Diferencia entre revisiones de «Ondas: conservacion»
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| Línea 3: | Línea 3: | ||
Borowitz, Fundamentals of Quantum Mechanics, W.A. Benjamin, NY (1967), | Borowitz, Fundamentals of Quantum Mechanics, W.A. Benjamin, NY (1967), | ||
pp.71-74}} | pp.71-74}} | ||
<math>\frac{\partial ^2\psi }{\partial z^2}-\frac{1}{v^2}\frac{\partial ^2\psi | <math>\frac{\partial ^2\psi }{\partial z^2}-\frac{1}{v^2}\frac{\partial ^2\psi | ||
}{\partial t^2}=0</math> | }{\partial t^2}=0</math> | ||
multiply by | multiply by <math>\frac{\partial \psi }{\partial t}=\dot {\psi }</math>, then <math>\dot | ||
{\psi }\left( {\frac{\partial ^2\psi }{\partial | {\psi }\left( {\frac{\partial ^2\psi }{\partial | ||
z^2}-\frac{1}{v^2}\frac{\partial \dot {\psi }}{\partial t}} \right)=0 | z^2}-\frac{1}{v^2}\frac{\partial \dot {\psi }}{\partial t}} \right)=0</math>. | ||
Notice that | Notice that <math>\frac{\partial }{\partial t}\left( {\frac{\partial \psi | ||
}{\partial z}} \right)^2=2\frac{\partial \psi }{\partial z}\frac{\partial | }{\partial z}} \right)^2=2\frac{\partial \psi }{\partial z}\frac{\partial | ||
\dot {\psi }}{\partial z} | \dot {\psi }}{\partial z}</math> and <math>\frac{\partial }{\partial z}\left( {\dot | ||
{\psi }\frac{\partial \psi }{\partial z}} \right)=\dot {\psi }\frac{\partial | {\psi }\frac{\partial \psi }{\partial z}} \right)=\dot {\psi }\frac{\partial | ||
^2\psi }{\partial z^2}+\frac{\partial \psi }{\partial z}\frac{\partial \dot | ^2\psi }{\partial z^2}+\frac{\partial \psi }{\partial z}\frac{\partial \dot | ||
{\psi }}{\partial z} | {\psi }}{\partial z}</math> so that their difference yields <math>\textstyle{1 \over | ||
2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} | 2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} | ||
\right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial | \right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial | ||
\psi }{\partial z}} \right)=-\dot {\psi }\frac{\partial ^2\psi }{\partial | \psi }{\partial z}} \right)=-\dot {\psi }\frac{\partial ^2\psi }{\partial | ||
z^2} | z^2}</math>; whereas the second time derivative may be written as <math>\dot {\psi | ||
}\frac{\partial \dot {\psi }}{\partial t}=\textstyle{1 \over | }\frac{\partial \dot {\psi }}{\partial t}=\textstyle{1 \over | ||
2}\frac{\partial \dot {\psi }^2}{\partial t} | 2}\frac{\partial \dot {\psi }^2}{\partial t}</math>, so that the equation is: | ||
\textstyle{1 \over 2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi | <math>\textstyle{1 \over 2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi | ||
}{\partial z}} \right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi | }{\partial z}} \right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi | ||
}\frac{\partial \psi }{\partial z}} \right)+\frac{1}{v^2}\textstyle{1 \over | }\frac{\partial \psi }{\partial z}} \right)+\frac{1}{v^2}\textstyle{1 \over | ||
2}\frac{\partial \dot {\psi }^2}{\partial t}=0 | 2}\frac{\partial \dot {\psi }^2}{\partial t}=0</math> | ||
grouping temporal and spatial derivatives, we obtain a continuity type | grouping temporal and spatial derivatives, we obtain a continuity type | ||
equation: | equation: | ||
<math>\frac{\partial }{\partial t}\textstyle{1 \over 2}\left[ {\left( | |||
\frac{\partial }{\partial t}\textstyle{1 \over 2}\left[ {\left( | |||
{\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} | {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} | ||
\right]-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi | \right]-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi | ||
}{\partial z}} \right)=0 | }{\partial z}} \right)=0</math> | ||
and thus a density and flux may be defined as ${e}'=\textstyle{1 \over | and thus a density and flux may be defined as ${e}'=\textstyle{1 \over | ||
2}\left[ {\left( {\frac{\partial \psi }{\partial z}} | 2}\left[ {\left( {\frac{\partial \psi }{\partial z}} | ||
| Línea 43: | Línea 42: | ||
units of velocity) are translated into energy density and flow multiplying | units of velocity) are translated into energy density and flow multiplying | ||
by $\sigma =\rho v^2$, where $\rho $ is the mass per unit length. | by $\sigma =\rho v^2$, where $\rho $ is the mass per unit length. | ||
<math>e=\textstyle{1 \over 2}\rho v^2\left[ {\left( {\frac{\partial \psi | |||
e=\textstyle{1 \over 2}\rho v^2\left[ {\left( {\frac{\partial \psi | }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]</math> | ||
}{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right] | |||
<math>S=\rho v^2\dot {\psi }\frac{\partial \psi }{\partial z}</math> | |||
S=\rho v^2\dot {\psi }\frac{\partial \psi }{\partial z} | |||
the wave function may be real or complex. If it is complex, the energy and | the wave function may be real or complex. If it is complex, the energy and | ||
flow are also complex. | flow are also complex. | ||
Revisión del 14:52 10 sep 2007
\section{Energy of a wave according to Borowitz} \subsection{Energy and flow of real wave equation in one dimension\footnote{ S. Borowitz, Fundamentals of Quantum Mechanics, W.A. Benjamin, NY (1967), pp.71-74}}
multiply by , then . Notice that and so that their difference yields ; whereas the second time derivative may be written as , so that the equation is:
grouping temporal and spatial derivatives, we obtain a continuity type equation:
![\frac{\partial }{\partial t}\textstyle{1 \over 2}\left[ {\left(
{\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2}
\right]-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi
}{\partial z}} \right)=0](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d4a2b2ba63953ba1e839e4fff9032c313c78bdbb)
and thus a density and flux may be defined as ${e}'=\textstyle{1 \over 2}\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]$, ${S}'=\dot {\psi }\frac{\partial \psi }{\partial z}$. The dimensionless density or flow (with units of velocity) are translated into energy density and flow multiplying by $\sigma =\rho v^2$, where $\rho $ is the mass per unit length.
![e=\textstyle{1 \over 2}\rho v^2\left[ {\left( {\frac{\partial \psi
}{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d83d96a00fe0a82b15906265d97d73bd4971bc51)
the wave function may be real or complex. If it is complex, the energy and flow are also complex.
\section{Energy of a wave according to Borowitz} \subsection{Energy and flow of real wave equation in one dimension\footnote{ S. Borowitz, Fundamentals of Quantum Mechanics, W.A. Benjamin, NY (1967), pp.71-74}} \[</span> <span style="color: #606000;">\frac</span><span style="color: #00a000;">{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> ^2</span><span style="color: #606000;">\psi</span><span style="color: #00a000;"> }{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> z^2}-</span><span style="color: #606000;">\frac</span><span style="color: #00a000;">{1}{v^2}</span><span style="color: #606000;">\frac</span><span style="color: #00a000;">{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> ^2</span><span style="color: #606000;">\psi</span><span style="color: #00a000;"> </span> <span style="color: #00a000;">}{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> t^2}=0</span> <span style="color: #00a000;">\] multiply by $\frac{\partial \psi }{\partial t}=\dot {\psi }$, then $\dot {\psi }\left( {\frac{\partial ^2\psi }{\partial z^2}-\frac{1}{v^2}\frac{\partial \dot {\psi }}{\partial t}} \right)=0$. Notice that $\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} \right)^2=2\frac{\partial \psi }{\partial z}\frac{\partial \dot {\psi }}{\partial z}$ and $\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=\dot {\psi }\frac{\partial ^2\psi }{\partial z^2}+\frac{\partial \psi }{\partial z}\frac{\partial \dot {\psi }}{\partial z}$ so that their difference yields $\textstyle{1 \over 2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} \right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=-\dot {\psi }\frac{\partial ^2\psi }{\partial z^2}$; whereas the second time derivative may be written as $\dot {\psi }\frac{\partial \dot {\psi }}{\partial t}=\textstyle{1 \over 2}\frac{\partial \dot {\psi }^2}{\partial t}$, so that the equation is: \[</span> <span style="color: #606000;">\textstyle</span><span style="color: #00a000;">{1 </span><span style="color: #606000;">\over</span><span style="color: #00a000;"> 2}</span><span style="color: #606000;">\frac</span><span style="color: #00a000;">{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> }{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> t}</span><span style="color: #606000;">\left</span><span style="color: #00a000;">( {</span><span style="color: #606000;">\frac</span><span style="color: #00a000;">{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> </span><span style="color: #606000;">\psi</span><span style="color: #00a000;"> </span> <span style="color: #00a000;">}{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> z}} </span><span style="color: #606000;">\right</span><span style="color: #00a000;">)^2-</span><span style="color: #606000;">\frac</span><span style="color: #00a000;">{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> }{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> z}</span><span style="color: #606000;">\left</span><span style="color: #00a000;">( {</span><span style="color: #606000;">\dot</span><span style="color: #00a000;"> {</span><span style="color: #606000;">\psi</span><span style="color: #00a000;"> </span> <span style="color: #00a000;">}</span><span style="color: #606000;">\frac</span><span style="color: #00a000;">{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> </span><span style="color: #606000;">\psi</span><span style="color: #00a000;"> }{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> z}} </span><span style="color: #606000;">\right</span><span style="color: #00a000;">)+</span><span style="color: #606000;">\frac</span><span style="color: #00a000;">{1}{v^2}</span><span style="color: #606000;">\textstyle</span><span style="color: #00a000;">{1 </span><span style="color: #606000;">\over</span><span style="color: #00a000;"> </span> <span style="color: #00a000;">2}</span><span style="color: #606000;">\frac</span><span style="color: #00a000;">{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> </span><span style="color: #606000;">\dot</span><span style="color: #00a000;"> {</span><span style="color: #606000;">\psi</span><span style="color: #00a000;"> }^2}{</span><span style="color: #606000;">\partial</span><span style="color: #00a000;"> t}=0</span> <span style="color: #00a000;">\] grouping temporal and spatial derivatives, we obtain a continuity type equation: \begin{equation} \label{eq1} \frac{\partial }{\partial t}\textstyle{1 \over 2}\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=0 \end{equation} and thus a density and flux may be defined as ${e}'=\textstyle{1 \over 2}\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]$, ${S}'=\dot {\psi }\frac{\partial \psi }{\partial z}$. The dimensionless density or flow (with units of velocity) are translated into energy density and flow multiplying by $\sigma =\rho v^2$, where $\rho $ is the mass per unit length. \begin{equation} \label{eq2} e=\textstyle{1 \over 2}\rho v^2\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right] \end{equation} \begin{equation} \label{eq3} S=\rho v^2\dot {\psi }\frac{\partial \psi }{\partial z} \end{equation} the wave function may be real or complex. If it is complex, the energy and flow are also complex.
