Editing Klein Gordon 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 1: | Line 1: | ||
<noinclude>{{ScriptProf|Kapitel=1|Abschnitt=1|Prof= | <noinclude>{{ScriptProf|Kapitel=1|Abschnitt=1|Prof=Brandes|Thema=Quantenmechanik|Schreiber=Moritz Schubotz}}</noinclude> | ||
Ein quantenmechanisches {{FB|Wellenpaket}} hat die Form | |||
{{NumBlk|:|<math>\Psi \left( \underline{x},t \right)={{\left( 2\pi \right)}^{-{}^{d}\!\!\diagup\!\!{}_{2}\;}}\int{\varphi \left( \underline{k} \right){{e}^{-\mathfrak{i} \omega \left( \underline{k} \right)t+\mathfrak{i} \underline{k}.\underline{x}}}{{d}^{d}}\underline{k}}</math> | <FONT COLOR="#FFBF00">'''LITERATUR: SKRIPT FREDENHAGEN QMII, HAMBURG'''</FONT> | ||
Ein quantenmechanisches Wellenpaket{{FB|Wellenpaket}} hat die Form | |||
{{NumBlk||Nach Schrödinger (nicht relativistisch) <math>\omega \left( \underline{k} \right)=\frac{{{k}^{2}}}{2m}\quad \text{mit }\hbar =1</math>|(1.2)}} | |||
{{NumBlk|:| | |||
<math>\Psi \left( \underline{x},t \right)={{\left( 2\pi \right)}^{-{}^{d}\!\!\diagup\!\!{}_{2}\;}}\int{\varphi \left( \underline{k} \right){{e}^{-\mathfrak{i} \omega \left( \underline{k} \right)t+\mathfrak{i} \underline{k}.\underline{x}}}{{d}^{d}}\underline{k}}</math> | |||
: |(1.1)}} | |||
wobei d die Raumdimension angibt. | |||
{{NumBlk|:|Nach Schrödinger (nicht relativistisch) | |||
<math>\omega \left( \underline{k} \right)=\frac{{{k}^{2}}}{2m}\quad \text{mit }\hbar =1</math> | |||
: |(1.2)}} | |||
was auf die Schrödingergleichung{{FB|Schrödingergleichung:freies Teilchen}} | |||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\mathfrak{i} {{\partial }_{t}}\Psi =\hat{H}\Psi ,\quad \hat{H}=-\frac{\Delta }{2m}</math> | |||
: |(1.3)}} | : |(1.3)}} | ||
führt. | |||
Relativistisch (SRT) gilt | Relativistisch (SRT) gilt | ||
{{NumBlk|:| <math>\omega \left( \underline{k} \right)=\sqrt{{{{\underline{k}}}^{2}}+{{m}^{2}}}</math> |(1.4)}} | {{NumBlk|:| <math>\omega \left( \underline{k} \right)=\sqrt{{{{\underline{k}}}^{2}}+{{m}^{2}}}</math> |(1.4)}} | ||
< | wegen <math>E=\sqrt{{{m}^{2}}{{c}^{4}}+{{{\underline{p}}}^{2}}{{c}^{2}}}</math> und <math>\underline{p}=\hbar k</math>. | ||
Ab jetzt gilt <math>c=1</math>. | |||
Mit (1.4) erfüllt Ψ jetzt die {{FB|Klein-Gordon-Gleichung}}: | Mit (1.4) erfüllt Ψ jetzt die {{FB|Klein-Gordon-Gleichung}}: | ||
: |(1.5)| | {{NumBlk|:|Klein-Gordon-Gleichung | ||
<math>\left( \partial _{t}^{2}-\Delta +{{m}^{2}} \right)\Psi \left( \underline{x},t \right)=0</math> | |||
: |(1.5)}} | |||
Es gilt die <font color="#FFFF00">'''''(AUFGABE)'''''</FONT> | |||
{{NumBlk|:|Kontinuitätsgleichung{{FB|Kontinuitätsgleichung}} | |||
<math>{{\partial }_{t}}\rho +\nabla .\underline{j}=0</math> | |||
: |(1.6)}} | |||
mit | |||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\begin{align} | |||
& \underline{j}=\frac{1}{2\mathfrak{i} m}\left( {{\Psi }^{*}}\nabla \Psi -\Psi \nabla {{\Psi }^{*}} \right) \\ | & \underline{j}=\frac{1}{2\mathfrak{i} m}\left( {{\Psi }^{*}}\nabla \Psi -\Psi \nabla {{\Psi }^{*}} \right) \\ | ||
& \rho \equiv \frac{1}{2m}\left( {{\Psi }^{*}}{{\partial }_{t}}\Psi -\Psi {{\partial }_{t}}{{\Psi }^{*}} \right) \\ | & \rho \equiv \frac{1}{2m}\left( {{\Psi }^{*}}{{\partial }_{t}}\Psi -\Psi {{\partial }_{t}}{{\Psi }^{*}} \right) \\ | ||
\end{align}</math> | \end{align}</math> | ||
: |(1.7)}} | : |(1.7)}} | ||
Line 35: | Line 70: | ||
Allerdings gilt | Allerdings gilt | ||
<math>\begin{align} | |||
& \int{\rho \left( \underline{x},t \right){{d}^{d}}\underline{x}}={{\left( \frac{1}{2\pi } \right)}^{d}}\frac{1}{m}\int{\int{\int{{{\varphi }^{*}}\left( {\underline{k}} \right)\varphi \left( {{\underline{k}}'} \right){{e}^{i\left( \underline{k}-{\underline{k}}' \right)\underline{x}}}\omega \left( {{\underline{k}}'} \right){{d}^{d}}x}{{d}^{d}}k}{{d}^{d}}{k}'} \\ | & \int{\rho \left( \underline{x},t \right){{d}^{d}}\underline{x}}={{\left( \frac{1}{2\pi } \right)}^{d}}\frac{1}{m}\int{\int{\int{{{\varphi }^{*}}\left( {\underline{k}} \right)\varphi \left( {{\underline{k}}'} \right){{e}^{i\left( \underline{k}-{\underline{k}}' \right)\underline{x}}}\omega \left( {{\underline{k}}'} \right){{d}^{d}}x}{{d}^{d}}k}{{d}^{d}}{k}'} \\ | ||
& =\frac{1}{m}\int{\omega \left( {\underline{k}} \right){{\left| \varphi \left( {\underline{k}} \right) \right|}^{2}}{{d}^{d}}\underline{k}}>0 | & =\frac{1}{m}\int{\omega \left( {\underline{k}} \right){{\left| \varphi \left( {\underline{k}} \right) \right|}^{2}}{{d}^{d}}\underline{k}}>0 | ||
\end{align}</math> für<math>\omega \left( {\underline{k}} \right)>0</math>. | \end{align}</math> für<math>\omega \left( {\underline{k}} \right)>0</math>. | ||
Line 45: | Line 82: | ||
* Schreibweise | * Schreibweise | ||
{{NumBlk|:| | {{NumBlk|:| | ||
<math>\left( \square +\frac{{{m}^{2}}{{c}^{2}}}{{{\hbar }^{2}}} \right)\Psi =0</math> | |||
: |(1.8)}} | : |(1.8)}} | ||
mit <math>\frac{\hbar }{mc}</math>der {{FB|Compton-Wellenlänge}} als charakteristische Längenskala. | mit <math>\frac{\hbar }{mc}</math>der <u>Compton-Wellenlänge{{FB|Compton-Wellenlänge}}</u> als charakteristische Längenskala. | ||
Hier ist <math>\square ={{\partial }_{\mu }}{{\partial }^{\mu }}={{c}^{-2}}\partial _{t}^{2}-\Delta </math> der {{FB|d’Alambert-Operator}}. | Hier ist <math>\square ={{\partial }_{\mu }}{{\partial }^{\mu }}={{c}^{-2}}\partial _{t}^{2}-\Delta </math> der d’Alambert-Operator{{FB|d’Alambert-Operator}}. | ||