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Diffstat (limited to 'source/know/concept/einstein-coefficients')
| -rw-r--r-- | source/know/concept/einstein-coefficients/index.md | 13 |
1 files changed, 7 insertions, 6 deletions
diff --git a/source/know/concept/einstein-coefficients/index.md b/source/know/concept/einstein-coefficients/index.md index ad75ce5..2367abc 100644 --- a/source/know/concept/einstein-coefficients/index.md +++ b/source/know/concept/einstein-coefficients/index.md @@ -157,10 +157,10 @@ $$\begin{aligned} = - q E_0 z \cos(\omega t) \end{aligned}$$ -This form of $$\hat{H}_1$$ is a well-known case for -[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/), -which tells us that the transition probability -from $$\ket{a}$$ to $$\ket{b}$$ is (to first order): +From the derivation of +[Fermi's golden rule](/know/concept/fermis-golden-rule/), +we then know that the probability $$P_{ab}$$ of a transition +from $$\ket{a}$$ to $$\ket{b}$$ is as follows (to first order): $$\begin{aligned} P_{ab} @@ -210,6 +210,7 @@ However, the result would not be constant in time $$t$$, so is not really useful. + ## Polarized light To solve this "problem", we generalize to (incoherent) polarized polychromatic light. @@ -238,8 +239,7 @@ $$\begin{aligned} \int_0^\infty \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} u(\omega) \dd{\omega} \end{aligned}$$ -From here, the derivation is similar to that of -[Fermi's golden rule](/know/concept/fermis-golden-rule/), +From here, the derivation is similar to that of Fermi's golden rule, despite the distinction that we are integrating over frequencies rather than states. At sufficiently large $$t$$, the integrand is sharply peaked at $$\omega = \omega_0$$ @@ -276,6 +276,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Unpolarized light We can generalize the above result even further to unpolarized light. |