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diff --git a/content/know/concept/electric-dipole-approximation/index.pdc b/content/know/concept/electric-dipole-approximation/index.pdc index 265babf..67c73ee 100644 --- a/content/know/concept/electric-dipole-approximation/index.pdc +++ b/content/know/concept/electric-dipole-approximation/index.pdc @@ -20,120 +20,142 @@ The general Hamiltonian of an electron in such a wave is given by: $$\begin{aligned} \hat{H} - &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{2 m} (\vec{A} \cdot \vec{P} + \vec{P} \cdot \vec{A}) + \frac{q^2 \vec{A}{}^2}{2m} + V + &= \frac{(\vu{P} - q \vb{A})^2}{2 m} + q \varphi + \\ + &= \frac{\vu{P}{}^2}{2 m} - \frac{q}{2 m} (\vb{A} \cdot \vu{P} + \vu{P} \cdot \vb{A}) + \frac{q^2 \vb{A}^2}{2m} + q \varphi +\end{aligned}$$ + +With charge $q = - e$, +canonical momentum operator $\vu{P} = - i \hbar \nabla$, +and magnetic vector potential $\vb{A}(\vb{x}, t)$. +We reduce this by fixing the Coulomb gauge $\nabla \cdot \vb{A} = 0$, +so that $\vb{A} \cdot \vu{P} = \vu{P} \cdot \vb{A}$: + +$$\begin{aligned} + \comm*{\vb{A}}{\vu{P}} \psi + &= -i \hbar \vb{A} \cdot (\nabla \psi) + i \hbar \nabla \cdot (\vb{A} \psi) + \\ + &= i \hbar (\nabla \cdot \vb{A}) \psi + = 0 \end{aligned}$$ -With charge $q = - e$ -and electromagnetic vector potential $\vec{A}(\vec{r}, t)$. -We reduce this by fixing the Coulomb gauge $\nabla \cdot \vec{A} = 0$, -so that $\vec{A} \cdot \vec{P} = \vec{P} \cdot \vec{A}$, -and assume that $\vec{A}{}^2$ is negligible: +Where $\psi$ is an arbitrary test function. +Assuming $\vb{A}$ is so small that $\vb{A}{}^2$ is negligible, we split $\hat{H}$ as follows, +where $\hat{H}_1$ can be regarded as a perturbation to $\hat{H}_0$: $$\begin{aligned} \hat{H} = \hat{H}_0 + \hat{H}_1 \qquad \quad \hat{H}_0 - \equiv \frac{\vec{P}{}^2}{2 m} + V + \equiv \frac{\vu{P}{}^2}{2 m} + q \varphi \qquad \quad \hat{H}_1 - \equiv - \frac{q}{m} \vec{P} \cdot \vec{A} + \equiv - \frac{q}{m} \vu{P} \cdot \vb{A} \end{aligned}$$ -We have split $\hat{H}$ into $\hat{H}_0$ -and a perturbation $\hat{H}_1$, since $\vec{A}$ is small. -In an electromagnetic wave, -$\vec{A}$ is oscillating sinusoidally in time and space as follows: +In an electromagnetic wave, $\vb{A}$ is oscillating sinusoidally in time and space: $$\begin{aligned} - \vec{A}(\vec{r}, t) = - i \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) + \vb{A}(\vb{x}, t) = \vb{A}_0 \sin\!(\vb{k} \cdot \vb{x} - \omega t) \end{aligned}$$ -The corresponding perturbative -[electric field](/know/concept/electric-field/) $\vec{E}$ -points in the same direction: +Mathematically, it is more convenient to represent this with a complex exponential, +whose real part should be taken at the end of the calculation: $$\begin{aligned} - \vec{E}(\vec{r}, t) - = - \pdv{\vec{A}}{t} - = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) + \vb{A}(\vb{x}, t) = - i \vb{A}_0 \exp\!(i \vb{k} \cdot \vb{x} - i \omega t) \end{aligned}$$ -Where $\vec{E}_0 = \omega \vec{A}_0$. +The corresponding perturbative [electric field](/know/concept/electric-field/) $\vb{E}$ is then given by: + +$$\begin{aligned} + \vb{E}(\vb{x}, t) + = - \pdv{\vb{A}}{t} + = \vb{E}_0 \exp\!(i \vb{k} \cdot \vb{x} - i \omega t) +\end{aligned}$$ + +Where $\vb{E}_0 = \omega \vb{A}_0$. Let us restrict ourselves to visible light, -whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$. -Meanwhile, an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$, -so $\vec{k} \cdot \vec{r}$ is negligible: +whose wavelength $2 \pi / |\vb{k}| \sim 10^{-6} \:\mathrm{m}$. +Meanwhile, an atomic orbital is several Bohr $\sim 10^{-10} \:\mathrm{m}$, +so $\vb{k} \cdot \vb{x}$ is negligible: $$\begin{aligned} \boxed{ - \vec{E}(\vec{r}, t) - \approx \vec{E}_0 \exp\!(- i \omega t) + \vb{E}(\vb{x}, t) + \approx \vb{E}_0 \exp\!(- i \omega t) } \end{aligned}$$ This is the **electric dipole approximation**: -we ignore all spatial variation of $\vec{E}$, +we ignore all spatial variation of $\vb{E}$, and only consider its temporal oscillation. Also, since we have not used the word "photon", we are implicitly treating the radiation classically, and the electron quantum-mechanically. -Next, we want to convert $\hat{H}_1$ -to use the electric field $\vec{E}$ instead of the potential $\vec{A}$. -To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$ +Next, we want to rewrite $\hat{H}_1$ +to use the electric field $\vb{E}$ instead of the potential $\vb{A}$. +To do so, we use that $\vu{P} = m \: \dv*{\vu{x}}{t}$ and evaluate this in the [interaction picture](/know/concept/interaction-picture/): $$\begin{aligned} - \matrixel{2}{\dv*{\vec{r}}{t}}{1} - &= \frac{i}{\hbar} \matrixel{2}{[\hat{H}_0, \vec{r}]}{1} - = \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vec{r} - \vec{r} \hat{H}_0}{1} - \\ - &= \frac{i}{\hbar} (E_2 - E_1) \matrixel{2}{\vec{r}}{1} - = i \omega_0 \matrixel{2}{\vec{r}}{1} + \vu{P} + = m \dv*{\vu{x}}{t} + = m \frac{i}{\hbar} \comm*{\hat{H}_0}{\vu{x}} + = m \frac{i}{\hbar} (\hat{H}_0 \vu{x} - \vu{x} \hat{H}_0) +\end{aligned}$$ + +Taking the off-diagonal inner product with +the two-level system's states $\ket{1}$ and $\ket{2}$ gives: + +$$\begin{aligned} + \matrixel{2}{\vu{P}}{1} + = m \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vu{x} - \vu{x} \hat{H}_0}{1} + = m i \omega_0 \matrixel{2}{\vu{x}}{1} \end{aligned}$$ -Therefore, $\vec{P} / m = i \omega_0 \vec{r}$, -where $\omega_0 \equiv (E_2 - E_1) / \hbar$ is the resonance frequency of the transition, -close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating, i.e. $\omega \approx \omega_0$. +Therefore, $\vu{P} / m = i \omega_0 \vu{x}$, +where $\omega_0 \equiv (E_2 \!-\! E_1) / \hbar$ is the resonance of the energy gap, +close to which we assume that $\vb{A}$ and $\vb{E}$ are oscillating, i.e. $\omega \approx \omega_0$. We thus get: $$\begin{aligned} \hat{H}_1(t) - &= - \frac{q}{m} \vec{P} \cdot \vec{A} - = - (- i i) q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t) + &= - \frac{q}{m} \vu{P} \cdot \vb{A} + = - (- i i) q \omega_0 \vu{x} \cdot \vb{A}_0 \exp\!(- i \omega t) \\ - &\approx - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t) - = - \vec{d} \cdot \vec{E}_0 \exp\!(- i \omega t) + &\approx - q \vu{x} \cdot \vb{E}_0 \exp\!(- i \omega t) + = - \vu{d} \cdot \vb{E}_0 \exp\!(- i \omega t) \end{aligned}$$ -Where $\vec{d} \equiv q \vec{r} = - e \vec{r}$ is +Where $\vu{d} \equiv q \vu{x} = - e \vu{x}$ is the **transition dipole moment operator** of the electron, hence the name **electric dipole approximation**. -Finally, since electric fields are actually real -(we let it be complex for mathematical convenience), -we take the real part, yielding: +Finally, we take the real part, yielding: $$\begin{aligned} \boxed{ \hat{H}_1(t) - = - q \vec{r} \cdot \vec{E}_0 \cos\!(\omega t) + = - \vu{d} \cdot \vb{E}(t) + = - q \vu{x} \cdot \vb{E}_0 \cos\!(\omega t) } \end{aligned}$$ If this approximation is too rough, -$\vec{E}$ can always be Taylor-expanded in $(i \vec{k} \cdot \vec{r})$: +$\vb{E}$ can always be Taylor-expanded in $(i \vb{k} \cdot \vb{x})$: $$\begin{aligned} - \vec{E}(\vec{r}, t) - = \vec{E}_0 \Big( 1 + (i \vec{k} \cdot \vec{r}) + \frac{1}{2} (i \vec{k} \cdot \vec{r})^2 + \: ... \Big) \exp\!(- i \omega t) + \vb{E}(\vb{x}, t) + = \vb{E}_0 \Big( 1 + (i \vb{k} \cdot \vb{x}) + \frac{1}{2} (i \vb{k} \cdot \vb{x})^2 + \: ... \Big) \exp\!(- i \omega t) \end{aligned}$$ Taking the real part then yields the following series of higher-order correction terms: $$\begin{aligned} - \vec{E}(\vec{r}, t) - = \vec{E}_0 \Big( \cos\!(\omega t) + (\vec{k} \cdot \vec{r}) \sin\!(\omega t) - \frac{1}{2} (\vec{k} \cdot \vec{r})^2 \cos\!(\omega t) + \: ... \Big) + \vb{E}(\vb{x}, t) + = \vb{E}_0 \Big( \cos\!(\omega t) + (\vb{k} \cdot \vb{x}) \sin\!(\omega t) - \frac{1}{2} (\vb{k} \cdot \vb{x})^2 \cos\!(\omega t) + \: ... \Big) \end{aligned}$$ diff --git a/content/know/concept/fabry-perot-cavity/cavity.png b/content/know/concept/fabry-perot-cavity/cavity.png Binary files differnew file mode 100644 index 0000000..547e3e9 --- /dev/null +++ b/content/know/concept/fabry-perot-cavity/cavity.png diff --git a/content/know/concept/fabry-perot-cavity/index.pdc b/content/know/concept/fabry-perot-cavity/index.pdc index d40852f..50b7c62 100644 --- a/content/know/concept/fabry-perot-cavity/index.pdc +++ b/content/know/concept/fabry-perot-cavity/index.pdc @@ -14,45 +14,154 @@ markup: pandoc # Fabry-Pérot cavity In its simplest form, a **Fabry-Pérot cavity** -is a region of light-transmitting medium -surrounded by two mirrors, +is a region of light-transmitting medium surrounded by two mirrors, which may transmit some of the incoming light. Such a setup can be used as e.g. an interferometer or a laser cavity. +Below, we calculate its quasinormal modes in 1D. +We divide the $x$-axis into three domains: left $L$, center $C$, and right $R$. +The cavity $C$ has length $\ell$ and is centered on $x = 0$. +Let $n_L$, $n_C$ and $n_R$ be the respective domains' refractive indices: -## Modes of macroscopic cavity +<a href="cavity.png"> +<img src="cavity.png" style="width:70%;display:block;margin:auto;"> +</a> -Consider a Fabry-Pérot cavity large enough -that we can neglect the mirrors' thicknesses, -which have reflection coefficients $r_L$ and $r_R$. -Let $\tilde{n}_C$ be the complex refractive index inside, -and $\tilde{n}_L$ and $\tilde{n}_R$ be the indices outside. -The cavity has length $L$, centered on $x = 0$. -To find the quasinormal modes, -we make the following ansatz, with mode number $m$: +## Microscopic cavity + +In its simplest "microscopic" form, the reflection at the boundaries +is simply caused by the index differences there. +Consider this ansatz for the [electric field](/know/concept/electric-field/) $E_m(x)$, +where $m$ is the mode: + +$$\begin{aligned} + E_m(x) + = \begin{cases} + A_1 e^{- i k_m n_L x} & \mathrm{for}\; x < -\ell/2 \\ + A_2 e^{- i k_m n_C x} + A_3 e^{i k_m n_C x} & \mathrm{for}\; \!-\!\ell/2 < x < \ell/2 \\ + A_4 e^{i k_m n_R x} & \mathrm{for}\; x > \ell/2 + \end{cases} +\end{aligned}$$ + +The goal is to find the modes' wavenumbers $k_m$. +First, we demand that $E_m$ and its derivative $\dv*{E_m}{x}$ +are continuous at the boundaries $x = \pm \ell/2$: + +$$\begin{aligned} + A_1 e^{i k_m n_L \ell/2} + &= A_2 e^{i k_m n_C \ell/2} + A_3 e^{- i k_m n_C \ell/2} + \\ + A_4 e^{i k_m n_R \ell/2} + &= A_2 e^{- i k_m n_C \ell/2} + A_3 e^{i k_m n_C \ell/2} +\end{aligned}$$ +$$\begin{aligned} + - i k_m n_L A_1 e^{i k_m n_L \ell/2} + &= - i k_m n_C A_2 e^{i k_m n_C \ell/2} + i k_m n_C A_3 e^{- i k_m n_C \ell/2} + \\ + i k_m n_R A_4 e^{i k_m n_R \ell/2} + &= - i k_m n_C A_2 e^{- i k_m n_C \ell/2} + i k_m n_C A_3 e^{i k_m n_C \ell/2} +\end{aligned}$$ + +Rearranging the four equations above yields the following linear system: + +$$\begin{aligned} + 0 + &= A_1 - A_2 e^{i k_m (n_C - n_L) \ell/2} - A_3 e^{- i k_m (n_C + n_L) \ell/2} + \\ + 0 + &= A_2 e^{- i k_m (n_C + n_R) \ell/2} + A_3 e^{i k_m (n_C - n_R) \ell/2} - A_4 + \\ + 0 + &= n_L A_1 + n_C \big( A_3 e^{- i k_m (n_C + n_L) \ell/2} - A_2 e^{i k_m (n_C - n_L) \ell/2} \big) + \\ + 0 + &= n_C \big( A_3 e^{i k_m (n_C - n_R) \ell/2} - A_2 e^{- i k_m (n_C + n_R) \ell/2} \big) - n_R A_4 +\end{aligned}$$ + +Which can be rewritten in matrix form as follows, with the system matrix on the left: + +$$\begin{aligned} + \begin{bmatrix} + 1 & -e^{i k_m (n_C - n_L) \ell/2} & -e^{- i k_m (n_C + n_L) \ell/2} & 0 \\ + 0 & e^{- i k_m (n_C + n_R) \ell/2} & e^{i k_m (n_C - n_R) \ell/2} & -1 \\ + n_L & -n_C e^{i k_m (n_C - n_L) \ell/2} & n_C e^{- i k_m (n_C + n_L) \ell/2} & 0 \\ + 0 & -n_C e^{- i k_m (n_C + n_R) \ell/2} & n_C e^{i k_m (n_C - n_R) \ell/2} & -n_R + \end{bmatrix} + \cdot + \begin{bmatrix} + A_1 \\ A_2 \\ A_3 \\ A_4 + \end{bmatrix} + = + \begin{bmatrix} + 0 \\ 0 \\ 0 \\ 0 + \end{bmatrix} +\end{aligned}$$ + +We want non-trivial solutions, where we +cannot simply satisfy the system by setting $A_1$, $A_2$, $A_3$ and +$A_4$; this constraint will give us an equation for $k_m$. Therefore, we +demand that the system matrix is singular, i.e. its determinant is zero: + +$$\begin{aligned} + 0 = + &- n_C (n_L + n_R) \big( e^{i k_m (2 n_C - n_L - n_R) \ell/2} + e^{- i k_m (2 n_C + n_L + n_R) \ell/2} \big) + \\ + &+ (n_C^2 + n_L n_R) \big( e^{i k_m (2 n_C - n_L - n_R) \ell/2} - e^{- i k_m (2 n_C + n_L + n_R) \ell/2} \big) +\end{aligned}$$ + +We multiply by $e^{i k_m (n_L + n_R) \ell / 2}$ and +decompose the exponentials into sines and cosines: + +$$\begin{aligned} + 0 + = i 2 (n_C^2 + n_L n_R) \sin\!(k_m n_C \ell) + - 2 n_C (n_L + n_R) \cos\!(k_m n_C \ell) +\end{aligned}$$ + +Finally, some further rearranging gives a convenient transcendental equation: + +$$\begin{aligned} + \boxed{ + 0 + = \tan\!(k_m n_C \ell) + i \frac{n_C (n_L + n_R)}{n_C^2 + n_L n_R} + } +\end{aligned}$$ + +Thanks to linearity, we can choose one of the amplitudes +$A_1$, $A_2$, $A_3$ or $A_4$ freely, +and then the others are determined by $k_m$ and the field's continuity. + + +## Macroscopic cavity + +Next, consider a "macroscopic" Fabry-Pérot cavity +with complex mirror structures at boundaries, e.g. Bragg reflectors. +If the cavity is large enough, we can neglect the mirrors' thicknesses, +and just use their reflection coefficients $r_L$ and $r_R$. +We use the same ansatz: $$\begin{aligned} E_m(x) = \begin{cases} - A_m \exp\!(-i \tilde{n}_L \tilde{k}_m x) & \mathrm{if}\; x < -L/2 \\ - B_m \exp\!(i \tilde{n}_C \tilde{k}_m x) + C_m \exp\!(-i \tilde{n}_C \tilde{k}_m x) & \mathrm{if}\; -\!L/2 < x < L/2 \\ - D_m \exp\!(i \tilde{n}_R \tilde{k}_m x) & \mathrm{if}\; L/2 < x + A_1 e^{-i k_m n_L x} & \mathrm{for}\; x < -\ell/2 \\ + A_2 e^{-i k_m n_C x} + A_3 e^{i k_m n_C x} & \mathrm{for}\; \!-\!\ell/2 < x < \ell/2 \\ + A_4 e^{i k_m n_R x} & \mathrm{for}\; \ell/2 < x \end{cases} \end{aligned}$$ -On the left, $B_m$ is the reflection of $C_m$, -and on the right, $C_m$ is the reflection of $B_m$, -where the reflected amplitude is determined -by the coefficients $r_L$ and $r_L$, respectively: +On the left, $A_3$ is the reflection of $A_2$, +and on the right, $A_2$ is the reflection of $A_3$, +where the reflected amplitudes are determined +by the coefficients $r_L$ and $r_R$, respectively: $$\begin{aligned} - B_m \exp\!(-i \tilde{n}_C \tilde{k}_m L/2) - &= r_L C_m \exp\!(i \tilde{n}_C \tilde{k}_m L/2) + A_3 e^{- i k_m n_C \ell/2} + &= r_L A_2 e^{i k_m n_C \ell/2} \\ - C_m \exp\!(-i \tilde{n}_C \tilde{k}_m L/2) - &= r_R B_m \exp\!(i \tilde{n}_C \tilde{k}_m L/2) + A_2 e^{-i k_m n_C \ell/2} + &= r_R A_3 e^{i k_m n_C \ell/2} \end{aligned}$$ These equations might seem to contradict each other. @@ -60,12 +169,12 @@ We recast them into matrix form: $$\begin{aligned} \begin{bmatrix} - 1 & - r_L \exp\!(i \tilde{n}_C \tilde{k}_m L) \\ - - r_R \exp\!(i \tilde{n}_C \tilde{k}_m L) & 1 + 1 & - r_R e^{i k_m n_C \ell} \\ + - r_L e^{i k_m n_C \ell} & 1 \end{bmatrix} \cdot \begin{bmatrix} - B_m \\ C_m + A_2 \\ A_3 \end{bmatrix} = \begin{bmatrix} @@ -73,58 +182,53 @@ $$\begin{aligned} \end{bmatrix} \end{aligned}$$ -Now, we do not want to be able to find values for $B_m$ and $C_m$ -that satisfy this for a given $\tilde{k}_m$. -Instead, we only want specific values of $\tilde{k}_m$ to be allowed, -corresponding to the cavity's modes. -We thus demand that the determinant to zero: +Again, we demand that the determinant is zero, in order to get non-trivial solutions: $$\begin{aligned} 0 - &= 1 - r_L r_R \exp\!(i 2 \tilde{n}_C \tilde{k}_m L) + &= 1 - r_L r_R e^{i 2 k_m n_C \ell} \end{aligned}$$ -Isolating this for $\tilde{k}_m$ yields the following modes, +Isolating this for $k_m$ yields the following modes, where $m$ is an arbitrary integer: $$\begin{aligned} \boxed{ - \tilde{k}_m - = - \frac{\ln\!(r_L r_R) + i 2 \pi m}{i 2 \tilde{n}_C L} + k_m + = - \frac{\ln\!(r_L r_R) + i 2 \pi m}{i 2 n_C \ell} } \end{aligned}$$ -These $\tilde{k}_m$ satisfy the matrix equation above. -Thanks to linearity, we can choose one of $B_m$ or $C_m$, -and then the other is determined by the corresponding equation. +These $k_m$ satisfy the matrix equation above. +Thanks to linearity, we can choose one of $A_2$ or $A_3$, +and then the other is determined by the corresponding reflection equation. Finally, we look at the light transmitted through the mirrors, according to $1 \!-\! r_L$ and $1 \!-\! r_R$: $$\begin{aligned} - A_m \exp\!(i \tilde{n}_L \tilde{k}_m L/2) - &= (1 - r_L) C_m \exp\!(i \tilde{n}_C \tilde{k}_m L/2) + A_1 e^{i k_m n_L \ell/2} + &= (1 - r_L) A_2 e^{i k_m n_C \ell/2} \\ - D_m \exp\!(i \tilde{n}_R \tilde{k}_m L/2) - &= (1 - r_R) B_m \exp\!(i \tilde{n}_C \tilde{k}_m L/2) + A_4 e^{i k_m n_R \ell/2} + &= (1 - r_R) A_3 e^{i k_m n_C \ell/2} \end{aligned}$$ -We simply isolate for $A_m$ and $D_m$ respectively, +We simply isolate for $A_1$ and $A_4$ respectively, yielding the following amplitudes: $$\begin{aligned} - A_m - &= (1 - r_L) C_m \exp\!\big( i (\tilde{n}_C \!-\! \tilde{n}_L) \tilde{k}_m L/2 \big) + A_1 + &= (1 - r_L) A_2 e^{i k_m (n_C - n_L) \ell/2} \\ - D_m - &= (1 - r_R) B_m \exp\!\big( i (\tilde{n}_C \!-\! \tilde{n}_R) \tilde{k}_m L/2 \big) + A_4 + &= (1 - r_R) A_3 e^{i k_m (n_C - n_R) \ell/2} \end{aligned}$$ Note that we have not demanded continuity of the electric field. This is because the mirrors are infinitely thin "magic" planes; -if we had instead used a full physical mirror structure, -then the we would have demanded continuity, -as you might have expected. +had we instead used the full mirror structure, +then we would have demanded continuity, as you maybe expected. diff --git a/content/know/concept/feynman-diagram/index.pdc b/content/know/concept/feynman-diagram/index.pdc index dfb63c1..600be61 100644 --- a/content/know/concept/feynman-diagram/index.pdc +++ b/content/know/concept/feynman-diagram/index.pdc @@ -284,12 +284,12 @@ involving the [Matsubara Green's function](/know/concept/matsubara-greens-functi $$\begin{aligned} i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1) \:\: &\longrightarrow \:\: - -\!\hbar G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + \hbar G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) = \expval{\mathcal{T} \Big\{ \hat{\Psi}_I(\vb{r}_2, \tau_2) \hat{\Psi}_I^\dagger(\vb{r}_1, \tau_1) \Big\}} \\ i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1) \:\: &\longrightarrow \:\: - -\!\hbar G_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) + \hbar G_{s_2 s_1}(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) = \expval{\mathcal{T} \Big\{ \hat{\Psi}_H(\vb{r}_2, \tau_2) \hat{\Psi}_H^\dagger(\vb{r}_1, \tau_1) \Big\}} \end{aligned}$$ @@ -312,7 +312,8 @@ and a distinction must be made between fermionic Matsubara frequencies $i \omega_n^f$ (for $G$ and $G^0$) and bosonic Matsubara ones $i \omega_n^b$ (for $W$). This distinction is compatible with frequency conservation, -since a sum of two fermionic frequencies is always bosonic: +since a sum of two fermionic frequencies is always bosonic. +We have: $$\begin{aligned} G_{s_2 s_1}^0(\vb{r}_2, \tau_2; \vb{r}_1, \tau_1) diff --git a/content/know/concept/random-phase-approximation/index.pdc b/content/know/concept/random-phase-approximation/index.pdc index 970a884..ed85106 100644 --- a/content/know/concept/random-phase-approximation/index.pdc +++ b/content/know/concept/random-phase-approximation/index.pdc @@ -77,7 +77,7 @@ i.e. the ones where all $n$ interaction lines carry the same momentum and energy: <a href="rpasigma.png"> -<img src="rpasigma.png" style="width:91%;display:block;margin:auto;"> +<img src="rpasigma.png" style="width:92%;display:block;margin:auto;"> </a> Where we have defined the **screened interaction** $W^\mathrm{RPA}$, diff --git a/content/know/concept/random-phase-approximation/rpasigma.png b/content/know/concept/random-phase-approximation/rpasigma.png Binary files differindex 87ba3cc..c9587b8 100644 --- a/content/know/concept/random-phase-approximation/rpasigma.png +++ b/content/know/concept/random-phase-approximation/rpasigma.png diff --git a/content/know/concept/self-energy/dyson.png b/content/know/concept/self-energy/dyson.png Binary files differindex f576632..168505c 100644 --- a/content/know/concept/self-energy/dyson.png +++ b/content/know/concept/self-energy/dyson.png diff --git a/content/know/concept/self-energy/fullgf.png b/content/know/concept/self-energy/fullgf.png Binary files differindex 5767dba..0ea6958 100644 --- a/content/know/concept/self-energy/fullgf.png +++ b/content/know/concept/self-energy/fullgf.png diff --git a/content/know/concept/self-energy/index.pdc b/content/know/concept/self-energy/index.pdc index 7e67143..935cca8 100644 --- a/content/know/concept/self-energy/index.pdc +++ b/content/know/concept/self-energy/index.pdc @@ -27,7 +27,7 @@ and $\beta = 1 / (k_B T)$: $$\begin{aligned} G_{s_b s_a}(\vb{r}_b, \tau_b; \vb{r}_a, \tau_a) - = - \frac{\expval{\mathcal{T}\Big\{ \hat{K}(\hbar \beta, 0) \hat{\Psi}_{s_b}(\vb{r}_b, \tau_b) \hat{\Psi}_{s_a}^\dagger(\vb{r}_a, \tau_a) \Big\}}} + = \frac{\expval{\mathcal{T}\Big\{ \hat{K}(\hbar \beta, 0) \hat{\Psi}_{s_b}(\vb{r}_b, \tau_b) \hat{\Psi}_{s_a}^\dagger(\vb{r}_a, \tau_a) \Big\}}} {\hbar \expval{\hat{K}(\hbar \beta, 0)}} \end{aligned}$$ @@ -50,7 +50,7 @@ and $\hat{\Psi}_a \equiv \hat{\Psi}_{s_a}(\vb{r}_a, \tau_a)$: $$\begin{aligned} G_{ba} - &= - \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\!\frac{1}{\hbar} \Big)^n \idotsint_0^{\hbar \beta} + &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\!\frac{1}{\hbar} \Big)^n \idotsint_0^{\hbar \beta} \expval{\mathcal{T}\Big\{ \hat{W}(\tau_1) \cdots \hat{W}(\tau_n) \hat{\Psi}_b \hat{\Psi}_a^\dagger \Big\}} \dd{\tau_1} \cdots \dd{\tau_n}} {\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\!\frac{1}{\hbar} \Big)^n \idotsint_0^{\hbar \beta} \expval{\mathcal{T}\Big\{ \hat{W}(\tau_1) \cdots \hat{W}(\tau_n) \Big\}} \dd{\tau_1} \cdots \dd{\tau_n}} @@ -84,7 +84,7 @@ The full $G_{ba}$ thus becomes: $$\begin{aligned} G_{ba} - &= - \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{1}{2 \hbar} \Big)^n (-\hbar)^{2n+1} + &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{1}{2 \hbar} \Big)^n (-\hbar)^{2n+1} \idotsint W_{1'1} \cdots W_{n'n} \: \Big( G^0_\mathrm{num} \Big) \dd{1'} \dd{1} \cdots \dd{n'} \dd{n}} {\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{1}{2 \hbar} \Big)^n (-\hbar)^{2n} \idotsint W_{1'1} \cdots W_{n'n} \: \Big( G^0_\mathrm{den} \Big) \dd{1'} \dd{1} \cdots \dd{n'} \dd{n}} @@ -131,7 +131,7 @@ times $(-1)^p$ to account for swaps of fermionic operators: $$\begin{aligned} G_{ba} - &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{\hbar}{2} \Big)^n + &= -\frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{\hbar}{2} \Big)^n \idotsint W_{1'1} \cdots W_{n'n} \: \Big( \sum_{p} (-1)^p \prod_{m = 1}^{2 n + 1} G^0_{(p,m)} \Big) \dd{1}' \dd{1} \cdots \dd{n'} \dd{n}} {\displaystyle\sum_{n = 0}^\infty \frac{1}{n!} \Big( \!-\! \frac{\hbar}{2} \Big)^n \idotsint W_{1'1} \cdots W_{n'n} \: \Big( \sum_{p} (-1)^p \prod_{m = 1}^{2 n} G^0_{(p,m)} \Big) \dd{1'} \dd{1} \cdots \dd{n'} \dd{n}} @@ -172,7 +172,7 @@ $$\begin{aligned} &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \bigg[ \sum_{m = 0}^{n} \frac{n!}{m! (n \!-\! m)!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; (n \!-\! m)}_{\!\Sigma\mathrm{all}} \bigg]} - {-\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} + {\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} \end{aligned}$$ Where the total order is the sum of the orders of all considered diagrams, @@ -186,7 +186,7 @@ $$\begin{aligned} &= \frac{\displaystyle\sum_{m = 0}^{\infty} \frac{1}{2^m m!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}} \bigg[ \sum_{n = 0}^\infty \frac{1}{2^{n-m} (n \!-\! m)!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; (n \!-\! m)}_{\!\Sigma\mathrm{all}} \bigg]} - {-\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} + {\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} \end{aligned}$$ Since both $n$ and $m$ start at zero, @@ -194,7 +194,7 @@ and the sums include all possible diagrams, we see that the second sum in the numerator does not actually depend on $m$: $$\begin{aligned} - -\hbar G_{ba} + \hbar G_{ba} &= \frac{\displaystyle\sum_{m = 0}^{\infty} \frac{1}{2^m m!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}} \bigg[ \sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}} \bigg]} {\displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}} diff --git a/content/know/concept/self-energy/selfenergy.png b/content/know/concept/self-energy/selfenergy.png Binary files differindex 55e182e..84d3651 100644 --- a/content/know/concept/self-energy/selfenergy.png +++ b/content/know/concept/self-energy/selfenergy.png diff --git a/sources/know/concept/random-phase-approximation/main.tex b/sources/know/concept/random-phase-approximation/main.tex index 11c9bde..f20cf33 100644 --- a/sources/know/concept/random-phase-approximation/main.tex +++ b/sources/know/concept/random-phase-approximation/main.tex @@ -16,7 +16,7 @@ \noindent Definition of RPA self-energy \begin{align*} -\frac{\Sigma_s^\mathrm{RPA}(\vb{k}, i \omega_n^F)}{-\hbar} +\frac{\Sigma_s^\mathrm{RPA}(\vb{k}, i \omega_n^F)}{\hbar} &\equiv \:\: \begin{fmffile}{sigma1order} \begin{fmfgraph*}(30,20) diff --git a/sources/know/concept/self-energy/main.tex b/sources/know/concept/self-energy/main.tex index e359053..0e5e4f5 100644 --- a/sources/know/concept/self-energy/main.tex +++ b/sources/know/concept/self-energy/main.tex @@ -16,7 +16,7 @@ \noindent Full Green's function \begin{align*} --\hbar G(b,a) +\hbar G(b,a) &= \quad \begin{fmffile}{0order1} \begin{fmfgraph*}(30,20) @@ -70,7 +70,7 @@ Full Green's function \end{align*} Self-energy definition \begin{align*} -\frac{\Sigma(y,x)}{-\hbar} +\frac{\Sigma(y,x)}{\hbar} =\!\!\!\! \begin{fmffile}{selfenergy} \begin{fmfgraph*}(20,20) @@ -126,7 +126,7 @@ Self-energy definition \end{align*} Dyson equation \begin{align*} --\hbar G(b,a) +\hbar G(b,a) = \quad \begin{fmffile}{fullgf} \begin{fmfgraph*}(50,20) |
