Editing
Weitere Eigenschaften der Dirac-Gleichung
(section)
Jump to navigation
Jump to search
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== ''Lorentz''-Transformation == Koordinaten <math>x{{'}^{\mu }}={{L}^{\mu }}_{\nu }{{x}^{\nu }}</math> 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> Ruhemasse ist dieselbe <math>m'=m</math> Selbe Ableitung der Dirac-Gleichung :<math>\gamma {{'}^{\nu }}={{\gamma }^{\nu }}</math> 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 Vergleich mit (1.57) <math>{{\left( {{L}^{-1}} \right)}^{\mu }}_{\nu }{{S}^{-1}}{{\gamma }^{\nu }}S={{\gamma }^{\mu }}</math> {{NumBlk|:| :<math>\Rightarrow {{S}^{-1}}{{\gamma }^{\alpha }}S={{L}^{\alpha }}_{\mu }{{\gamma }^{\mu }}</math> : |(1.58)|RawN=.}} Wenn (1.58) erfüllt ist, folgt relativistische Invarianz. * Konstriktion der Matrix S: Für kleine <math>\beta :=\frac{v}{c}\ll 1</math> {{NumBlk|:| <math>S\left( \beta \right)=\underline{\underline{1}}+\frac{\beta }{2}{{\gamma }^{1}}{{\gamma }^{0}}+O\left( {{\beta }^{2}} \right)=\left( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{matrix} \right)+\frac{\beta }{2}\left( \begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{matrix} \right)+O\left( {{\beta }^{2}} \right)</math> : |(1.59)|RawN=.}} Für beliebige ß durch Exponenten (wichtiger Trick, steckt natürlich tiefere Mathematik dahinter: Liegruppen, Lie-Algebra…) {{NumBlk|:| :<math>\left( {{\gamma }^{\mu }}{{k}_{\mu }}-m \right)\underbrace{\left( {{\gamma }^{\nu }}{{k}_{\nu }}+m \right)\left( \begin{align} & 0 \\ & 0 \\ & {{u}_{1}} \\ & {{u}_{2}} \\ \end{align} \right)}_{{{{\tilde{\phi }}}_{-}}}=0</math> :<math>\begin{align} & -{{{\tilde{\phi }}}_{-}}=-\left( E+m \right)\left( \begin{align} & {{u}_{1}} \\ & {{u}_{2}} \\ & 0 \\ & 0 \\ \end{align} \right)-{{k}_{x}}\left( \begin{matrix} 0 & {{\sigma }_{x}} \\ -{{\sigma }_{x}} & 0 \\ \end{matrix} \right)\left( \begin{align} & {{u}_{1}} \\ & {{u}_{2}} \\ & 0 \\ & 0 \\ \end{align} \right)-{{k}_{y}}... \\ & =-\left( \begin{align} & \underline{k}.\underline{\sigma }\left( \begin{align} & {{u}_{1}} \\ & {{u}_{2}} \\ \end{align} \right) \\ & \left( E+m \right)\left( \begin{align} & {{u}_{1}} \\ & {{u}_{2}} \\ \end{align} \right) \\ \end{align} \right) \end{align}</math> |(1.60)|RawN=.}} Berechnung <font color="#33FF99">'''''(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> |(1.61)|RawN=.}} * Kontinuitätsgleichung, Viererstromdichte (1.37) {{NumBlk|:|(Viererstromdichte{{FB|Viererstromdichte}}) :<math>{{j}^{\mu }}={{\Psi }^{+}}{{\gamma }^{0}}{{\gamma }^{\mu }}\Psi </math> : |(1.62)|RawN=.}} {{NumBlk|:|(Kontinuitätsgleichung{{FB|Kontinuitätsgleichung}}) :<math>{{\partial }_{\mu }}{{j}^{\mu }}=0</math> : |(1.63)|RawN=.}} Lorentz-Invarianz von <math>{{\partial }_{\mu }}{{j}^{\mu }}</math>: zeige <math>\partial {{'}_{\mu }}j{{'}^{\mu }}=0</math> wobei {{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=.}} {{NumBlk|:|Außerdem <font color="#3399FF">'''''(AUFGABE) </font>'''''''''''(Vierstrom transformiert sich wie kontravarianter Vektor)<math>j{{'}^{\mu }}={{L}^{\mu }}_{\nu }{{j}^{\nu }}</math> : |(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 :<math>{{\partial }_{\mu }}{{j}^{\mu }}</math>
Summary:
Please note that all contributions to testwiki are considered to be released under the Creative Commons Attribution (see
Testwiki:Copyrights
for details). If you do not want your writing to be edited mercilessly and redistributed at will, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource.
Do not submit copyrighted work without permission!
Cancel
Editing help
(opens in new window)
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Log in
Namespaces
Page
Discussion
English
Views
Read
Edit
Edit source
View history
More
Search
Navigation
Main page
Recent changes
Random page
Physikerwelt
Tools
What links here
Related changes
Special pages
Page information