LaTeX vs MathPix image — AKolbe-BA

\begin{aligned} c(|\phi\rangle+|\psi\rangle) & =c|\phi\rangle+c|\psi\rangle \\ (c+d)|\phi\rangle & =c|\phi\rangle+d|\phi\rangle \\ c(d|\phi\rangle) & =c d|\phi\rangle \end{aligned}

\begin{aligned} c(\left| \phi \right>+\left| \psi \right>)=&\,c\left| \phi \right>+c\left| \psi \right> \qquad \text{}\\ (c+d)\left| \phi \right>=&\,c\left| \phi \right>+d\left| \phi \right>\\ c(d\left| \phi \right>)=&\,cd\left| \phi \right> \end{aligned}

\vec{J}=\vec{r} \times \vec{p}

\vec{J}=\vec{r}\times\vec{p}

\hat{J}=\frac{\mathrm{i}}{\hbar} \hat{r} \times \nabla

\begin{aligned} \hat{J}= & \dfrac{\mathrm{i}}{\hbar}\hat{r}\times\nabla \end{aligned}

\hat{J}_{x}=\frac{\mathrm{i}}{\hbar}\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}\right) \quad ; \quad \hat{J}_{y}=\frac{\mathrm{i}}{\hbar}\left(z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}\right) \quad ; \quad \hat{J}_{z}=\frac{\mathrm{i}}{\hbar}\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right)

\hat{J}_x=\dfrac{\mathrm{i}}{\hbar}\left(y\dfrac{\partial}{\partial z}-z\dfrac{\partial}{\partial y}\right)\,\,\,\, ;\,\,\,\, \hat{J}_y= \dfrac{\mathrm{i}}{\hbar}\left(z\dfrac{\partial}{\partial x}-x\dfrac{\partial}{\partial z}\right)\,\,\,\, ;\,\,\,\,\hat{J}_z=\dfrac{\mathrm{i}}{\hbar}\left(x\dfrac{\partial}{\partial y}-y\dfrac{\partial}{\partial x}\right)

\left[\hat{J}_{x}, \hat{J}_{y}\right]=\mathrm{i} \hbar \hat{J}_{z} \quad ; \quad\left[\hat{J}_{y}, \hat{J}_{z}\right]=\mathrm{i} \hbar \hat{J}_{x} \quad ; \quad\left[\hat{J}_{z}, \hat{J}_{x}\right]=\mathrm{i} \hbar \hat{J}_{y}

[\hat{J}_x,\hat{J}_y]=\mathrm{i}\hbar\hat{J}_z \,\,\,\, ;\,\,\,\, [\hat{J}_y,\hat{J}_z]=\mathrm{i}\hbar\hat{J}_x \,\,\,\, ;\,\,\,\, [\hat{J}_z,\hat{J}_x]=\mathrm{i}\hbar\hat{J}_y

\epsilon_{i j k}=\left\{\begin{aligned} +1, & \text { wenn ijk eine gerade Permutation von } 123 \text { ist } \\ -1, & \text { wen ijk eine ungerade Permutation von } 123 \text { ist } \\ 0, & \text { wenn mindestens zwei Indizes gleich sind } \end{aligned}\right.

\begin{aligned} \epsilon_{ijk} = \begin{cases} +1, & \text{ wenn ijk eine gerade Permutation von 123 ist} \\ -1, & \text{wen ijk eine ungerade Permutation von 123 ist} \\ \,\,\,\,\,0, & \text{wenn mindestens zwei Indizes gleich sind} \end{cases} \end{aligned}

\left[\hat{J}_{i}, \hat{J}_{j}\right]=\mathrm{i} \hbar \sum_{k} \hat{J}_{k} \epsilon_{i j k}

[\hat{J}_i,\hat{J}_j]=\mathrm{i}\hbar\sum_k\hat{J}_k\epsilon_{ijk}

\left[\hat{J}_{i}, \hat{J}^{2}\right]=0 \quad i=x, y, z

[\hat{J}_i,\hat{J}^2]=0 \,\,\,\,\,\,\,\, i= x,y,z

\begin{aligned} \hat{\mathbf{J}}^{2}|j, m\rangle & =\hbar^{2} j(j+1)|j, m\rangle \\ \hat{J}_{z}|j, m\rangle & =\hbar m|j, m\rangle \end{aligned}

\begin{aligned} \hat{\mathbf{J}}^2\left| j,m \right>=&\,\hbar^2 j(j+1)\left| j,m \right>\\ \hat{J}_z\left| j,m \right>=&\, \hbar m \left| j,m \right> \end{aligned}

\begin{aligned} \hat{J}^{+} & =\left(\hat{J}_{x}+\mathrm{i} \hat{J}_{y}\right) \\ \hat{J}^{-} & =\left(\hat{J}_{x}-\mathrm{i} \hat{J}_{y}\right) \end{aligned}

\begin{aligned} \hat{J}^+= & \left(\hat{J}_x+\mathrm{i}\hat{J}_y\right) \\ \hat{J}^-= & \left(\hat{J}_x-\mathrm{i}\hat{J}_y\right) \end{aligned}

\left[\hat{J}_{z}, \hat{J}^{+}\right]=\hbar \hat{J}^{+} \quad, \quad\left[\hat{J}_{z}, \hat{J}^{-}\right]=-\hbar \hat{J}^{-} \quad, \quad\left[\hat{J}^{+}, \hat{J}^{-}\right]=2 \hbar \hat{J}_{z}

[\hat{J}_z,\hat{J}^+]=\hbar\hat{J}^+ \,\,\,\, ,\,\,\,\, [\hat{J}_z,\hat{J}^-]=-\hbar\hat{J}^- \,\,\,\, ,\,\,\,\, [\hat{J}^+,\hat{J}^-]=2\hbar\hat{J}_z

|s\rangle=\alpha|\uparrow\rangle+\beta|\downarrow\rangle=\binom{\alpha}{\beta}

\left| s \right>=\alpha\left| \uparrow \right>+\beta\left| \downarrow \right>=\vector{\alpha}{\beta}

\hat{\sigma}_{x}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \quad ; \quad \hat{\sigma}_{y}=\left(\begin{array}{cc} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{array}\right) \quad ; \quad \hat{\sigma}_{z}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right) \quad ; \quad \hat{\sigma}_{0}=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)

\begin{aligned} \hat{\sigma}_x=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}\,\,\,\,;\,\,\,\,\hat{\sigma}_y=\begin{pmatrix} 0 & -\mathrm{i}\\ \mathrm{i} & 0 \end{pmatrix}\,\,\,\,;\,\,\,\,\hat{\sigma}_z=\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}\,\,\,\,;\,\,\,\,\hat{\sigma}_0=\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \end{aligned}

\begin{aligned} \vec{S} & =\frac{\hbar}{2} \vec{\sigma} \\ \hat{S}_{i} & =\frac{\hbar}{2} \hat{\sigma}_{i} \quad i=x, y, z \end{aligned}

\begin{aligned} \vec{S}=&\,\dfrac{\hbar}{2}\vec{\sigma}\\ \hat{S_i}=&\,\dfrac{\hbar}{2}\hat{\sigma}_i\,\,\,\,\,\, i=x,y,z \end{aligned}

\begin{aligned} \mathbf{1}_{2} & =\hat{\sigma}_{x}^{2}+\hat{\sigma}_{y}^{2}+\hat{\sigma}_{z}^{2} \\ 2 \mathrm{i} \hat{\sigma}_{z} & =\left[\hat{\sigma}_{x}, \hat{\sigma}_{y}\right] \quad \text { und zyklisch } \\ \left\{\hat{\sigma}_{x}, \hat{\sigma}_{y}\right\} & =\left\{\hat{\sigma}_{y}, \hat{\sigma}_{z}\right\}=\left\{\hat{\sigma}_{z}, \hat{\sigma}_{x}\right\}=0 \end{aligned}

\begin{aligned} \mathbf{1}_2=&\,\,\hat{\sigma}_x^2+\hat{\sigma}_y^2+\hat{\sigma}_z^2\\ 2\mathrm{i}\hat{\sigma}_z=&\,\,[\hat{\sigma}_x,\hat{\sigma}_y] \,\,\,\, \text{und zyklisch}\\ \{\hat{\sigma}_x,\hat{\sigma}_y\}=&\,\,\{\hat{\sigma}_y,\hat{\sigma}_z\}=\{\hat{\sigma}_z,\hat{\sigma}_x\}=0 \end{aligned}

\begin{aligned} & \hat{S}_{x}=\frac{1}{2}\left(\hat{S}^{+}+\hat{S}^{-}\right) \\ & \hat{S}_{y}=\frac{1}{2 \mathrm{i}}\left(\hat{S}^{+}-\hat{S}^{-}\right) \end{aligned}

\begin{aligned} \hat{S}_x=&\,\,\dfrac{1}{2}(\hat{S}^++\hat{S}^-)\\ \hat{S}_y=&\,\,\dfrac{1}{2\mathrm{i}}(\hat{S}^+-\hat{S}^-) \end{aligned}

\hat{S}^{+}=\hbar\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right) \quad ; \quad \hat{S}^{-}=\hbar\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right)

\begin{aligned} \hat{S}^+=\hbar \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} \,\,\,\,;\,\,\, \hat{S}^-=\hbar \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix} \end{aligned}

\begin{aligned} \hat{S}^{+}|\downarrow\rangle & =\hbar|\uparrow\rangle & \hat{S}^{+}|\uparrow\rangle & =0 \\ \hat{S}^{-}|\uparrow\rangle & =\hbar|\downarrow\rangle & \hat{S}^{-}|\downarrow\rangle & =0 \\ \hat{S}_{z}|\uparrow\rangle & =\frac{\hbar}{2}|\uparrow\rangle & \hat{S}_{z}|\downarrow\rangle & =-\frac{\hbar}{2}|\downarrow\rangle \end{aligned}

\begin{aligned} \hat{S}^{+}|\downarrow\rangle & =\hbar|\uparrow\rangle & \hat{S}^{+}|\uparrow\rangle & =0 \\ \hat{S}^{-}|\uparrow\rangle & =\hbar|\downarrow\rangle & \hat{S}^{-}|\downarrow\rangle & =0 \\ \hat{S}_{z}|\uparrow\rangle & =\frac{\hbar}{2}|\uparrow\rangle & \hat{S}_{z}|\downarrow\rangle & =-\frac{\hbar}{2}|\downarrow\rangle \end{aligned}

\begin{aligned} -1 & <\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}}<0 & & \text { antikorreliert. } \\ 0 & <\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}}<1 & & \text { korreliert. } \\ 0 & =\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}} & & \text { unkorreliert. } \\ 1 & =\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}} & & \text { maximal korreliert. } \end{aligned}

\begin{aligned} -1<&\langle \hat{S}_i\hat{S}_j\rangle_{\psi_g}<0\quad\text{antikorreliert.}\\ 0<&\langle \hat{S}_i\hat{S}_j\rangle_{\psi_g}<1\quad\text{korreliert.}\\ 0=&\langle \hat{S}_i\hat{S}_j\rangle_{\psi_g}\quad\qquad\text{unkorreliert.}\\ 1=&\langle \hat{S}_i\hat{S}_j\rangle_{\psi_g}\quad\qquad\text{maximal korreliert.} \end{aligned}

\left\langle\vec{S}_{i} \vec{S}_{j}\right\rangle_{T}=\frac{1}{Z} \sum_{n} e^{\beta E_{n}} \sum_{l=1}^{g_{n}}\left\langle\Psi_{E_{n}, l}\right| \vec{S}_{i} \vec{S}_{j}\left|\Psi_{E_{n}, l}\right\rangle

\langle \vec{S}_i\vec{S}_j\rangle_T=\dfrac{1}{Z}\sum_n e^{\beta E_n}\sum_{l=1}^{g_n} \braket{\Psi_{E_n,l}}{\vec{S}_i\vec{S}_j}{\Psi_{E_n,l}}

\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}}=\frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \hat{S}_{i} \hat{S}_{j}\left|\psi^{0, l}\right\rangle

\langle \hat{S}_i\hat{S}_j\rangle_{\psi_g}=\dfrac{1}{g_0}\sum_{l=0}^{g_0}\left< \psi^{0,l} \right|\hat{S}_i\hat{S}_j\left| \psi^{0,l} \right>

\mathcal{H}=\mathcal{H}_{1} \otimes \mathcal{H}_{2} \otimes \ldots \otimes \mathcal{H}_{N-1} \otimes \mathcal{H}_{N}

\mathcal{H}=\mathcal{H}_1\otimes\mathcal{H}_2\otimes\,....\,\otimes\mathcal{H}_{N-1}\otimes\mathcal{H}_N

|\psi\rangle=|\psi\rangle_{1} \otimes|\psi\rangle_{2} \otimes \ldots \otimes|\psi\rangle_{N-1} \otimes|\psi\rangle_{N}

\mid\psi\rangle=\mid\psi\rangle_1\otimes\mid\psi\rangle_2\otimes \,...\,\otimes\mid\psi\rangle_{N-1}\otimes\mid\psi\rangle_N

\hat{\mathrm{A}}_{i}: \mathcal{H}_{i} \rightarrow \mathcal{H}_{i}

\hat{\mathrm{A}}_i : \mathcal{H}_i\rightarrow\mathcal{H}_i

\overline{\mathrm{A}}_{1}+\ldots+\overline{\mathrm{A}}_{N}=\hat{\mathrm{A}}_{1} \otimes \hat{\mathrm{I}} \otimes \ldots \otimes \hat{\mathrm{I}}+\hat{\mathrm{I}} \otimes \hat{\mathrm{~A}}_{2} \otimes \ldots \otimes \hat{\mathrm{I}}+\ldots+\hat{\mathrm{I}} \otimes \ldots \otimes \hat{\mathrm{~A}_{N}}

\overline{\mathrm{A}}_1+\,...\,+ \overline{\mathrm{A}}_N= \hat{\mathrm{A}}_1\otimes\hat{\mathrm{I}}\otimes ...\otimes\hat{\mathrm{I}} +\hat{\mathrm{I}}\otimes\hat{\mathrm{A}}_2\otimes ...\otimes\hat{\mathrm{I}} +...+\hat{\mathrm{I}}\otimes ...\otimes\hat{\mathrm{A}_N}

\mathcal{H}=\mathcal{H}_{1}^{(1)} \otimes \mathcal{H}_{1}^{(2)} \otimes \ldots \otimes \mathcal{H}_{1}^{(N)}

\mathcal{H}= \mathcal{H}_1^{(1)}\otimes\mathcal{H}_1^{(2)}\otimes...\otimes\mathcal{H}_1^{(N)}

\mathcal{V}\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\sum_{\left\{\alpha_{i}\right\}} a_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}} \nu_{\alpha_{1}}\left(x_{1}\right) \nu_{\alpha_{2}}\left(x_{2}\right) \ldots \nu_{\alpha_{N}}\left(x_{N}\right), \quad a_{\alpha_{i}} \in \mathbb{C}

\mathcal{V}(x_1,x_2,...,x_N)=\sum_{\{\alpha_i\}}a_{\alpha_1\alpha_2...\alpha_N}\nu_{\alpha_1}(x_1)\nu_{\alpha_2}(x_2)...\nu_{\alpha_N}(x_N),\quad a_{\alpha_i}\in\mathbb{C}

\begin{array}{lr} \Psi\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\operatorname{sign}(\pi) \Psi\left(x_{\pi(1)}, x_{\pi(2)}, \ldots, x_{\pi(N)}\right) & \text { für Fermionen } \\ \Phi\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\Phi\left(x_{\pi(1)}, x_{\pi(2)}, \ldots, x_{\pi(N)}\right) & \text { für Bosonen } \end{array}

\begin{aligned} \Psi(x_1,x_2,...,x_N)=&\,\,\text{sign}(\pi)\Psi(x_{\pi(1)},x_{\pi(2)},...,x_{\pi(N)})& \text{für Fermionen}\\ \Phi(x_1,x_2,...,x_N)=&\Phi(x_{\pi(1)},x_{\pi(2)},...,x_{\pi(N)})& \text{für Bosonen} \end{aligned}

\Psi\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\sum_{\left\{\alpha_{i}\right\}} a_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}} \psi_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}}\left(x_{1}, x_{2}, \ldots, x_{N}\right), \quad a_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}} \in \mathbb{C}

\Psi(x_1,x_2,...,x_N)=\sum_{\{\alpha_i\}}a_{\alpha_1\alpha_2...\alpha_N}\psi_{\alpha_1\alpha_2...\alpha_N}(x_1,x_2,...,x_N),\,\,\,\,\,\,a_{\alpha_1\alpha_2...\alpha_N}\in\mathbb{C}

\psi_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}}\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\frac{1}{\sqrt{N!}} \operatorname{det}\left(\begin{array}{ccc} \psi_{\alpha_{1}}\left(x_{1}\right) & \ldots & \psi_{\alpha_{N}}\left(x_{1}\right) \\ \vdots & \ddots & \vdots \\ \psi_{\alpha_{1}}\left(x_{N}\right) & \ldots & \psi_{\alpha_{N}}\left(x_{N}\right) \end{array}\right)

\begin{aligned} \psi_{\alpha_1\alpha_2...\alpha_N}(x_1,x_2,...,x_N)=\dfrac{1}{\sqrt{N!}}det\begin{pmatrix} \psi_{\alpha_1}(x_1) & \dots & \psi_{\alpha_N}(x_1)\\ \vdots & \ddots & \vdots \\ \psi_{\alpha_1}(x_N) & \dots & \psi_{\alpha_N}(x_N)\end{pmatrix} \end{aligned}

\left\langle n_{1 \sigma}, n_{2 \sigma}, \ldots \mid n_{1 \sigma}^{\prime}, n_{2 \sigma}^{\prime}, \ldots\right\rangle=\prod_{i, \sigma} \delta_{n_{i \sigma}, n_{i \sigma}^{\prime}}

\bracket{n_{1\sigma},n_{2\sigma},\dots}{n_{1\sigma}',n_{2\sigma}',\dots}=\prod_{i,\sigma}\delta_{n_{i\sigma},n_{i\sigma}'}

|\Psi\rangle=\sum_{n_{1 \sigma}, n_{2 \sigma}, \ldots} a_{n_{1 \sigma}, n_{2 \sigma}, \ldots}\left|n_{1 \sigma}, n_{2 \sigma}, \ldots\right\rangle, \quad \text { wobei } 2 N=\sum_{i, \sigma} n_{i, \sigma}

\left| \Psi \right>=\sum_{n_{1\sigma},n_{2\sigma},\dots}a_{n_{1\sigma},n_{2\sigma},\dots}\left| n_{1\sigma},n_{2\sigma},\dots \right>,\,\,\,\,\text{wobei}\,\, 2N=\sum_{i,\sigma}n_{i,\sigma}

\left.F^{(2 N)}=\mathcal{H}^{(+)}=\{|\Psi\rangle \in \mathcal{H}|\hat{\pi}| \Psi\rangle=(-1)^{\pi}|\Psi\rangle, \forall \pi \in S_{n}\right\}

F^{(2N)}=\mathcal{H}^{(+)}=\{\left| \Psi \right>\in\mathcal{H}\mid \hat{\pi}\left| \Psi \right>=(-1)^{\pi}\left| \Psi \right>,\forall\pi\in S_n\}

c_{i \sigma}^{\dagger}\left|n_{1 \sigma}, \ldots, n_{i \sigma}, \ldots\right\rangle=\left(1-n_{i \sigma}(-1)^{\sum_{\sigma, j<i} n_{j \sigma}}\left|n_{1 \sigma}, \ldots, n_{i \sigma}+1, \ldots\right\rangle\right.

c_{i\sigma}\left| n_{1\sigma},\dots,n_{i\sigma},\dots \right>=n_{i\sigma}(-1)^{\sum_{\sigma,j<i}2n_{j\sigma}}\left| n_{1\sigma},\dots,n_{i\sigma}-1,\dots \right>

\left|n_{1 \uparrow} n_{1 \downarrow} \ldots\right\rangle=\prod_{i, \sigma} \frac{1}{\sqrt{n_{i \sigma!}}}\left(c_{i, \sigma}^{\dagger}\right)^{n_{i \sigma}}|0\rangle

\left| n_{1\uparrow}n_{1\downarrow}\dots \right>=\prod_{i,\sigma}\dfrac{1}{\sqrt{n_{i\sigma!}}}(c\;\!\!^\dagger_{i,\sigma})^{n_{i\sigma}}\left| 0 \right>

c_{i \sigma}\left|n_{1 \sigma}, \ldots, n_{i \sigma}, \ldots\right\rangle=n_{i \sigma}(-1)^{\sum_{\sigma, j<i} 2 n_{j \sigma}}\left|n_{1 \sigma}, \ldots, n_{i \sigma}-1, \ldots\right\rangle

c\;\!\!^\dagger_{i\sigma}\left| n_{1\sigma},\dots,n_{i\sigma},\dots \right>=(1-n_{i\sigma}(-1)^{\sum_{\sigma,j<i}n_{j\sigma}}\left| n_{1\sigma},\dots,n_{i\sigma}+1,\dots \right>

\begin{aligned} {\left[c_{i \sigma}, c_{j \sigma^{\prime}}^{\dagger}\right] } & =\delta_{i j} \delta_{\sigma \sigma^{\prime}} \\ {\left[c_{i \sigma}, c_{j \sigma^{\prime}}\right] } & =0 \\ {\left[c_{i \sigma}^{\dagger}, c_{j \sigma^{\prime}}^{\dagger}\right] } & =0 \end{aligned}

\begin{aligned} [c_{i\sigma},c\;\!\!^\dagger_{j\sigma'}]=&\,\,\delta_{ij}\delta_{\sigma\sigma'}\\ [c_{i\sigma},c_{j\sigma'}]=&\,\,0\\ [c\;\!\!^\dagger_{i\sigma},c\;\!\!^\dagger_{j\sigma'}]=&\,\,0 \end{aligned}

\hat{n}_{i \sigma}=c_{i \sigma}^{\dagger} c_{i \sigma}

\hat{n}_{i\sigma}=c\;\!\!^\dagger_{i\sigma}c_{i\sigma}

\hat{n}_{i \sigma}\left|n_{1 \sigma}, \ldots, n_{i \sigma}, \ldots\right\rangle=n_{i \sigma}\left|n_{1 \sigma}, \ldots, n_{i \sigma}, \ldots\right\rangle

\hat{n}_{i\sigma}\left| n_{1\sigma},\dots,n_{i\sigma},\dots \right>=n_{i\sigma}\left| n_{1\sigma},\dots,n_{i\sigma},\dots \right>

\mathbf{H}_{\mathrm{Heis}}=-J \sum_{\langle i, j\rangle} \vec{S}_{i} \cdot \vec{S}_{j}

\displaystyle \mathbf{H}_{\text{Heis}}=-J\sum_{\langle i,j\rangle }\vec{S_{i}}\cdot \vec{S_{j}}

\vec{S}_{i} \cdot \vec{S}_{j}=S_{i}^{x} S_{j}^{x}+S_{i}^{y} S_{j}^{y}+S_{i}^{z} S_{j}^{z}

\displaystyle \vec{S_{i}}\cdot\vec{S_{j}} = S^x_i S^x_j+ S^y_iS^y_j+S^z_iS^z_j

\mathbf{H}_{\mathrm{Heis}}=-J \sum_{\langle i, j\rangle} \frac{\hat{S}_{i}^{+} \hat{S}_{j}^{-}}{2}+\frac{\hat{S}_{i}^{-} \hat{S}_{j}^{+}}{2}+\hat{S}_{i}^{z} \hat{S}_{j}^{z}

\displaystyle \mathbf{H}_{\text{Heis}}=-J\sum_{\langle i, j\rangle}\tfrac{\hat{S}^+_i\hat{S}^-_j}{2}+\tfrac{\hat{S}^-_i\hat{S}^+_j}{2}+\hat{S}^z_i\hat{S}^z_j

\mathbf{H}_{\mathrm{Hub}}=\mathbf{H}_{U}+\mathbf{H}_{\epsilon}+\mathbf{H}_{t}

\mathbf{H}_{\text{Hub}}=\mathbf{H}_U+\mathbf{H}_{\epsilon}+\mathbf{H}_t

\mathbf{H}_{\mathrm{Hub}}=U \sum_{i=1}^{N} \hat{c}_{i, \uparrow}^{\dagger} \hat{c}_{i, \uparrow} \hat{c}_{i, \downarrow}^{\dagger} \hat{c}_{i, \downarrow}+\sum_{\sigma}\left(\sum_{i=1}^{N} \epsilon_{i} \hat{c}_{i, \sigma}^{\dagger} \hat{c}_{i, \sigma}-t \sum_{\langle i, j\rangle}\left(\hat{c}_{i, \sigma}^{\dagger} \hat{c}_{j, \sigma}+\hat{c}_{j, \sigma}^{\dagger} \hat{c}_{i, \sigma}\right)\right)

\displaystyle \mathbf{H}_{\text{Hub}}=U\sum_{i=1}^N\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i,\uparrow}\hat{c}^{\dagger}_{i,\downarrow}\hat{c}_{i,\downarrow} +\sum_{\sigma}\left( \sum_{i=1}^{N}\epsilon_i \hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i,\sigma}-t\sum_{\langle i,j\rangle}( \hat{c}^{\dagger}_{i,\sigma}\hat{c}_{j,\sigma}+\hat{c}^{\dagger}_{j,\sigma}\hat{c}_{i,\sigma})\right)

\hat{S}_{i}^{+} \hat{S}_{j}^{-}|Z u s t a n d\rangle \equiv \operatorname{SpinPlus}(i, \operatorname{SpinMinus}(j, Z u s t a n d))

\hat{S}^+_i\hat{S}^-_j\left| Zustand \right>\equiv SpinPlus(i,SpinMinus(j,Zustand))

\left(\begin{array}{lll} 0 & \mathrm{~J} & 0 \\ 0 & 0 & \mathrm{~J} \\ \mathrm{~J} & 0 & 0 \end{array}\right)

\left( \begin{tabular}{ccc} 0 & J & 0 \\ 0 & 0 & J \\ J & 0 & 0 \end{tabular} \right)

\left\langle\hat{S}_{1} \hat{S}_{j}\right\rangle_{\psi_{g}}=\frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{\hat{S}_{1}^{+} \hat{S}_{j}^{-}}{2}+\frac{\hat{S}_{1}^{-} \hat{S}_{j}^{+}}{2}+\hat{S}_{1}^{z} \hat{S}_{j}^{z}\left|\psi^{0, l}\right\rangle

\langle \hat{S}_1\hat{S}_j\rangle_{\psi_g}=\dfrac{1}{g_0}\sum_{l=0}^{g_0}\left< \psi^{0,l} \right|\tfrac{\hat{S}^+_1\hat{S}^-_j}{2}+\tfrac{\hat{S}^-_1\hat{S}^+_j}{2}+\hat{S}^z_1\hat{S}^z_j\left| \psi^{0,l} \right>

\left|\psi_{g}\right\rangle=\sum_{k=1}^{2^{N}} \alpha_{k}|k\rangle \quad \text { Heisenberg-Modell }

\begin{aligned} \displaystyle \mid\psi_g\rangle=&\sum_{k=1}^{2^N}\alpha_k \mid k\rangle\quad\text{Heisenberg-Modell} \end{aligned}

\hat{n}_{\uparrow}|Z u s t a n d\rangle=\hat{c}_{i \uparrow}^{\dagger} \hat{c}_{j \uparrow}|Z u s t a n d\rangle \Leftrightarrow \operatorname{CupDagger}(i, \operatorname{Cup}(j, \text { Zustand }))

\hat{n}_{\uparrow}\left| Zustand \right>=\hat{c}_{i\uparrow}\;\!\!^\dagger\hat{c}_{j\uparrow}\left| Zustand \right> \Leftrightarrow CupDagger(i,Cup(j,Zustand))

\left(\begin{array}{lll} 0 & \mathrm{t} & 0 \\ 0 & 0 & \mathrm{t} \\ 0 & 0 & 0 \end{array}\right)

\left( \begin{tabular}{ccc} 0 & t & 0 \\ 0 & 0 & t \\ 0 & 0 & 0 \end{tabular} \right)

\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}^{\dagger}|0\rangle_{j}=|\uparrow \downarrow\rangle_{j} \quad \quad \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}^{\dagger}|0\rangle_{j}=-\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}^{\dagger}|0\rangle_{j}=-|\uparrow \downarrow\rangle_{j}

\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}^{\dagger}|0\rangle_{j}=|\uparrow \downarrow\rangle_{j} \quad \quad \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}^{\dagger}|0\rangle_{j}=-\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}^{\dagger}|0\rangle_{j}=-|\uparrow \downarrow\rangle_{j}

|0,0\rangle=[0,0] \quad|\downarrow, \downarrow\rangle=[1,1] \quad|\uparrow, \uparrow\rangle=[2,2] \quad|\uparrow \downarrow, \uparrow \downarrow\rangle=[3,3]

\left| 0,0 \right>=[0,0]\quad\left| \downarrow,\downarrow \right>=[1,1]\quad\left| \uparrow,\uparrow \right>=[2,2]\quad\left| \uparrow \downarrow,\uparrow\downarrow \right>=[3,3]

\begin{array}{rll} |0, \downarrow\rangle=[0,1] & \rightarrow & |\downarrow, 0\rangle=[1,0] \\ |\downarrow, 0\rangle=[1,0] & \rightarrow & |0, \downarrow\rangle=[0,1] \\ |0, \uparrow\rangle=[0,2] & \rightarrow & |\uparrow, 0\rangle=[2,0] \\ |\uparrow, 0\rangle=[2,0] & \rightarrow & |0, \uparrow\rangle=[0,2] \end{array}

\begin{array}{rll} |0, \downarrow\rangle=[0,1] & \rightarrow & |\downarrow, 0\rangle=[1,0] \\ |\downarrow, 0\rangle=[1,0] & \rightarrow & |0, \downarrow\rangle=[0,1] \\ |0, \uparrow\rangle=[0,2] & \rightarrow & |\uparrow, 0\rangle=[2,0] \\ |\uparrow, 0\rangle=[2,0] & \rightarrow & |0, \uparrow\rangle=[0,2] \end{array}

\begin{array}{lll} |\uparrow, \uparrow \downarrow\rangle=[1,3] & \rightarrow & |\uparrow \downarrow, \uparrow\rangle=[3,1] \\ |\downarrow, \downarrow \uparrow\rangle=[1,3] & \rightarrow & |\uparrow \downarrow, \downarrow\rangle=[3,2] \\ |\uparrow \downarrow, \uparrow\rangle=[3,1] & \rightarrow & |\uparrow, \uparrow \downarrow\rangle=[1,3] \\ |\uparrow \downarrow, \downarrow\rangle=[3,2] & \rightarrow & |\downarrow, \uparrow \downarrow\rangle=[2,3] \end{array}

\begin{array}{lll} |\uparrow, \uparrow \downarrow\rangle=[1,3] & \rightarrow & |\uparrow \downarrow, \uparrow\rangle=[3,1] \\ |\downarrow, \downarrow \uparrow\rangle=[1,3] & \rightarrow & |\uparrow \downarrow, \downarrow\rangle=[3,2] \\ |\uparrow \downarrow, \uparrow\rangle=[3,1] & \rightarrow & |\uparrow, \uparrow \downarrow\rangle=[1,3] \\ |\uparrow \downarrow, \downarrow\rangle=[3,2] & \rightarrow & |\downarrow, \uparrow \downarrow\rangle=[2,3] \end{array}

\begin{aligned} |\uparrow, \downarrow\rangle=[1,2] & \rightarrow & |\uparrow \downarrow, 0\rangle+|0, \uparrow \downarrow\rangle=[3,0]+[0,3] \\ |\downarrow, \uparrow\rangle=[2,1] & \rightarrow & -|\uparrow \downarrow, 0\rangle-|0, \uparrow \downarrow\rangle=-[3,0]-[0,3] \\ |\uparrow \downarrow, 0\rangle=[3,0] & \rightarrow & |\uparrow, \downarrow\rangle-|\downarrow, \uparrow\rangle=[2,1]-[1,2] \\ |0, \uparrow \downarrow\rangle=[0,3] & \rightarrow & |\uparrow, \downarrow\rangle-|\downarrow, \uparrow\rangle=[1,2]-[2,1] \end{aligned}

\begin{aligned} |\uparrow, \downarrow\rangle=[1,2] & \rightarrow & |\uparrow \downarrow, 0\rangle+|0, \uparrow \downarrow\rangle=[3,0]+[0,3] \\ |\downarrow, \uparrow\rangle=[2,1] & \rightarrow & -|\uparrow \downarrow, 0\rangle-|0, \uparrow \downarrow\rangle=-[3,0]-[0,3] \\ |\uparrow \downarrow, 0\rangle=[3,0] & \rightarrow & |\uparrow, \downarrow\rangle-|\downarrow, \uparrow\rangle=[2,1]-[1,2] \\ |0, \uparrow \downarrow\rangle=[0,3] & \rightarrow & |\uparrow, \downarrow\rangle-|\downarrow, \uparrow\rangle=[1,2]-[2,1] \end{aligned}

\mathbf{H}_{\epsilon}+\mathbf{H}_{U}=\sum_{i=1}^{N}\left(\epsilon_{i} \cdot\left(\hat{n}_{i, \uparrow}+\hat{n}_{i, \downarrow}\right)+U_{i} \cdot \hat{n}_{i, \uparrow} \hat{n}_{i, \downarrow}\right)

\mathbf{H}_{\epsilon}+\mathbf{H}_{U}=\sum_{i=1}^N\left(\epsilon_i\cdot(\hat{n}_{i,\uparrow}+\hat{n}_{i,\downarrow})+U_i\cdot\hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}\right)

\langle l| \mathbf{H}_{\epsilon}+\mathbf{H}_{U}|m\rangle= \begin{cases}\epsilon_{l m}+U_{l m} & \text { wenn } l=m \text { ist. } \\ 0 & \text { wenn } l \neq m \text { ist. }\end{cases}

\begin{aligned} \left< l \right|\,\mathbf{H}_{\epsilon}+\mathbf{H}_U\,\left| m \right>= \begin{cases} \epsilon_{lm}+U_{lm} & \quad \text{wenn $l=m$ ist.}\\ 0 & \quad \text{wenn $l\ne m$ ist.} \end{cases} \end{aligned}

\hat{S}_{i}^{+}=\hat{c}_{i, \uparrow}^{\dagger} \hat{c}_{i, \downarrow} \quad ; \quad \hat{S}_{i}^{-}=\hat{c}_{i, \downarrow}^{\dagger} \hat{c}_{i, \uparrow} \quad ; \quad \hat{S}_{i}^{z}=\frac{1}{2}\left(\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right)

\hat{S}_{i}^{+}=\hat{c}_{i, \uparrow}^{\dagger} \hat{c}_{i, \downarrow} \quad ; \quad \hat{S}_{i}^{-}=\hat{c}_{i, \downarrow}^{\dagger} \hat{c}_{i, \uparrow} \quad ; \quad \hat{S}_{i}^{z}=\frac{1}{2}\left(\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right)

\begin{aligned} C_{1 j}= & \frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{1}{2}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \downarrow} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \uparrow} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}\right) \\ & +\frac{1}{4}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow}-\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow}\right)\left(\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right)\left|\psi^{0, l}\right\rangle \\ = & \frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{1}{2}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \downarrow} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \uparrow} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}\right) \\ & +\frac{1}{4}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow} \cdot \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow} \cdot \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}-\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow} \cdot \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow} \cdot \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right)\left|\psi^{0, l}\right\rangle \end{aligned}

\begin{aligned} C_{1 j}= & \frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{1}{2}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \downarrow} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \uparrow} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}\right) \\ & +\frac{1}{4}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow}-\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow}\right)\left(\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right)\left|\psi^{0, l}\right\rangle \\ = & \frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{1}{2}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \downarrow} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \uparrow} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}\right) \\ & +\frac{1}{4}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow} \cdot \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow} \cdot \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}-\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow} \cdot \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow} \cdot \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right)\left|\psi^{0, l}\right\rangle \end{aligned}

\begin{aligned} \left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}}= & \frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{1}{2}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \downarrow} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \uparrow} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}\right) \\ & +\frac{1}{4}\left(\hat{n}_{1, \uparrow} \hat{n}_{j, \uparrow}-\hat{n}_{1, \uparrow} \hat{n}_{j, \downarrow}-\hat{n}_{1, \downarrow} \hat{n}_{j, \uparrow}+\hat{n}_{1, \downarrow} \hat{n}_{j, \downarrow}\right)\left|\psi^{0, l}\right\rangle \end{aligned}

\begin{aligned} \langle \hat{S}_i\hat{S}_j\rangle_{\psi_g}=&\,\dfrac{1}{g_0}\sum_{l=0}^{g_0}\left< \psi^{0,l} \right| \tfrac{1}{2}\left(\hat{c}^{\dagger}_{1,\uparrow}\hat{c}_{1,\downarrow}\hat{c}^{\dagger}_{j,\downarrow}\hat{c}_{j,\uparrow}+\hat{c}^{\dagger}_{1,\downarrow}\hat{c}_{1,\uparrow}\hat{c}^{\dagger}_{j,\uparrow}\hat{c}_{j,\downarrow}\right)\\ &\,+\tfrac14\left(\hat{n}_{1,\uparrow}\hat{n}_{j,\uparrow}-\hat{n}_{1,\uparrow}\hat{n}_{j,\downarrow}-\hat{n}_{1,\downarrow}\hat{n}_{j,\uparrow}+\hat{n}_{1,\downarrow}\hat{n}_{j,\downarrow} \right)\left| \psi^{0,l} \right> \end{aligned}

\left|\psi_{g}\right\rangle=\sum_{k=1}^{4^{N}} \alpha_{k}|k\rangle \quad \text { Hubbard-Modell }

\begin{aligned} \displaystyle \mid\psi_g\rangle=&\sum_{k=1}^{4^N}\alpha_k \mid k\rangle\quad\text{Hubbard-Modell} \end{aligned}