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 7: | Line 7: | ||
:was auf die {{FB|Schrödingergleichung|freies Teilchen}} | :was auf die {{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. | :führt. | ||
Line 26: | Line 26: | ||
:mit | :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) \\ | ||
Line 36: | Line 36: | ||
Allerdings gilt | Allerdings gilt | ||
:<math>\begin{align} | :<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 47: | ||
* 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 {{FB|Compton-Wellenlänge}} als charakteristische Längenskala. |