Formula Report

Source: cocalc/akolbe/AKolbe-BA-May-2018/AKolbe-BA.lines.json  |  MathExpressions: 238  |  Equations: 61

Inline Math — MathExpression tiddlers (238)

Display Equations (61)

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AKolbe-BA_EQ0001_p007	\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}
AKolbe-BA_EQ0002_p008	\vec{J}=\vec{r} \times \vec{p}
AKolbe-BA_EQ0003_p008	\hat{J}=\frac{\mathrm{i}}{\hbar} \hat{r} \times \nabla
AKolbe-BA_EQ0004_p008	\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)
AKolbe-BA_EQ0005_p008	\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}
AKolbe-BA_EQ0006_p009	\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.
AKolbe-BA_EQ0007_p009	\left[\hat{J}_{i}, \hat{J}_{j}\right]=\mathrm{i} \hbar \sum_{k} \hat{J}_{k} \epsilon_{i j k}
AKolbe-BA_EQ0008_p009	\left[\hat{J}_{i}, \hat{J}^{2}\right]=0 \quad i=x, y, z
AKolbe-BA_EQ0009_p009	\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}
AKolbe-BA_EQ0010_p009	\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}
AKolbe-BA_EQ0011_p009	\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}
AKolbe-BA_EQ0012_p010	|s\rangle=\alpha|\uparrow\rangle+\beta|\downarrow\rangle=\binom{\alpha}{\beta}
AKolbe-BA_EQ0013_p010	\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)
AKolbe-BA_EQ0014_p010	\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}
AKolbe-BA_EQ0015_p010	\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}
AKolbe-BA_EQ0016_p011	\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}
AKolbe-BA_EQ0017_p011	\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)
AKolbe-BA_EQ0018_p011	\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}
AKolbe-BA_EQ0019_p011	\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}
AKolbe-BA_EQ0020_p011	\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
AKolbe-BA_EQ0021_p011	\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
AKolbe-BA_EQ0022_p012	\mathcal{H}=\mathcal{H}_{1} \otimes \mathcal{H}_{2} \otimes \ldots \otimes \mathcal{H}_{N-1} \otimes \mathcal{H}_{N}
AKolbe-BA_EQ0023_p012	|\psi\rangle=|\psi\rangle_{1} \otimes|\psi\rangle_{2} \otimes \ldots \otimes|\psi\rangle_{N-1} \otimes|\psi\rangle_{N}
AKolbe-BA_EQ0024_p012	\hat{\mathrm{A}}_{i}: \mathcal{H}_{i} \rightarrow \mathcal{H}_{i}
AKolbe-BA_EQ0025_p012	\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}}
AKolbe-BA_EQ0026_p013	\mathcal{H}=\mathcal{H}_{1}^{(1)} \otimes \mathcal{H}_{1}^{(2)} \otimes \ldots \otimes \mathcal{H}_{1}^{(N)}
AKolbe-BA_EQ0027_p013	\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}
AKolbe-BA_EQ0028_p013	\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}
AKolbe-BA_EQ0029_p013	\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}
AKolbe-BA_EQ0030_p013	\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)
AKolbe-BA_EQ0031_p014	\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}}
AKolbe-BA_EQ0032_p014	|\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}
AKolbe-BA_EQ0033_p014	\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\}
AKolbe-BA_EQ0034_p014	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.
AKolbe-BA_EQ0035_p014	\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
AKolbe-BA_EQ0036_p015	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
AKolbe-BA_EQ0037_p015	\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}
AKolbe-BA_EQ0038_p015	\hat{n}_{i \sigma}=c_{i \sigma}^{\dagger} c_{i \sigma}
AKolbe-BA_EQ0039_p015	\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
AKolbe-BA_EQ0040_p017	\mathbf{H}_{\mathrm{Heis}}=-J \sum_{\langle i, j\rangle} \vec{S}_{i} \cdot \vec{S}_{j}
AKolbe-BA_EQ0041_p017	\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}
AKolbe-BA_EQ0042_p017	\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}
AKolbe-BA_EQ0043_p019	\mathbf{H}_{\mathrm{Hub}}=\mathbf{H}_{U}+\mathbf{H}_{\epsilon}+\mathbf{H}_{t}
AKolbe-BA_EQ0044_p019	\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)
AKolbe-BA_EQ0045_p021	\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))
AKolbe-BA_EQ0046_p021	\left(\begin{array}{lll} 0 & \mathrm{~J} & 0 \\ 0 & 0 & \mathrm{~J} \\ \mathrm{~J} & 0 & 0 \end{array}\right)
AKolbe-BA_EQ0047_p021	\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
AKolbe-BA_EQ0048_p022	\left|\psi_{g}\right\rangle=\sum_{k=1}^{2^{N}} \alpha_{k}|k\rangle \quad \text { Heisenberg-Modell }
AKolbe-BA_EQ0049_p023	\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 }))
AKolbe-BA_EQ0050_p023	\left(\begin{array}{lll} 0 & \mathrm{t} & 0 \\ 0 & 0 & \mathrm{t} \\ 0 & 0 & 0 \end{array}\right)
AKolbe-BA_EQ0051_p023	\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}
AKolbe-BA_EQ0052_p023	|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]
AKolbe-BA_EQ0053_p023	\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}
AKolbe-BA_EQ0054_p024	\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}
AKolbe-BA_EQ0055_p024	\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}
AKolbe-BA_EQ0056_p024	\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)
AKolbe-BA_EQ0057_p024	\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}
AKolbe-BA_EQ0058_p025	\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)
AKolbe-BA_EQ0059_p025	\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}
AKolbe-BA_EQ0060_p025	\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}
AKolbe-BA_EQ0061_p025	\left|\psi_{g}\right\rangle=\sum_{k=1}^{4^{N}} \alpha_{k}|k\rangle \quad \text { Hubbard-Modell }

TikZ & Tables (13; 0 rendered to SVG)

#	type	caption	LaTeX source	SVG render
1	Diagram
2	Diagram	Abbildung 1.1: Darstellung einer Heisenberg-Kette mit Übernächster-Nachbar-Wechselwirkung und sieben Plätzen
3	Diagram	Abbildung 2.1: Ein Sketch zu dem Phänomen der Frustration.
4	Diagram	Abbildung 3.1: Eine lineare Heisenberg-Kette mit N Plätzen.
5	Diagram	Abbildung 3.2: Lineare Hubbard-Kette mit N Plätzen.
6	Diagram		``` function spin_state_generator(N) x=collect(1:N) states=[] for j in 0:2^N-1 stat=copy(digits!(x,j,2)) push!(states, stat) end return (states) end ```
7	Diagram		``` function SpinPlus(i,state) zustand=copy(state) if(state==0) zustand=0 else if(zustand[i]==0) zustand=0 elseif(zustand[i]==1) zustand[i]=0 end ```
8	Diagram		``` end return zustand end ```
9	Diagram		``` function fermion_state_generator(N) x=collect(1:N) states=[] for j in 0:4^N-1 stat=copy(digits!(x, j,4)) push!(states, stat) end return states end ```
10	Diagram		``` function CupDagger(i, vec) #0=0 2=up 1=down 3= updown states=copy (vec) if states==0 return 0 else if(states[i]==0) states[i]=2 elseif(states[i]==1) states[i]=3 else return 0 end end return states ```
11	Diagram		``` end ```
12	Diagram	Abbildung 5.3: Eine Schematische Darstellung einer Heisenberg-Kette mit Über-nächster Nachbar Wechselwirkung.
13	Diagram	Abbildung 5.6: Ein Sketch einer Heisenberg-Kette mit Über-nächster Nachbar Wechselwirkung und sechs Plätzen.