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Diffstat (limited to 'source/know/concept/kubo-formula/index.md')
| -rw-r--r-- | source/know/concept/kubo-formula/index.md | 23 |
1 files changed, 12 insertions, 11 deletions
diff --git a/source/know/concept/kubo-formula/index.md b/source/know/concept/kubo-formula/index.md index 4cb39ac..41fda3c 100644 --- a/source/know/concept/kubo-formula/index.md +++ b/source/know/concept/kubo-formula/index.md @@ -27,14 +27,15 @@ respectively refer to the Schrödinger and [interaction pictures](/know/concept/interaction-picture/): $$\begin{aligned} - \expval{\hat{A}}(t) + \expval{\hat{A}(t)} = \matrixel{\psi_S(t)}{\hat{A}_S}{\psi_S(t)} &= \matrixel{\psi_I(t)}{\hat{A}_I(t)}{\psi_I(t)} \\ &= \matrixel{\psi_I(t_0)\,}{\,\hat{K}_I^\dagger(t, t_0) \hat{A}_I(t) \hat{K}_I(t, t_0)\,}{\,\psi_I(t_0)} \end{aligned}$$ -Where the time evolution operator $$\hat{K}_I(t, t_0)$$ is as follows, +Where the [time evolution operator](/know/concept/time-evolution-operator/) +$$\hat{K}_I(t, t_0)$$ is as follows, which we Taylor-expand: $$\begin{aligned} @@ -71,18 +72,18 @@ where $$\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) + \expval{\hat{A}(t)} = \expval{\hat{K}_I^\dagger \hat{A}_I \hat{K}_I}_0 = \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)$$ +Now we define $$\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)$$: +and insert $$\expval{\hat{A}(t)}$$: $$\begin{aligned} - \delta\!\expval{\hat{A}}\!(t) - \equiv \expval{\hat{A}}(t) - \expval{\hat{A}_I}_0 + \delta\!\expval{\hat{A}(t)} + \equiv \expval{\hat{A}(t)} - \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}$$ @@ -94,7 +95,7 @@ describing the response of $$\expval{\hat{A}}$$ to first order in $$\hat{H}_1$$: $$\begin{aligned} \boxed{ - \delta\!\expval{\hat{A}}\!(t) + \delta\!\expval{\hat{A}(t)} = \int_{t_0}^\infty C^R_{A H_1}(t, t') \dd{t'} } \end{aligned}$$ @@ -142,7 +143,7 @@ With this, the Kubo formula can be written as follows, where we have set $$t_0 = - \infty$$: $$\begin{aligned} - \delta\!\expval{A}\!(t) + \delta\!\expval{\hat{A}(t)} = \int_{-\infty}^\infty C^R_{A B}(t - t') f(t') \dd{t'} = (C^R_{A B} * f)(t) \end{aligned}$$ @@ -150,12 +151,12 @@ $$\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 $$\delta\!\expval{\hat{A}(t)}$$ is simply the product of the transforms of $$C^R_{AB}$$ and $$f$$: $$\begin{aligned} \boxed{ - \delta\!\expval{\hat{A}}\!(\omega) + \delta\!\expval{\hat{A}(\omega)} = \tilde{C}{}^R_{A B}(\omega) \: \tilde{f}(\omega) } \end{aligned}$$ |