Editing Weitere Eigenschaften der Dirac-Gleichung
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 3: | Line 3: | ||
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 \\ | ||
Line 23: | Line 25: | ||
\end{align}</math> | \end{align}</math> | ||
mit der {{FB|Wahrscheinlichkeitsdichte}} ρ und der {{FB|Wahrscheinlichkeitsstromdichte}} j<sub>k.</sub> | |||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\begin{align} | |||
& \rho :={{\Psi }^{+}}\Psi =\sum\limits_{k=1}^{4}{\Psi _{k}^{*}{{\Psi }_{k}}} \\ | & \rho :={{\Psi }^{+}}\Psi =\sum\limits_{k=1}^{4}{\Psi _{k}^{*}{{\Psi }_{k}}} \\ | ||
Line 39: | Line 41: | ||
{{NumBlk|:|(Kontinuitätsgleichung) | {{NumBlk|:|(Kontinuitätsgleichung) | ||
<math>\Rightarrow {{\partial }_{t}}\rho +\underline{\nabla }\underline{j}=0</math> | |||
: |(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 | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>{{\gamma }^{0}}:=\beta =\left( \begin{matrix} | |||
{\underline{\underline{1}}} & {\underline{\underline{0}}} \\ | {\underline{\underline{1}}} & {\underline{\underline{0}}} \\ | ||
Line 62: | Line 64: | ||
-{{\sigma }_{k}} & 0 \\ | -{{\sigma }_{k}} & 0 \\ | ||
\end{matrix} \right)</math> |(1.48)|RawN=.}} | \end{matrix} \right)</math> | ||
: |(1.48)|RawN=.}} | |||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\begin{align} | |||
& {{\left( {{\gamma }^{0}} \right)}^{+}}={{\gamma }^{0}},{{\left( {{\gamma }^{0}} \right)}^{2}}=1 \\ | & {{\left( {{\gamma }^{0}} \right)}^{+}}={{\gamma }^{0}},{{\left( {{\gamma }^{0}} \right)}^{2}}=1 \\ | ||
Line 80: | Line 84: | ||
(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 | ||
Line 87: | Line 90: | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>{{x}^{\mu }}\leftrightarrow \left( {{x}^{0}},{{x}^{1}},{{x}^{2}},{{x}^{3}} \right):=\left( ct,x,y,z \right)=\left( ct,\underline{x} \right)</math> | |||
: |(1.50)|RawN=.}} | : |(1.50)|RawN=.}} | ||
Line 95: | Line 98: | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>{{x}_{\mu }}=\left( {{x}_{0}},{{x}_{1}},{{x}_{2}},{{x}_{3}} \right):=\left( ct,-x,-y,-z \right)=\left( ct,-\underline{x} \right)</math> | |||
: |(1.51)|RawN=.}} | : |(1.51)|RawN=.}} | ||
Line 102: | Line 105: | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>{{x}_{\mu }}{{x}^{\mu }}=\sum\limits_{\mu =0}^{4}{{{x}_{\mu }}{{x}^{\mu }}={{c}^{2}}{{t}^{2}}-{{{\underline{x}}}^{2}}}</math> | |||
: |(1.52)|RawN=.}} | : |(1.52)|RawN=.}} | ||
Line 114: | Line 117: | ||
* Lorentz-Transformation wie in (1.11) (Bewegung in x-Richtung) | * Lorentz-Transformation wie in (1.11) (Bewegung in x-Richtung) | ||
* | * | ||
<math>\begin{align} | |||
& ct'=\gamma ct-\gamma \beta x \\ | & ct'=\gamma ct-\gamma \beta x \\ | ||
Line 137: | Line 140: | ||
\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: | ||
Line 149: | Line 154: | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\begin{align} | |||
& {{\partial }^{\nu }}=\frac{\partial }{\partial {{x}_{\nu }}}\quad \text{kontravarianter Vierergradient} \\ | & {{\partial }^{\nu }}=\frac{\partial }{\partial {{x}_{\nu }}}\quad \text{kontravarianter Vierergradient} \\ | ||
Line 161: | Line 166: | ||
Die Dirac-Gleichung folgt aus | Die Dirac-Gleichung folgt aus | ||
<math>\begin{align} | |||
& \left( \mathfrak{i} {{\partial }_{t}}-\underline{\alpha }\frac{1}{\mathfrak{i} }\underline{\nabla }-\beta m \right)\Psi =0\quad |\centerdot \beta \\ | & \left( \mathfrak{i} {{\partial }_{t}}-\underline{\alpha }\frac{1}{\mathfrak{i} }\underline{\nabla }-\beta m \right)\Psi =0\quad |\centerdot \beta \\ | ||
Line 171: | Line 176: | ||
{{NumBlk|:|{{FB|Dirac-Gleichung}} | {{NumBlk|:|{{FB|Dirac-Gleichung}} | ||
<math>\left( \mathfrak{i} {{\gamma }^{\mu }}{{\partial }_{\mu }}-m \right)\Psi =0</math> | |||
: |(1.56)|RawN=.|Border=1}} | : |(1.56)|RawN=.|Border=1}} | ||
Line 180: | Line 185: | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\left( \mathfrak{i} {{\gamma }^{\nu }}{{\partial }_{\nu }}-m \right)\Psi =0\ \left( \text{in S} \right)\quad \left( \mathfrak{i} \gamma {{'}^{\nu }}\partial {{'}_{\nu }}-m' \right)\Psi '=0\ \left( \text{in S }\!\!'\!\!\text{ } \right)</math> | |||
: |(1.57)|RawN=.}} | : |(1.57)|RawN=.}} | ||
Line 186: | Line 191: | ||
(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> | ||
Line 193: | Line 197: | ||
Ableitung | Ableitung | ||
<math>\partial {{'}_{\mu }}=\frac{\partial }{\partial x{{'}^{\mu }}}=\frac{\partial x{{'}^{\nu }}}{\partial {{x}^{\mu }}}\frac{\partial }{\partial {{x}^{\nu }}}={{\left( {{L}^{-1}} \right)}^{\nu }}_{\mu }{{\partial }_{\nu }}</math> | |||
Wellenfunktion (4er Spinor) <math>\Psi '\left( x' \right)=\underbrace{S}_{\in {{M}^{4x4}}}\Psi \left( x \right)</math> | Wellenfunktion (4er Spinor) <math>\Psi '\left( x' \right)=\underbrace{S}_{\in {{M}^{4x4}}}\Psi \left( x \right)</math> | ||
Line 201: | Line 205: | ||
Selbe Ableitung der Dirac-Gleichung | Selbe Ableitung der Dirac-Gleichung | ||
<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> | |||
---- | |||
Multiplikation von S<sup>-1</sup> von links | Multiplikation von S<sup>-1</sup> von links | ||
Line 215: | Line 224: | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\Rightarrow {{S}^{-1}}{{\gamma }^{\alpha }}S={{L}^{\alpha }}_{\mu }{{\gamma }^{\mu }}</math> | |||
: |(1.58)|RawN=.}} | : |(1.58)|RawN=.}} | ||
Line 249: | Line 258: | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\left( {{\gamma }^{\mu }}{{k}_{\mu }}-m \right)\underbrace{\left( {{\gamma }^{\nu }}{{k}_{\nu }}+m \right)\left( \begin{align} | |||
& 0 \\ | & 0 \\ | ||
Line 261: | Line 270: | ||
\end{align} \right)}_{{{{\tilde{\phi }}}_{-}}}=0</math> | \end{align} \right)}_{{{{\tilde{\phi }}}_{-}}}=0</math> | ||
<math>\begin{align} | |||
& -{{{\tilde{\phi }}}_{-}}=-\left( E+m \right)\left( \begin{align} | & -{{{\tilde{\phi }}}_{-}}=-\left( E+m \right)\left( \begin{align} | ||
Line 315: | Line 324: | ||
|(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> | ||
Line 324: | Line 333: | ||
{{NumBlk|:|(Viererstromdichte{{FB|Viererstromdichte}}) | {{NumBlk|:|(Viererstromdichte{{FB|Viererstromdichte}}) | ||
<math>{{j}^{\mu }}={{\Psi }^{+}}{{\gamma }^{0}}{{\gamma }^{\mu }}\Psi </math> | |||
: |(1.62)|RawN=.}} | : |(1.62)|RawN=.}} | ||
Line 330: | Line 339: | ||
{{NumBlk|:|(Kontinuitätsgleichung{{FB|Kontinuitätsgleichung}}) | {{NumBlk|:|(Kontinuitätsgleichung{{FB|Kontinuitätsgleichung}}) | ||
<math>{{\partial }_{\mu }}{{j}^{\mu }}=0</math> | |||
: |(1.63)|RawN=.}} | : |(1.63)|RawN=.}} | ||
Line 338: | Line 347: | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\partial {{'}_{\mu }}=\frac{\partial }{\partial x{{'}^{\mu }}}=\frac{\partial x{{'}^{\nu }}}{\partial {{x}^{\mu }}}\frac{\partial }{\partial {{x}^{\nu }}}={{\left( {{L}^{-1}} \right)}^{\nu }}_{\mu }{{\partial }_{\nu }}</math> | |||
: |(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> | ||
: |(1.65)|RawN=.}} | : |(1.65)|RawN=.}} | ||
<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> |