diff options
Diffstat (limited to 'source/know/concept/kubo-formula/index.md')
| -rw-r--r-- | source/know/concept/kubo-formula/index.md | 56 |
1 files changed, 28 insertions, 28 deletions
diff --git a/source/know/concept/kubo-formula/index.md b/source/know/concept/kubo-formula/index.md index 80309c5..4cb39ac 100644 --- a/source/know/concept/kubo-formula/index.md +++ b/source/know/concept/kubo-formula/index.md @@ -10,19 +10,19 @@ layout: "concept" --- Consider the following quantum Hamiltonian, -split into a main time-independent term $\hat{H}_{0,S}$ -and a small time-dependent perturbation $\hat{H}_{1,S}$, -which is turned on at $t = t_0$: +split into a main time-independent term $$\hat{H}_{0,S}$$ +and a small time-dependent perturbation $$\hat{H}_{1,S}$$, +which is turned on at $$t = t_0$$: $$\begin{aligned} \hat{H}_S(t) = \hat{H}_{0,S} + \hat{H}_{1,S}(t) \end{aligned}$$ -And let $\Ket{\psi_S(t)}$ be the corresponding solutions to the Schrödinger equation. -Then, given a time-independent observable $\hat{A}$, -its expectation value $\expval{\hat{A}}$ evolves like so, -where the subscripts $S$ and $I$ +And let $$\Ket{\psi_S(t)}$$ be the corresponding solutions to the Schrödinger equation. +Then, given a time-independent observable $$\hat{A}$$, +its expectation value $$\expval{\hat{A}}$$ evolves like so, +where the subscripts $$S$$ and $$I$$ respectively refer to the Schrödinger and [interaction pictures](/know/concept/interaction-picture/): @@ -34,7 +34,7 @@ $$\begin{aligned} &= \matrixel{\psi_I(t_0)\,}{\,\hat{K}_I^\dagger(t, t_0) \hat{A}_I(t) \hat{K}_I(t, t_0)\,}{\,\psi_I(t_0)} \end{aligned}$$ -Where the time evolution operator $\hat{K}_I(t, t_0)$ is as follows, +Where the time evolution operator $$\hat{K}_I(t, t_0)$$ is as follows, which we Taylor-expand: $$\begin{aligned} @@ -56,7 +56,7 @@ $$\begin{aligned} \end{aligned}$$ Where we have dropped the last term, -because $\hat{H}_{1}$ is assumed to be so small +because $$\hat{H}_{1}$$ is assumed to be so small that it only matters to first order. Here, we notice a commutator, so we can rewrite: @@ -65,10 +65,10 @@ $$\begin{aligned} &= \hat{A}_I(t) - \frac{i}{\hbar} \int_{t_0}^t \Comm{\hat{A}_I(t)}{\hat{H}_{1,I}(t')} \dd{t'} \end{aligned}$$ -Returning to $\expval{\hat{A}}$, +Returning to $$\expval{\hat{A}}$$, we have the following formula, -where $\Expval{}$ is the expectation value for $\Ket{\psi(t)}$, -and $\Expval{}_0$ is the expectation value for $\Ket{\psi_I(t_0)}$: +where $$\Expval{}$$ is the expectation value for $$\Ket{\psi(t)}$$, +and $$\Expval{}_0$$ is the expectation value for $$\Ket{\psi_I(t_0)}$$: $$\begin{aligned} \expval{\hat{A}}(t) @@ -76,9 +76,9 @@ $$\begin{aligned} = \expval{\hat{A}_I(t)}_0 - \frac{i}{\hbar} \int_{t_0}^t \Expval{\Comm{\hat{A}_I(t)}{\hat{H}_{1,I}(t')}}_0 \dd{t'} \end{aligned}$$ -Now we define $\delta\!\expval{\hat{A}}\!(t)$ -as the change of $\expval{\hat{A}}$ due to the perturbation $\hat{H}_1$, -and insert $\expval{\hat{A}}(t)$: +Now we define $$\delta\!\expval{\hat{A}}\!(t)$$ +as the change of $$\expval{\hat{A}}$$ due to the perturbation $$\hat{H}_1$$, +and insert $$\expval{\hat{A}}(t)$$: $$\begin{aligned} \delta\!\expval{\hat{A}}\!(t) @@ -87,10 +87,10 @@ $$\begin{aligned} \end{aligned}$$ Finally, we introduce -a [Heaviside step function](/know/concept/heaviside-step-function) $\Theta$ +a [Heaviside step function](/know/concept/heaviside-step-function) $$\Theta$$ and change the integration limit accordingly, leading to the **Kubo formula** -describing the response of $\expval{\hat{A}}$ to first order in $\hat{H}_1$: +describing the response of $$\expval{\hat{A}}$$ to first order in $$\hat{H}_1$$: $$\begin{aligned} \boxed{ @@ -99,7 +99,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where we have defined the **retarded correlation function** $C^R_{A H_1}(t, t')$ as follows: +Where we have defined the **retarded correlation function** $$C^R_{A H_1}(t, t')$$ as follows: $$\begin{aligned} \boxed{ @@ -115,10 +115,10 @@ of particle creation/annihiliation operators. Therefore, this correlation function is a two-particle [Green's function](/know/concept/greens-functions/). -A common situation is that $\hat{H}_1$ consists of -a time-independent operator $\hat{B}$ -and a time-dependent function $f(t)$, -allowing us to split $C^R_{A H_1}$ as follows: +A common situation is that $$\hat{H}_1$$ consists of +a time-independent operator $$\hat{B}$$ +and a time-dependent function $$f(t)$$, +allowing us to split $$C^R_{A H_1}$$ as follows: $$\begin{aligned} \hat{H}_{1,S}(t) @@ -128,10 +128,10 @@ $$\begin{aligned} = C^R_{A B}(t, t') f(t') \end{aligned}$$ -Since $C_{AB}^R$ is a Green's function, -we know that it only depends on the difference $t - t'$, +Since $$C_{AB}^R$$ is a Green's function, +we know that it only depends on the difference $$t - t'$$, as long as the system was initially in thermodynamic equilibrium, -and $\hat{H}_{0,S}$ is time-independent: +and $$\hat{H}_{0,S}$$ is time-independent: $$\begin{aligned} C^R_{A B}(t, t') @@ -139,7 +139,7 @@ $$\begin{aligned} \end{aligned}$$ With this, the Kubo formula can be written as follows, -where we have set $t_0 = - \infty$: +where we have set $$t_0 = - \infty$$: $$\begin{aligned} \delta\!\expval{A}\!(t) @@ -150,8 +150,8 @@ $$\begin{aligned} This is a convolution, so the [convolution theorem](/know/concept/convolution-theorem/) states that the [Fourier transform](/know/concept/fourier-transform/) -of $\delta\!\expval{\hat{A}}\!(t)$ is simply the product -of the transforms of $C^R_{AB}$ and $f$: +of $$\delta\!\expval{\hat{A}}\!(t)$$ is simply the product +of the transforms of $$C^R_{AB}$$ and $$f$$: $$\begin{aligned} \boxed{ |