Editing Weitere Eigenschaften der Dirac-Gleichung
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Latest revision | Your text | ||
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Wir starten von | Wir starten von | ||
{{NumBlk|:|<math>i{{\partial }_{t}}\Psi =\left( \underline{a}.\underline{\hat{p}}+\beta m \right)\Psi </math> | {{NumBlk|:| <math>i{{\partial }_{t}}\Psi =\left( \underline{a}.\underline{\hat{p}}+\beta m \right)\Psi </math> | ||
|(1.45)|RawN=.}} | |||
*# Kontinuitätsgleichung mit <math>{{\Psi }^{+}}</math>(1.45) und (1.45)+<math>\Psi </math> | |||
<math>\begin{align} | |||
& \mathfrak{i} {{\Psi }^{+}}\dot{\Psi }\quad ={{\Psi }^{+}}\left( \underline{\alpha }.\hat{\underline{p}}+\beta m \right)\Psi \\ | & \mathfrak{i} {{\Psi }^{+}}\dot{\Psi }\quad ={{\Psi }^{+}}\left( \underline{\alpha }.\hat{\underline{p}}+\beta m \right)\Psi \\ | ||
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\end{align}</math> | \end{align}</math> | ||
mit der {{FB|Wahrscheinlichkeitsdichte}} ρ und der {{FB|Wahrscheinlichkeitsstromdichte}} j<sub>k.</sub> | |||
{{NumBlk|:| | {{NumBlk|:| | ||
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: |(1.47)|RawN=.}} | : |(1.47)|RawN=.}} | ||
Die Wahrscheinlichkeitsdichte setzt sich aus den 4 Komponenten des Spinors <math>\Psi </math> zusammen. | |||
# | *# Lorentz-Invarianz | ||
Umdefinieren der Matrizen <math>{{\underline{\underline{\alpha }}}_{k}},\underline{\underline{\beta }}</math>als | Umdefinieren der Matrizen <math>{{\underline{\underline{\alpha }}}_{k}},\underline{\underline{\beta }}</math>als | ||
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-{{\sigma }_{k}} & 0 \\ | -{{\sigma }_{k}} & 0 \\ | ||
\end{matrix} \right)</math> |(1.48)|RawN=.}} | \end{matrix} \right)</math> | ||
: |(1.48)|RawN=.}} | |||
{{NumBlk|:| | {{NumBlk|:| | ||
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(z.B. <math>{{\gamma }^{k}}{{\gamma }^{j}}+{{\gamma }^{j}}{{\gamma }^{k}}=\beta {{\alpha }_{k}}\beta {{\alpha }_{j}}+\beta {{\alpha }_{j}}\beta {{\alpha }_{k}}\underbrace{=}_{1.32}-{{\alpha }_{k}}{{\beta }^{2}}{{\alpha }_{j}}-{{\alpha }_{j}}{{\beta }^{2}}{{\alpha }_{k}}=-2{{\delta }_{jk}}</math>) | (z.B. <math>{{\gamma }^{k}}{{\gamma }^{j}}+{{\gamma }^{j}}{{\gamma }^{k}}=\beta {{\alpha }_{k}}\beta {{\alpha }_{j}}+\beta {{\alpha }_{j}}\beta {{\alpha }_{k}}\underbrace{=}_{1.32}-{{\alpha }_{k}}{{\beta }^{2}}{{\alpha }_{j}}-{{\alpha }_{j}}{{\beta }^{2}}{{\alpha }_{k}}=-2{{\delta }_{jk}}</math>) | ||
<u>Relativistische Notation:</u> | |||
kontravarianter Vierervektor{{FB|Vierervektor}} mit Index oben | kontravarianter Vierervektor{{FB|Vierervektor}} mit Index oben | ||
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\end{matrix} \right)</math>. | \end{matrix} \right)</math>. | ||
---- | |||
* Invarianz von <math>{{x}_{\mu }}{{x}^{\mu }}</math>unter Lorentz-Transformationen: | * Invarianz von <math>{{x}_{\mu }}{{x}^{\mu }}</math>unter Lorentz-Transformationen: | ||
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(Hier ohne Vektorpotential, mit Vektorpotential A analog, vgl. Rollnik II) | (Hier ohne Vektorpotential, mit Vektorpotential A analog, vgl. Rollnik II) | ||
<u>''Lorentz''-Transformation</u> | |||
Koordinaten <math>x{{'}^{\mu }}={{L}^{\mu }}_{\nu }{{x}^{\nu }}</math> | Koordinaten <math>x{{'}^{\mu }}={{L}^{\mu }}_{\nu }{{x}^{\nu }}</math> | ||
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:<math>\gamma {{'}^{\nu }}={{\gamma }^{\nu }}</math> | :<math>\gamma {{'}^{\nu }}={{\gamma }^{\nu }}</math> | ||
---- | |||
Also muss gelten | Also muss gelten | ||
---- | |||
:<math>\left( \mathfrak{i} \gamma {{'}^{\nu }}\partial {{'}_{\nu }}-m' \right)\Psi '=0\Rightarrow \left( \mathfrak{i} {{\gamma }^{\nu }}{{\left( {{L}^{-1}} \right)}^{\mu }}_{\nu }{{\partial }_{\mu }}-m \right)S\Psi =0</math> | :<math>\left( \mathfrak{i} \gamma {{'}^{\nu }}\partial {{'}_{\nu }}-m' \right)\Psi '=0\Rightarrow \left( \mathfrak{i} {{\gamma }^{\nu }}{{\left( {{L}^{-1}} \right)}^{\mu }}_{\nu }{{\partial }_{\mu }}-m \right)S\Psi =0</math> | ||
---- | |||
Multiplikation von S<sup>-1</sup> von links | Multiplikation von S<sup>-1</sup> von links | ||
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|(1.60)|RawN=.}} | |(1.60)|RawN=.}} | ||
Berechnung <font color="# | Berechnung <font color="#FFFF00">'''''(AUFGABE)''''' </font>ergibt | ||
{{NumBlk|:| <math>S\left( \beta \right)=\cosh \frac{\beta }{2}+\sinh \left( \frac{\beta }{2} \right){{\underline{\underline{\gamma }}}^{1}}{{\underline{\underline{\gamma }}}^{0}}</math> | {{NumBlk|:| <math>S\left( \beta \right)=\cosh \frac{\beta }{2}+\sinh \left( \frac{\beta }{2} \right){{\underline{\underline{\gamma }}}^{1}}{{\underline{\underline{\gamma }}}^{0}}</math> | ||
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: |(1.64)|RawN=.}} | : |(1.64)|RawN=.}} | ||
{{NumBlk|:|Außerdem <font color="# | {{NumBlk|:|Außerdem <font color="#FFFF00">'''''(AUFGABE) | ||
</font>'''''''''''(Vierstrom transformiert sich wie kontravarianter Vektor)<math>j{{'}^{\mu }}={{L}^{\mu }}_{\nu }{{j}^{\nu }}</math> | </font>'''''''''''(Vierstrom transformiert sich wie kontravarianter Vektor)<math>j{{'}^{\mu }}={{L}^{\mu }}_{\nu }{{j}^{\nu }}</math> | ||
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:<math>\partial {{'}_{\mu }}j{{'}^{\mu }}=\underbrace{{{\left( {{L}^{-1}} \right)}^{\nu }}_{\mu }{{\partial }_{\nu }}{{L}^{\mu }}_{\alpha }}_{{{\delta }^{\nu }}_{\alpha }}{{j}^{\alpha }}={{\partial }_{\nu }}{{j}^{\nu }}=0</math> | :<math>\partial {{'}_{\mu }}j{{'}^{\mu }}=\underbrace{{{\left( {{L}^{-1}} \right)}^{\nu }}_{\mu }{{\partial }_{\nu }}{{L}^{\mu }}_{\alpha }}_{{{\delta }^{\nu }}_{\alpha }}{{j}^{\alpha }}={{\partial }_{\nu }}{{j}^{\nu }}=0</math> | ||
Lorentz-Invarianz von | |||
Lorentz-Invarianz von | |||
:<math>{{\partial }_{\mu }}{{j}^{\mu }}</math> | :<math>{{\partial }_{\mu }}{{j}^{\mu }}</math> |