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Diffstat (limited to 'source/know/concept/rabi-oscillation/index.md')
| -rw-r--r-- | source/know/concept/rabi-oscillation/index.md | 21 |
1 files changed, 10 insertions, 11 deletions
diff --git a/source/know/concept/rabi-oscillation/index.md b/source/know/concept/rabi-oscillation/index.md index 2fcdea8..9497ad4 100644 --- a/source/know/concept/rabi-oscillation/index.md +++ b/source/know/concept/rabi-oscillation/index.md @@ -10,12 +10,12 @@ categories: layout: "concept" --- -In quantum mechanics, from the derivation of -[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/), -we know that a time-dependent term $$\hat{H}_1$$ in the Hamiltonian +In quantum mechanics, we know from the +[amplitude rate equations](/know/concept/amplitude-rate-equations/) +that a time-dependent term $$\hat{H}_1$$ in the Hamiltonian affects the state as follows, where $$c_n(t)$$ are the coefficients of the linear combination -of basis states $$\Ket{n} e^{-i E_n t / \hbar}$$: +of unperturbed basis states $$\ket{n} e^{-i E_n t / \hbar}$$: $$\begin{aligned} i \hbar \dv{c_m}{t} @@ -24,10 +24,8 @@ $$\begin{aligned} Where $$\omega_{mn} \equiv (E_m \!-\! E_n) / \hbar$$ for energies $$E_m$$ and $$E_n$$. -Note that this equation is exact, -despite being used for deriving perturbation theory. -Consider a two-level system where $$n \in \{a, b\}$$, -in which case the above equation can be expanded to the following: +Consider a two-level system $$\{\ket{a}, \ket{b}\}$$ with $$E_a < E_b$$, +in which case the above equation can be written out as: $$\begin{aligned} \dv{c_a}{t} @@ -169,10 +167,11 @@ $$\begin{aligned} \end{aligned}$$ Note that the period was halved by squaring. -This periodic "flopping" of the particle between $$\Ket{a}$$ and $$\Ket{b}$$ +This periodic "flopping" of the particle between $$\ket{a}$$ and $$\ket{b}$$ is known as **Rabi oscillation**, **Rabi flopping** or the **Rabi cycle**. This is a more accurate treatment -of the flopping found from first-order perturbation theory in textbooks. +of the flopping found from first-order +[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/). The name **generalized Rabi frequency** suggests that there is a non-general version. @@ -205,7 +204,7 @@ $$\begin{aligned} Where $$\vec{E}_0$$ is the [electric field](/know/concept/electric-field/) amplitude, and $$\vec{d} \equiv q \matrixel{b}{\vec{r}}{a}$$ is the transition dipole moment -of the electron between orbitals $$\Ket{a}$$ and $$\Ket{b}$$. +of the electron between orbitals $$\ket{a}$$ and $$\ket{b}$$. Apparently, some authors define $$\vec{d}$$ with the opposite sign, thereby departing from its classical interpretation. |