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Physik Physik Dynamik • Grundlagen • Subatomare Teilchen • Kovariante und Kontravariante Komponenten • Ricci-Kalkül • Wichtige Tensoren Mathematische Grundlagen Mathematik für ET Semester 1 • Semester 2 • Semester 3 Mathematik Kombinatorik und Permutationen • Ableitung • Dreiecksnotation • Gradient, Divergenz, Rotation • Metrischer Tensor • Christoffelsymbole • Krümmungstensor • Energie-Impuls-Tensor • Kosmologische Konstante Quantenmechanik mit Matrizen Imaginäre und komplexe Zahlen • Vektoren • Matrizen • Jacobi-Matrizen • Pauli-Matrizen • Hermitesche Matrizen Lineare Algebra und Tensoren Tensornotation • Tensoren • Dirac-Notation • Körper • Vektorraum Gruppentheorie Gruppen, Lie Gruppen, Lie Algebren Quantenphysik Spin und Magnetisches Moment • Quantenoperatoren • Schrödingergleichung Quantenfeldtheorie Einleitung Spezielle Relativitätstheorie • Fouriertransformationen • Elektromagnetismus Harmonische Oszillatoren Lagrange-Dichten Fermatsches Prinzip • Newtonsche Gesetze • Funktionale • Lagrange-Dichten und kleinste Wirkung Einfache Harmonische Oszillatoren  • Besetzungszahldarstellung  • Anwendung der Zweiten Quantisierung  • Spezielle Relativitätstheorie Viervektoren Elektrisches Dipolmoment $p=(E,\mathbf {p} )$ Elektrische Stromdichte $j=(p,\mathbf {j} )$ Magnetisches Vektorpotential $A=(V,\mathbf {A} )$ Ableitung $\partial _{\mu }\equiv \left({\frac {\partial }{\partial \,x^{\mu }}}\right)_{\mu }=\left({\frac {\partial }{\partial \,t}},\mathbf {\nabla } \right)_{\mu }=\left({\frac {\partial }{\partial \,t}},{\frac {\partial }{\partial \,x}},{\frac {\partial }{\partial \,y}},{\frac {\partial }{\partial \,z}}\right)_{\mu }$ Metrischer Tensor $g_{\mu \nu }={\begin{cases}1&{\text{if }}\mu =\nu =0\\-\delta _{\mu }^{\nu }&{\text{if }}(\mu \neq 0)\lor (\nu \neq 0)\end{cases}}$ $g_{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}$ Hoch- und Tiefstellen von Indizes: $a_{\mu }=g_{\mu \nu }\,a^{\nu }$ $a_{0}=a^{0}$ $a_{i}=-a^{i}$ Kronecker-Delta $\delta _{\mu }^{\nu }={\begin{cases}0{\text{ if }}\mu \neq \nu \\1{\text{ if }}\mu =\nu \end{cases}}$ $\delta _{\mu \nu }=g_{\mu \xi }\,\delta _{\nu }^{\xi }=g_{\nu \mathrm {o} }\,g_{\mu \xi }\,\delta ^{\xi \mathrm {o} }=\delta ^{\mu \nu }$ Inneres Produkt $a\cdot b=a^{0}\,b^{0}-\mathbf {a} \cdot \mathbf {b} =g_{\mu \nu }\,a^{\mu }\,b^{\nu }=a_{\mu }\,b^{\nu }=a^{\mu }\,b_{\nu }$ Beispiele $p\cdot p=p_{\mu }\,p^{\mu }=(E,\mathbf {p} )\cdot (E,\mathbf {p} )=E^{2}-\mathbf {p} ^{2}=m^{2}$ $\partial _{\mu }=x^{\nu }={\dfrac {\partial x^{\nu }}{\partial x^{\mu }}}=\delta _{\mu }^{\nu }=4$ (wg. Summenkonvention) d’Alembert-Operator: $\square =\partial ^{2}=\partial _{\mu }^{\mu }={\dfrac {\partial ^{2}}{\partial (ct)^{2}}}-{\dfrac {\partial ^{2}}{\partial x^{2}}}-{\dfrac {\partial ^{2}}{\partial y^{2}}}-{\dfrac {\partial ^{2}}{\partial z^{2}}}={\dfrac {\partial ^{2}}{c^{2}\partial t^{2}}}-{\boldsymbol {\nabla }}^{2}={\dfrac {\partial ^{2}}{c^{2}\partial t^{2}}}-{\boldsymbol {\Delta }}$ Koordinatentransformation Kontravariante Transformation Transformation von Vektor $r$ von Inertialsystem (Basis) ${\mathcal {S}}$ nach ${\mathcal {S'}}$ $r'^{\mu }=\left({\frac {\partial \,{\mathcal {S'}}^{\mu }}{\partial \,{\mathcal {S}}^{\nu }}}\right)\,r^{\nu }$ Kovariante Transformation Transformation von Gradientvektor ${\frac {\partial \,\phi }{\partial \,r^{\mu }}}\equiv \partial \,\phi ^{\mu }$ von Inertialsystem (Basis) ${\mathcal {S}}$ nach ${\mathcal {S'}}$ ${\frac {\partial \,\phi }{\partial \,{\mathcal {S'}}^{\mu }}}=\left({\frac {\partial \,{\mathcal {S}}^{\mu }}{\partial \,{\mathcal {S'}}^{\nu }}}\right)\,{\frac {\partial \,\phi }{\partial \,{\mathcal {S}}^{\nu }}}$ Lorentz-Transformation ${\begin{pmatrix}ct'\\r'^{i}\end{pmatrix}}={\begin{pmatrix}\gamma &-\gamma \beta _{i}\\-\gamma \beta _{j}&\delta _{ij}+(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}\\\end{pmatrix}}{\begin{pmatrix}ct\\r^{i}\end{pmatrix}}$ mit $\gamma ={\frac {1}{\left(1-\left(\mathbf {\beta } \right)^{2}\right)^{\frac {1}{2}}}}$ $\mathbf {\beta } ^{i}={\frac {\mathbf {v} ^{i}}{c}}$ $c=1$ als Tensor $r'^{\mu }={\Lambda ^{\mu }}_{\nu }\,r^{\nu }$ mit ${\Lambda ^{\mu }}_{\nu }={\frac {\partial \,{\mathcal {S}}^{\mu }}{\partial \,{\mathcal {S'}}^{\nu }}}$ als Matrix $\left[{\Lambda ^{\mu }}_{\nu }\right]_{\mathcal {S}}={\begin{pmatrix}\gamma &-\beta _{1}\,\gamma &-\beta _{2}\,\gamma &-\beta _{3}\,\gamma \\-\beta _{1}\,\gamma &\delta _{11}+\left(\gamma -1\right)\,{\dfrac {\beta _{1}\,\beta _{1}}{(\beta )^{2}}}&\delta _{12}+\left(\gamma -1\right)\,{\dfrac {\beta _{1}\,\beta _{2}}{(\beta )^{1}}}&\delta _{13}+\left(\gamma -1\right)\,{\dfrac {\beta _{1}\,\beta _{3}}{(\beta )^{2}}}\\-\beta _{2}\,\gamma &\delta _{21}+\left(\gamma -1\right)\,{\dfrac {\beta _{2}\,\beta _{1}}{(\beta )^{2}}}&\delta _{22}+\left(\gamma -1\right)\,{\dfrac {\beta _{2}\,\beta _{2}}{(\beta )^{2}}}&\delta _{23}+\left(\gamma -1\right)\,{\dfrac {\beta _{2}\,\beta _{3}}{(\beta )^{2}}}\\-\beta _{3}\,\gamma &\delta _{31}+\left(\gamma -1\right)\,{\dfrac {\beta _{3}\,\beta _{1}}{(\beta )^{2}}}&\delta _{32}+\left(\gamma -1\right)\,{\dfrac {\beta _{3}\,\beta _{2}}{(\beta )^{2}}}&\delta _{33}+\left(\gamma -1\right)\,{\dfrac {\beta _{3}\,\beta _{3}}{(\beta )^{2}}}\end{pmatrix}}$ mit $\Lambda _{00}=\gamma$ $\Lambda _{0i}=\Lambda _{i0}=-\gamma \beta _{i}$ $\Lambda _{ij}=\Lambda _{ji}=\delta _{ij}+(\gamma -1){\dfrac {\beta _{i}\beta _{j}}{\beta ^{2}}}$ Tranformation auf allgemeinem Tensor ${T'}_{l'\ldots n'}^{i'\ldots k'}={\dfrac {\partial {x'}^{i'}}{\partial {x}^{i}}}\cdots {\dfrac {\partial {x'}^{k'}}{\partial {x}^{k}}}\ {\dfrac {\partial {x}^{l}}{\partial {x'}^{l'}}}\cdots {\dfrac {\partial {x}^{n}}{\partial {x'}^{n'}}}\ T_{l\ldots n}^{i\ldots k}$ Beispiel ${\delta '}_{i}^{j}={\dfrac {\partial {x'}^{i}}{\partial {x}^{k}}}\ {\dfrac {\partial {x}^{l}}{\partial {x'}^{j}}}\ {\delta }_{k}^{l}={\dfrac {\partial {x'}^{i}}{\partial {x}^{k}}}\ {\dfrac {\partial {x}^{k}}{\partial {x'}^{j}}}={\delta }_{i}^{j}$ Levi-Civita-Symbol $\varepsilon ^{i_{1}\dots i_{n}}=\prod _{1\leq p<q\leq n}{\frac {i_{p}-i_{q}}{p-q}}$ Anwendung: $(\mathbf {b} \times \mathbf {c} )^{i}=\epsilon ^{ijk}\,b^{j}\,c^{k}$ $\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )^{i}=\epsilon ^{ijk}\,a^{i}\,b^{j}\,c^{k}$ $\det \mathbf {A} =\epsilon _{i_{1}\,i_{2}\,\ldots \,i_{n}}\,A_{1\,i_{1}}\,A_{2\,i_{2}}\,\ldots \,A_{n\,i_{n}}$ Beispiel $i$ $j$ $\varepsilon ^{ij}$ 0 1 +1 1 0 -1 1 1 0 0 0 0 $i$ $j$ $k$ $\varepsilon ^{ijk}$ 0 1 2 +1 1 0 2 -1 1 2 0 +1 2 1 0 -1 2 0 1 +1 0 2 1 -1 sonst. 0 $i$ $j$ $k$ $l$ $\varepsilon ^{ijkl}$ 0 1 2 3 +1 1 0 2 3 -1 1 2 0 3 +1 1 2 3 0 -1 2 1 3 0 +1 2 3 1 0 -1 2 3 0 1 +1 3 2 0 1 -1 3 0 2 1 +1 3 0 1 2 -1 0 3 1 2 +1 0 1 3 2 -1 sonst. 0