - &= \matrixel{g}{\hat{H}_1(t)}{g} \: c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} \: c_e(t) \: e^{- i \omega_0 t} + &= \matrixel{g}{\hat{H}_1(t)}{g} c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} c_e(t) \: e^{- i \omega_0 t} \\ i \hbar \dv{c_e}{t} - &= \matrixel{e}{\hat{H}_1(t)}{g} \: c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} \: c_e(t) + &= \matrixel{e}{\hat{H}_1(t)}{g} c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} c_e(t) \end{aligned}$$ Typically, $$\hat{V}$$ has odd spatial parity, in which case @@ -66,15 +66,10 @@ $$\begin{aligned} At last, here we make the **rotating wave approximation**: since $$\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$$) ones. -Furthermore, since they average out to zero over most realistic time intervals, -the fast terms can be dropped, leaving: +we argue that $$\omega \!+\! \omega_0$$ is much larger than $$\omega \!-\! \omega_0$$, +so that those oscillations average out to zero +when the system is observed over a realistic time interval. +Hence we drop those terms: $$\begin{aligned} \boxed{ @@ -103,13 +98,12 @@ $$\begin{aligned} This approximation's name is a bit confusing: the idea is that going from the Schrödinger to the [interaction picture](/know/concept/interaction-picture/) -has the effect of removing the exponentials of $$\omega_0$$ from the above equations, -i.e. multiplying them by $$e^{i \omega_0 t}$$ and $$e^{- i \omega_0 t}$$ +involves removing the exponentials of $$\omega_0$$ from the above equations, +i.e. they are multiplied by $$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)$$ -into two exponentials, one co-rotates, and the other counter-rotates. -We keep only the co-rotating waves, hence the name. +When we split the wave $$\cos(\omega t)$$ into two exponentials, +one co-rotates relative to this rotation, and the other counter-rotates. +We keep only the co-rotating terms, hence the name. The rotating wave approximation is usually used in the context of the two-level quantum system for light-matter interactions, |