% RIO 2003,   da lucidi biondini ....19 sett. 2002
%  12 luglio 2003
%  stampare in DIPartimento 
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%  definizioni specifiche per Kane

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%introdotte da Biondini
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% fine Biondini
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%  nuove per KP
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% 
\pagestyle{empty}
\date{}
\begin{document}
%\newcommand{\df}{\buildrel\rm def\over =}
% 
\title{\vskip55pt\bf
Quantum hydrodynamical model for the
two-band Kane system in a tunneling diode }


% 
\begin{slide}
%\begin{small}
%equazioni del momenmto di ordine 0
\begin{itemize}
	\item {\bf Zero Order Moments Equations}
\end{itemize}

$$
\left\{
\begin{array}{rcl}
%Prima equazione
\displaystyle     \frac{\ppt \ncc }{\ppt t}  \hskip-2mm & \!=\! & \hskip-2mm 
\displaystyle     \! \frac{1}{q}\frac{\ppt J_{cc}}{\ppt x} 
\displaystyle   + i \frac{\hbar P}{2 m}\frac{\ppt }{\ppt x}(\ncv - \nvc) 
\displaystyle    +  \frac{P}{q}   (J_{cv} + J_{vc}) 
\\
\\
\displaystyle      \frac{\ppt \ncv }{\ppt t}  \hskip-2mm  &\!=\!& \hskip-2mm
\displaystyle     \frac{1}{q}    \frac{\ppt \jcv}{\ppt x} 
\displaystyle    + \frac{i}{\hbar} [V_c(x,t) - V_v(x,t)]\ncv
\displaystyle   -  i \frac{\hbar P}{2 m}\frac{\ppt }{\ppt x}(\ncc + \nvv)
\displaystyle    +  \frac{P}{q}  (J_{cc} - J_{vv}) 
\\
\\
%Quarta Equazione
\displaystyle     \frac{\ppt \nvv }{\ppt t}  \hskip-2mm & \!=\! & \hskip-2mm 
\displaystyle       \frac{1}{q}\frac{\ppt J_{vv}}{\ppt x} 
\displaystyle   +  i \frac{\hbar P}{2m}\frac{\ppt }{\ppt x}(\ncv - \nvc) 
\displaystyle    -    \frac{P}{q}   (J_{cv} + J_{vc}) 

\end{array}
     \right.
$$
%\end{small}

\end{slide}

% Lucido numero 4
\begin{slide}
%\begin{small}
%equazioni del momenmto di ordine 1
\begin{itemize}
	\item {\bf First Order Moments Equations} 
\end{itemize}
%Equazioni di momento 1 
%
$$
\left\{
\begin{array}{rcl}
%Prima equazione
\displaystyle \frac{\ppt J_{cc}}{\ppt t}  \hskip-2mm & \!=\! & \hskip-2mm 
\displaystyle   \frac{2 q}{m}\frac{\ppt \Ener_{cc}}{\ppt x} 
\displaystyle   +  \frac{q}{m}V^{'}_c(x,t) \ncc 
\displaystyle   + i \frac{\hbar P}{2 m}\frac{\ppt}{\ppt x}  (J_{cv}- J_{vc} 
\displaystyle   + \frac{2 q P}{m} (\Ener_{cv} + \Ener_{vc})
\\
\\
%Seconda equazione
\displaystyle      \frac{\ppt J_{cv}}{\ppt t}  \hskip-2mm & \!=\! & \hskip-2mm 
\displaystyle     \! \frac{2 q}{m}\frac{\ppt \Ener_{cv}}{\ppt x} 
\displaystyle    +\frac{q}{2m}[V^{'}_{c}(x,t) + V^{'}_{v}(x,t)] \ncv 
\\
\\
& & \phantom{}
\displaystyle   + i\frac{1}{\hbar}[V^{}_{c}(x,t) - V^{}_{v}(x,t)] J_{cv}
\displaystyle    - i \frac{\hbar P}{2 m} \frac{\ppt }{\ppt x}(J_{cc}+J_{vv})
\displaystyle   -  i\frac{  2 q P }{m}(\Ener_{cc} - \Ener_{vv}) 
\\
\\
%Quarta Equazione
\displaystyle  \frac{\ppt J_{vv}}{\ppt t}  \hskip-2mm & \!=\! & \hskip-2mm 
\displaystyle   \!  \frac{2 q}{m}\frac{\ppt \Ener_{vv}}{\ppt x} 
\displaystyle   +  \frac{q}{m}V^{'}_v(x,t) \nvv 
\displaystyle   + i \frac{\hbar P}{2 m}\frac{\ppt}{\ppt x}  (J_{cv} - J_{vc}) 
\displaystyle   - \frac{2 q P}{m} (\Ener_{cv} + \Ener_{vc})
\end{array}
     \right.
\label{eq:M1}
$$

%\end{small}

\end{slide}


% Lucido numero 5
\begin{slide}

%\begin{small}
%equazioni del momenmto di ordine 2
\begin{itemize}
	\item {\bf Second Order Moments Equations} 
\end{itemize}
%Equazioni di momento 2 
%
%%%\end{small}
%%%\begin{tiny}
$$
%\left\{
\begin{array}{rcl}
%Prima equazione
&&
\displaystyle   \frac{\ppt \Ener_{cc}}{\ppt t}  = % \hskip-2mm  =  \hskip-2mm 
\displaystyle    - \frac{m}{2}\, \frac{\ppt M^{(3)}_{cc}}{\ppt x} 
\displaystyle   + \frac{1}{q} V_{c}^{'}(x ,t) J_{cc}
\displaystyle   + i \frac{\hbar P}{2 m}\frac{\ppt}{\ppt x}  (\Ener_{cv} - \Ener_{vc})
\displaystyle   - \frac{m P}{2} [M^{(3)}_{vc} + M^{(3)}_{cv}] 
\\[5mm]
\\
%Seconda equazione
&&
\displaystyle     \frac{\ppt \Ener_{cv}}{\ppt t} = % \hskip-2mm  = \hskip-2mm 
\displaystyle   - \frac{m}{2} \,\frac{\ppt M^{(3)}_{cv}}{\ppt x} 
\displaystyle    +  i\frac{ \hbar}{8 m}[V^{''}_{v}(x,t)-V^{''}_{c}(x,t)] \ncv 
\displaystyle   +\frac{1}{2 q} [V^{'}_{v}(x,t)-V^{'}_{c}(x,t) ]
\\[4mm]
\\
& & \phantom{x1234}
\displaystyle   - i \frac{1}{\hbar}[V^{}_{v}(x,t)+V^{}_{c}(x,t)] \Ener_{cv}
\displaystyle   - i  \frac{\hbar P}{2 m}\frac{\ppt }{\ppt x}(\Ener_{vv}+\Ener_{cc})
\displaystyle   - i\frac{  m P }{2}(M^{(3)}_{vv}-M^{(3)}_{cc}) 
\\[5mm]
\\
%Quarta Equazione
&&
\displaystyle    \frac{\ppt \Ener_{vv}}{\ppt t} = % \hskip-2mm  =  \hskip-2mm 
\displaystyle    \phantom{+} \frac{m}{2}\, \frac{\ppt M^{(3)}_{vv}}{\ppt x} 
\displaystyle   - \frac{1 }{q } V_{v}^{'}(x ,t) J_{vv}
\displaystyle   - i \frac{\hbar P}{2 m}\frac{\ppt}{\ppt x}  (\Ener_{cv} - \Ener_{vc})
\displaystyle   -  \frac{m P}{2}  [M^{(3)}_{cv} + M^{(3)}_{vc}] 
\end{array}
 %    \right.
$$
%\end{small} % \end{tiny}

\end{slide}

\end{document}