% rio2.tex
% 3 luglio 2003 - 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[a4, 12pt]{slides} %\documentclass[12pt]{article}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage[dvips,landscape]{geometry}
%
\topmargin-2.0cm       %-2
%\textwidth15.5cm   %15.5
\textheight 16.0cm  
%\oddsidemargin0.2cm        %0.5cm
%\topmargin-0.4cm    
\textheight23.9cm  
\oddsidemargin+0.9cm
%
\parindent0mm
\parskip2mm
\newcommand{\p}{\partial}
\newcommand{\pt}{\partial}
\newcommand{\nn}{\nonumber}
\renewcommand{\Im}{\hbox{ Im }}
\renewcommand{\Re}{\hbox{ Re }}
\newcommand{\itemitem}{\par\indent \hangindent2\parindent \textindent}

\newcommand{\za}{\alpha}
\newcommand{\zd}{\delta}
\newcommand{\zt}{\tau}
\newcommand{\ze}{\epsilon}
\newcommand{\zb}{\beta}
\newcommand{\zl}{\lambda}
\newcommand{\zs}{\sigma}
\newcommand{\zf}{\varphi}
\newcommand{\e}{\epsilon}
%
\newcommand{\baf}{\bar\zf}
\newcommand{\fibar}{\bar \varphi}
\newcommand{\psibar}{\bar \psi}
\newcommand{\fitil}{\tilde \varphi}
\newcommand{\psitil}{\tilde \psi}
\newcommand{\inti}{\int\limits_{-\infty}^{+\infty}}
\newcommand{\px}{\frac{\partial}{\partial x}}
\newcommand{\psx}{\frac{{\partial}^2}{\partial {x^2}}}
\newcommand{\prt}{\frac{\partial}{\partial t}}
%
%  definizioni specifiche per Kane
\newcommand{\ppt}{\partial}
%
\newcommand{\nc}{n_{c}}   %%%%%%%%%%%%%%%%%%%%%%%     PSI 
\newcommand{\nv}{n_{v}}
\newcommand{\jc}{J_{c}}
\newcommand{\jv}{J_{v}}
\newcommand{\jcc}{J_{c}}
\newcommand{\jvv}{J_{v}}
\newcommand{\jcv}{J_{cv}}
\newcommand{\jvc}{J_{vc}}
\newcommand{\psic}{\psi_{c}}
\newcommand{\psiv}{\psi_{v}}
\newcommand{\psicp}{\psi_{c}^\prime}
\newcommand{\psivp}{\psi_{v}^\prime}
\newcommand{\psicb}{\overline{\psi}_{c}}
\newcommand{\psivb}{\overline{\psi}_{v}}
\newcommand{\psicbp}{{\overline{\psi^\prime}}_{c}}
\newcommand{\psivbp}{\overline{\psi_{v}^\prime}}


\newcommand{\rdnc}{\sqrt{n_{c}}}
\newcommand{\rdnv}{\sqrt{n_{v}}}
\newcommand{\rdncv}{\sqrt{n_{c} \, n_{v}}}
\newcommand{\eisc}{{\rm e}^{i \frac{S_c (x,t)}{\ze} }}
\newcommand{\eisv}{{\rm e}^{i \frac{S_v (x,t)}{\ze} }}
\newcommand{\eiscv}{{\rm e}^{i \frac{S_c - S_v}{\ze} }}
\newcommand{\eisvc}{{\rm e}^{i \frac{S_v - S_c}{\ze} }}


\newcommand{\eiscmv}{{\rm e}^{i \frac{S_c (x,t)-S_v(x,t)}{\ze} }}
\newcommand{\eisvmc}{{\rm e}^{i \frac{S_v (x,t)-S_c(x,t)}{\ze} }}
\newcommand{\sincv}{{\rm S}_{cv}}
\newcommand{\coscv}{{\rm C}_{cv}}
\newcommand{\coscmv}{\cos{ \left( \frac{S_c-S_v}{\ze} \right)}}
\newcommand{\cosvmc}{\cos{ \left( \frac{S_v-S_c}{\ze} \right)}}
\newcommand{\sincmv}{\sin{ \left( \frac{S_c-S_v}{\ze} \right)}}
\newcommand{\sinvmc}{\sin{ \left( \frac{S_v-S_c}{\ze} \right)}}
%
\newcommand{\coscmvx}{\cos{ \left( \frac{S_c (x,t)-S_v(x,t)}{\ze} \right)}}
\newcommand{\cosvmcx}{\cos{ \left( \frac{S_v (x,t)-S_c(x,t)}{\ze} \right)}}
\newcommand{\sincmvx}{\sin{ \left( \frac{S_c (x,t)-S_v(x,t)}{\ze} \right)}}
\newcommand{\sinvmcx}{\sin{ \left( \frac{S_v (x,t)-S_c(x,t)}{\ze} \right)}}


\newcommand{\wcc}{w_{cc}}
\newcommand{\wcv}{w_{cv}}
\newcommand{\wvc}{w_{vc}}
\newcommand{\wvv}{w_{vv}}
%introdotte da Biondini
\newcommand{\wij}{w_{ij}}
\newcommand{\ncc}{n_{c}}
\newcommand{\ncv}{n_{cv}}
\newcommand{\nvc}{n_{vc}}
\newcommand{\nvv}{n_{v}}
\newcommand{\Ener}{\mathcal{E}}
% fine Biondini
\newcommand{\rcc}{\rho_{cc}}
\newcommand{\rcv}{\rho_{cv}}
\newcommand{\rvc}{\rho_{vc}}
\newcommand{\rvv}{\rho_{vv}}
%    Bold per lo stato misto
\newcommand{\bnc}{{\bf n_{cc}}}
\newcommand{\bncv}{{\bf n_{cv}}}
\newcommand{\bnv}{{\bf n_{vv}}}
\newcommand{\bjc}{{\bf J_{cc}}}
\newcommand{\bjv}{{\bf J_{vv}}}
\newcommand{\bj}{{\bf J_{cv}}}
\newcommand{\bjcv}{{\bf J_{cv}}}
\newcommand{\brej}{{\bf \Re J_{cv}}}
\newcommand{\bimj}{{\bf \Im J_{cv}}}
\newcommand{\tc}{{\bf T_{cc}}}
\newcommand{\tv}{{\bf T_{vv}}}
\newcommand{\tcv}{{\bf T_{cv}}}
%
\newcommand{\sumj}{{\sum_{j=0}^{\infty}}}
\newcommand{\sumk}{{\sum_{k=0}^{\infty}}}
\newcommand{\vuc}{{\bf u}_c}
\newcommand{\vuv}{{\bf u}_v}
\newcommand{\vucj}{{u}_{cj}}
\newcommand{\vuvj}{{u}_{vj}}
%
%fine stato misto
\renewcommand{\div}{\hbox{\,\rm div\,}}


\newcommand{\ktra}{(k_y^2+k_z^2)}
\newcommand{\ucc}{u_{cc}}
\newcommand{\ucv}{u_{cv}}
\newcommand{\uvc}{u_{vc}}
\newcommand{\uvv}{u_{vv}}
\newcommand{\mpv}{m_0 p v}
\newcommand{\mpsuh}{\frac{m_0 p}{\hbar}}
\newcommand{\dmsuh}{\frac{2 m_0}{\hbar}}
\newcommand{\hsm}{\frac{\hbar}{m_0}}
\newcommand{\hsi}{\frac{\hbar}{i}}
\newcommand{\usuh}{\frac{1}{\hbar}}
\newcommand{\usuih}{\frac{1}{i \hbar}}
\newcommand{\usui}{\frac{1}{i}}
\newcommand{\hdsqm}{\frac{\hbar}{2 m_0}}
\newcommand{\hsdm}{\frac{\hbar}{2 m_0}}
\newcommand{\hqkqsdm}{\frac{\hbar^2 \vk^2}{2 m_0}}
\newcommand{\hqsim}{\frac{\hbar^2 }{i m_*}}
\newcommand{\hsqm}{\frac{\hbar}{4 m_0}}
\newcommand{\mpsh}{\frac{m_0 p}{\hbar}}
\newcommand{\ish}{\frac{i}{\hbar}}
%**********************************************
%Equazione di Schroedinger per due bande SCALATA  da controllare per il p se e' puro
%**********************************************
\newcommand{\EqnSchro}[2]
		{
		\displaystyle     i \ze \frac{\ppt \psi _{{#1}} }{\ppt t}  & = & 
		\displaystyle    -\ze^2   \frac{\ppt^2\psi_{{#1}}}{\ppt x^2} 
		\displaystyle    +V_{#1}\psi_{{#1}}
		\displaystyle    +i \ze^2 P \frac{\ppt \psi_{{#2}}}{\ppt x} 
		}
%**********************************************
%Equazione di Schroedinger per due bande 
%**********************************************
\newcommand{\EqnSchrodascalare}[2]
		{
		\displaystyle     i \hbar \frac{\ppt \psi _{{#1}} }{\ppt t}  & = & 
		\displaystyle    -\frac{\hbar^2 }{2 m_0}  \frac{\ppt^2\psi_{{#1}}}{\ppt x^2} 
		\displaystyle    +V_{#1}\psi_{{#1}}
		\displaystyle    +i \hbar p \frac{\ppt \psi_{{#2}}}{\ppt x} 
		}
%**********************************************
%Equazione di Schroedinger per due bande Coniugata 
%**********************************************
\newcommand{\EqnSchroCon}[2]
		{
		\displaystyle     i \hbar \frac{\ppt \overline{\psi} _{{#1}} }{\ppt t}  & = & 
		\displaystyle    + \frac{\hbar^2 }{2 m_0}  \frac{\ppt^2\overline{\psi}_{{#1}}}{\ppt x^2} 
		\displaystyle    -V_{#1}\overline{\psi}_{{#1}}
		\displaystyle    +i \hbar p \frac{\ppt \overline{\psi}_{{#2}}}{\ppt x} 
		}
%**********************************************
%Equazione generica elementi della matrice densit�
%**********************************************
\newcommand{\EqnMatrixDensity}[4]
		{
		\displaystyle     i \hbar \frac{\ppt \rho_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle    \frac{\hbar^2 }{2 m_0} \left[ \frac{\ppt ^2 \rho_{{#1}{#2}}}{\ppt r^2} - \frac{\ppt ^2 \rho_{{#1}{#2}}}{\ppt s^2} \right] 
		\displaystyle    +\left[ V_{#2}(s,t)-V_{#1}(r,t)\right] \rho_{{#1}{#2}} 
		\displaystyle    +i  \hbar p \left[ \frac{\ppt \rho_{{#3}{#2}} }{\ppt r}+\frac{\ppt \rho_{{#1}{#4}} }{\ppt s}\right]
		}
%**********************************************
%Equazione Cambio di variabile
%**********************************************
\newcommand{\EqnChangeVar}[4]
		{
		\displaystyle     i \hbar \frac{\ppt u_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle    \frac{\hbar^2 }{2 m_0} \frac{2 m_0}{\hbar} \frac{\ppt ^2 u_{{#1}{#2}}}{\ppt x \ppt \eta}  
		\displaystyle    +\left[ V_{#2}(s,t)-V_{#1}(r,t)\right] u_{{#1}{#2}} \\
		\displaystyle    & & +i  \hbar p \left[   
		\displaystyle   \frac{1}{2}\frac{\ppt u_{{#3}{#2}}}{\ppt x}  
		\displaystyle   +\frac{m_0}{\hbar}\frac{\ppt u_{{#1}{#4}} }{\ppt \eta}  
		\displaystyle   +\frac{1}{2}\frac{\ppt u_{{#3}{#2}}}{\ppt x}  
		\displaystyle   -\frac{m_0}{\hbar}\frac{\ppt u_{{#1}{#4}} }{\ppt \eta}   
		\displaystyle	\right]
		}


%**********************************************
%Equazione di Kane con le funzioni di wigner generica
%**********************************************
\newcommand{\EqnWigner}[4]
		{
		\displaystyle     i \hbar \frac{\ppt w_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle    -i \hbar v  \frac{\ppt w_{{#1}{#2}}}{\ppt x} 
		\displaystyle    +\frac{1}{2\pi} \inti\inti \left[ V_{#2}(s,t)-V_{#1}(r,t)\right] w_{{#1}{#2}}e^{i(v-v')\eta}dv'd\eta \\
		&&
		\displaystyle    +i  \frac{\hbar p}{2}\left[ \frac{\ppt w_{{#3}{#2}} }{\ppt x}+\frac{\ppt w_{{#1}{#4}} }{\ppt x}\right]
		\displaystyle    +m_0 p v \left[w_{{#3}{#2}} -w_{{#1}{#4}}\right]
		}
%**********************************************
%Equazione di  Wigner: Momento di Ordine Zero1,4
%**********************************************
\newcommand{\EqnZeroP}[4]
		{
		\displaystyle     i \hbar \frac{\ppt n_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     i \frac{\hbar  }{q}\frac{\ppt J_{{#1}{#2}}}{\ppt x}	
		\displaystyle    +i  \frac{\hbar p}{2}\left[ \frac{\ppt n_{{#3}{#2}}}{\ppt x}+\frac{\ppt n_{{#1}{#4}} }{\ppt x}\right]
		\displaystyle    -\frac{m_0 p }{q} \left[J_{{#3}{#2}} -J_{{#1}{#4}}\right]
		}
%**********************************************
%Equazione di  Wigner: Momento di Ordine Zero Generiche 2,3
%**********************************************
\newcommand{\EqnZeroG}[4]
		{
		\displaystyle     i \hbar \frac{\ppt n_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     i \frac{\hbar  }{q}\frac{\ppt J_{{#1}{#2}}}{\ppt x} 
		\displaystyle    + \left[ V_{#2}(x,t)-V_{#1}(x,t)\right] n_{{#1}{#2}}\\
		&&
		\displaystyle    +i  \frac{\hbar p}{2}\left[ \frac{\ppt n_{{#3}{#2}} }{\ppt x}+\frac{\ppt n_{{#1}{#4}} }{\ppt x}\right]
		\displaystyle    -\frac{m_0 p }{q} \left[J_{{#3}{#2}} -J_{{#1}{#4}}\right]
		}

%**********************************************
%Equazione di  Wigner: Momento di Ordine Uno1,4
%**********************************************
\newcommand{\EqnUnoP}[4]
		{
		\displaystyle     -i \frac{\hbar }{q}\frac{\ppt J_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     -i \frac{2 \hbar }{m_0}\frac{\ppt \Ener_{{#1}{#2}}}{\ppt x} 
		\displaystyle     -i \frac{\hbar}{m_0} V'_{#1}(x,t) n_{{#1}{#2}}\\
		&&
		\displaystyle    -i  \frac{\hbar p}{2q}\left[ \frac{\ppt J_{{#3}{#2}} }{\ppt x}+\frac{\ppt J_{{#1}{#4}} }{\ppt x}\right]
		\displaystyle    +2p \left[\Ener_{{#3}{#2}} -\Ener_{{#1}{#4}}\right]
		}
%**********************************************
%Equazione di  Wigner: Momento di Ordine Uno Generiche 2,3
%**********************************************
\newcommand{\EqnUnoG}[4]
		{
		\displaystyle     -i \frac{\hbar }{q}\frac{\ppt J_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     -i \frac{2 \hbar }{m_0}\frac{\ppt \Ener_{{#1}{#2}}}{\ppt x} 
		\displaystyle     -i \frac{\hbar}{2m_0}\left[ V'_{#2}(x,t)+V'_{#1}(x,t)\right] n_{{#1}{#2}}\\
		&&
		\displaystyle     - \frac{1}{q}\left[ V_{#2}(x,t)-V_{#1}(x,t)\right] J_{{#1}{#2}}
		\displaystyle    -i  \frac{\hbar p}{2q}\left[ \frac{\ppt J_{{#3}{#2}} }{\ppt x}+\frac{\ppt J_{{#1}{#4}} }{\ppt x}\right]\\
		&&
		\displaystyle    +2p \left[\Ener_{{#3}{#2}} -\Ener_{{#1}{#4}}\right]
		}
%**********************************************
%Equazione di  Wigner: Momento di Ordine Due 1,4
%**********************************************
\newcommand{\EqnDueP}[4]
		{
		\displaystyle     i \frac{2\hbar }{m_0}\frac{\ppt \Ener_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     -i \hbar \frac{\ppt M^{(3)}_{{#1}{#2}}}{\ppt x} 
		\displaystyle     +i\frac{2 \hbar}{q m_0}\left[ V'_{#2}(x,t)\right] J_{{#1}{#2}}\\
		&&
		\displaystyle    +i  \frac{\hbar p}{m_0}\left[ \frac{\ppt \Ener_{{#3}{#2}} }{\ppt x}+\frac{\ppt\Ener_{{#1}{#4}} }{\ppt x}\right]
		\displaystyle    +m_0 p \left[M^{(3)}_{{#3}{#2}} -M^{(3)}_{{#1}{#4}}\right]
		}
%**********************************************
%Equazione di  Wigner: Momento di Ordine Due Generiche 2,3
%**********************************************
\newcommand{\EqnDueG}[4]
		{
		\displaystyle     i \frac{2\hbar }{m_0}\frac{\ppt \Ener_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     -i \hbar \frac{\ppt M^{(3)}_{{#1}{#2}}}{\ppt x} 
		\displaystyle     -\left(\frac{\hbar}{2m_0}\right)^2\left[ V''_{#2}(x,t)-V''_{#1}(x,t)\right] n_{{#1}{#2}}\\
		&&
		\displaystyle     +i\frac{\hbar}{q m_0}\left[ V'_{#2}(x,t)+V'_{#1}(x,t)\right] J_{{#1}{#2}}
		\displaystyle     +\frac{2}{ m_0}\left[ V_{#2}(x,t)-V_{#1}(x,t)\right]\Ener_{{#1}{#2}}\\
		&&
		\displaystyle    +i  \frac{\hbar p}{m_0}\left[ \frac{\ppt \Ener_{{#3}{#2}} }{\ppt x}+\frac{\ppt\Ener_{{#1}{#4}} }{\ppt x}\right]
		\displaystyle    +m_0 p \left[M^{(3)}_{{#3}{#2}} -M^{(3)}_{{#1}{#4}}\right]
		}

%%%
%%%
%%%
%%%
%%%
%%%
\begin{document}
\pagestyle{empty}
%
\oddsidemargin-0.0cm
\begin{slide}
\begin{center}
{\bf Zero-temperature QUANTUM HYDRODYNAMIC EQUATIONS (Madelung fluid equations)}
\end{center}
Direct approach from the Schr\"odinger-type equations.
\vskip -5pt
{\bf Special solution.} We look for solution of the (scaled) Kane model
\begin{equation}
	\left\{
	\begin{array}{lcl}
	%cc
	\EqnSchro{c}{v}\\ \\
	%cv
	\EqnSchro{v}{c}\\ \\		
	\end{array}
     	\right.
	\nn %label{eq:Schro}
\end{equation}
%
in the following form
\begin{equation}
	\begin{array}{lcl}
		\psi_{c}(x,t) &=& \sqrt{n_c(x,t)} \exp \left( \frac{i S_c(x,t)}{\ze} \right) \\
		\psi_{v}(x,t) &=& \sqrt{n_v(x,t)} \exp \left( \frac{i S_v(x,t)}{\ze} \right) \\
	\end{array} \nn   
\end{equation}
where 

$\ncc$ and $\nvv$ are the electron and hole densities

$S_c$ and $S_v$ are the phases of the "wave" functions

${J_{ij}(x,t)}    = \displaystyle   \ze \hbox{Im}\, \left[ \overline{\psi_i}
                         {\left( x ,t \right) } {\psi^{\,\prime}_j \left( x,t \right) } \right], \qquad i,j=c,v$ 

%\vskip 50pt
\end{slide}
\oddsidemargin-1.0cm
\begin{slide} 
The equations for the densities $\ncc$ and $\nvv$ and the phases $S_c$ and $S_v$ take the 
form 

%\small
\begin{equation}
\left\{
\begin{array}{ccl}
&&		\displaystyle       \frac{\ppt \nc }{\ppt t} 
                    	  = 
                           -   \frac{\ppt }{\ppt x}   \left(  \nc \frac{\ppt S_c }{\ppt x}  \right)
		\displaystyle    + \ze P  \frac{\rdnc }{ \rdnv} \frac{\ppt \nv }{\ppt x} \cosvmc \\
&&  \phantom{xxxx}	\displaystyle 
- 2 P  \rdnv \rdnc  \frac{\ppt S_v }{\ppt x} \sinvmc\\
\\
&&		\displaystyle      \frac{\ppt \nv }{\ppt t} 
	  =           -    \frac{\ppt  }{\ppt x}   \left(  \nv \frac{\ppt S_v }{\ppt x} \right)
		\displaystyle    + \ze P  \frac{\rdnv }{\rdnc} \frac{\ppt \nc }{\ppt x} \coscmv  \\
&&  \phantom{xxxx}	\displaystyle 
- 2 P \rdnc \rdnv \frac{\ppt S_c }{\ppt x} \sincmv \\
\\
&&        \displaystyle         \frac{\ppt S_c }{\ppt t} +  
                             \frac{1}{2} \left( \frac{\ppt S_c}{\ppt x} \right)^2  +  V_{c}  
	\displaystyle                              -
                             \frac{\ze^2}{2} \left(  
\frac{1 }{\rdnc}   \frac{\ppt^2 \rdnc }{\ppt x^2}  \right)     \\
&&   \phantom{xxxx}	\displaystyle                            -
                            \ze P \frac{\rdnv}{\rdnc}   \frac{\ppt S_v}{\ppt x} \cosvmc      \\
&& \phantom{xxxx}
- \ze^2 P \frac{1}{2}  \frac{1 }{\rdnv\rdnc}  \frac{\ppt \nv}{\ppt x} \sinvmc = 0\\
\\
&&        \displaystyle                \frac{\ppt S_v }{\ppt t} +
                            \frac{1}{2}  \left( \frac{\ppt S_v}{\ppt x} \right)^2  +  V_{v}  
	\displaystyle  -
                             \frac{\ze^2}{2} \left(  
\frac{1 }{\rdnv}   \frac{\ppt^2 \rdnv}{\ppt x^2} \right)  \\
&&   \phantom{xxxx}	\displaystyle                        -
                            \ze P  \frac{\rdnc}{\rdnv} \frac{\ppt S_c}{\ppt x} \coscmv    \\
&& \phantom{xxxx}
- \ze^2 p \frac{1}{2}  \frac{1 }{\rdnc\rdnv}  \frac{\ppt \nc}{\ppt x} \sincmv= 0 \,.
\end{array}
\right. \nn
\end{equation}
%

\end{slide}
%
%
%
%

\begin{slide}
The additional unknown
\begin{equation}
\sincv = \sincmv \, ,
\qquad
\coscv = \coscmv \,,   \label{sincos} \nn
\end{equation}
 can be manipulated in order to obtain links
with the current densities $\jc$, $\jv$,

$$
\px \sincv = 
 \frac{1}{\ze} \coscv
\left( \frac{\jc}{\nc} - \frac{\jv}{\nv}  \right) \,,
$$
$$
\px \coscv = -
 \frac{1}{\ze} \sincv
\left( \frac{\jc}{\nc} - \frac{\jv}{\nv}  \right) \,.
$$

%
%%%%%%%%%%%%%%
Summarizing,  we have the set of equations, which are the
{\bf zero-temperature quantum hydrodynamic equations}:
\end{slide}

\oddsidemargin-2.6cm
\begin{slide}
\begin{equation}
%\left\{
\begin{array}{ccl}
&&		\displaystyle       \frac{\ppt { \nc} }{\ppt t} 
                    	  = 
                          -   \frac{\ppt { \jcc}}{\ppt x}  + \ze P  \frac{ \sqrt{ \nc} }{\sqrt { \nv}} \frac{\ppt  \nv }{\ppt x} { \coscv} 
+ 2 P  \frac{ \sqrt{ \nc} }{\sqrt { \nv}}  \, { \jv} \, { \sincv} 
\\
\\[3mm]
&&		\displaystyle       \frac{\ppt { \nv} }{\ppt t} 
                    	  = 
                           -   \frac{\ppt { \jvv}}{\ppt x}  + \ze P  \frac{ \sqrt{ \nv} }{\sqrt { \nc}}  \frac{\ppt  \nc }{\ppt x} { \coscv} 
- 2 P \frac{ \sqrt{ \nv} }{\sqrt { \nc}}  \, {\jc} \, {\sincv} 
\\
\\[3mm]
%%%%%%%%%% 3^ equaz
&&        \displaystyle         \frac{\ppt { J_c} }{\ppt t} +   \frac{\ppt }{\ppt x} 
 \left( \frac{{ J_c}^2}{{ \nc}} \right) +  { \nc}    \frac{\ppt V_{c}}{\ppt x} 
	\displaystyle  -    \frac{\ze^2}{2} { \nc} \frac{\ppt }{\ppt x}  \left(  
\frac{1 }{ \rdnc}   \frac{\ppt^2  \rdnc}{\ppt x^2} \right)    
\\[8mm]
&& \phantom{xx}
-      \displaystyle            \ze P \, { \nc} \px \left( \frac{ \jv}{ \sqrt{ \nc \nv}}  \right) \coscv
	\displaystyle      
+ \ze^2 \frac{P}{2}  { \nc} \px \left(  \frac{1 }{ \sqrt{ \nc \nv}}  \frac{\ppt  \nv}{\ppt x} \right)  { \sincv} 
 \\[8mm]
&& \phantom{xx}
-        \displaystyle          \ze P   \frac{ \nc  \jv}{ \sqrt{ \nc \nv}}  \, \frac{\ppt  \coscv}{\ppt x} 
	\displaystyle      
+ \ze^2 P { \nc} \, \frac{1}{2}  \frac{1 }{ \sqrt{ \nc \nv}}  \frac{\ppt  \nv}{\ppt x} 
\, \frac{\ppt  \sincv}{\ppt x} 
= 0 \, 
\\
\\[3mm]
%%%%%%%%%% 4^ equaz
&&        \displaystyle         \frac{\ppt { J_v} }{\ppt t} +   \frac{\ppt }{\ppt x} 
 \left( \frac{{ J_v}^2}{{ \nc}} \right) +  { \nv}    \frac{\ppt V_{v}}{\ppt x} 
	\displaystyle  -    \frac{\ze^2}{2} { \nv} \frac{\ppt }{\ppt x}  \left(  
\frac{1 }{ \rdnv}   \frac{\ppt^2  \rdnv}{\ppt x^2} \right)    
\\[8mm]
&& 
\phantom{xx}
\displaystyle                        -
                            \ze P \, { \nv} \px \left( \frac{ \jc} { \sqrt{ \nc \nv}}   \right) {  \coscv}
	\displaystyle      
- \ze^2 \frac{P}{2}  {\nv}  \px \left( \frac{1 }{ \sqrt{ \nc \nv}}  \frac{\ppt  \nc}{\ppt x} \right) { \sincv} 
\\[8mm]
&& 
\phantom{xx}
\displaystyle                        -
                            \ze P \, { \nv} \frac{ \jc} { \sqrt{ \nc \nv}}   \frac{\ppt  \coscv }{\ppt x}
	\displaystyle      
- \ze^2 P \, { \nv} \frac{1}{2}  \frac{1}{ \sqrt{ \nc \nv}}  \frac{\ppt  \nc}{\ppt x}  \frac{\ppt  \sincv }{\ppt x}
= 0 
\end{array}
%\right.
\nn
\end{equation}

%\vskip+3pt
%\hskip 25pt These equations are the {\bf zero-temperature }
%\par \vskip -20pt  \hskip 98pt
%{\bf quantum  hydrodynamic equations.}
%\par \vskip -20pt  \hskip 28pt
%\vskip 20pt  \hskip 50pt For $\ze=0 (\hbar=0)$
%classical  {\bf Euler equations.} %zero-temperature equations.}
%\oddsidemargin+0.6cm
\hskip 20pt For $\ze=0 (\hbar=0)$  {\bf classical Euler equations.} 

\end{slide}


\oddsidemargin-0.6cm
\begin{slide}

{ It is easy to verify the }
{\bf CONTINUITY EQUATION}
%\vskip-30pt
$$
\frac{\pt \ {\bf n_{tot}(x,t)} }{\pt t} = - \div {\bf J_{tot}}
$$
where {\bf the total density} is
$$
{\bf n_{tot}(x,t)} = {\ncc (x,t)} + {\nvv (x,t)}  \,.
$$
and {\bf the total particle current density} is 
$$
{\bf J_{tot}}{(x,t)} = \jc {(x,t)} + \jv {(x,t)}
%J_{cc}(x) + J_{vv}(x)
- 2 \ze P \Re \left( \psibar_c \psi_v \right)
$$
with
$$
\Re \left( \psibar_c \psi_v \right) = {\rdncv}
                        \displaystyle \cosvmc 
$$
In terms  of "interband" densities and current densities
\vskip -20pt
\begin{small}
\begin{equation}
% \left\{
\begin{array}{ccl}
		\displaystyle       \frac{\ppt { \nc} }{\ppt t} 
                    	  &=& 	\displaystyle    
                           -   \frac{\ppt {\jc}}{\ppt x}  + \frac{ \ze p}{2} {\frac{\ppt }{{\ppt x}}}  \left( {n_{cv}} + {n_{vc}}\right)
+  i P \left[
{J_{cv}(x,t)} - {J_{vc}}{(x,t)} 
\right]
\\
\\
	\displaystyle       \frac{\ppt {\nv} }{\ppt t} 
                    	  &=& 	\displaystyle    
                           -   \frac{\ppt {\jv}}{\ppt x}  + \frac{ \ze p}{2}{\frac{\ppt }{{\ppt x}}} \left( {n_{vc}}+ {n_{cv}}\right)
-  i P \left[
 { J_{cv}(x,t)} - {J_{vc}}{(x,t)} 
\right]
\end{array}
% \right.
\nn
\end{equation}
\end{small}
\vskip -20pt
with ${\bf J_{tot}}{(x,t)} =$
$$
 \jc {(x,t)} + \jv {(x,t)}
- \ze\, P  \, \left[ {n_{cv}(x,t)} + {n_{vc}(x,t)}\right] 
$$
\end{slide}
\oddsidemargin-1.8cm
\begin{slide}
\small
%%%%%%%%%%%
\begin{equation}
\left\{
\begin{array}{ccl}
&&        \displaystyle         \frac{\ppt  }{\ppt t} \left({\bf \Re n_{cv}}\right)
=
-\frac{1}{2} \, \frac{1 }{\bf \nc}  \left({\bf \Re n_{cv}}\right)
\left[  \frac{\ppt {\bf J_{cc}}}{\ppt x}  + {\frac{\ppt }{{\ppt x}|_v}}  \left({\bf \Re n_{cv}}\right)
      \right]
\\
\\
&& \phantom{xxyyyix}  \displaystyle 
- \frac{1}{2} \, \frac{1 }{\bf \nv} \left({\bf \Re n_{cv}}\right)
\left[  \frac{\ppt {\bf J_{vv}}}{\ppt x}  + {\frac{\ppt }{{\ppt x}|_{c}}}  \left({\bf \Re n_{cv}}\right)
      \right]   
\\ \\
&& \phantom{xxyyyix}  \displaystyle 
+   \frac{1}{2 \ze} {\bf \Im n_{cv}}
 \left( \frac{{\bf J_{vv}}^2}{{\bf \nv}^2} -  \frac{{\bf n_{cc}}^2}{{\bf \nc}^2}   \right) 
+   \frac{1}{\ze} {\bf \Im n_{cv}}  \left( V_v - V_c \right) 
\\ \\
&& \phantom{xxyyyix}  \displaystyle 
+
 \frac{\ze}{2} {\bf \Im n_{cv}}  \left(  
\frac{1 }{\bf \rdnc}   \frac{\ppt^2 \bf \rdnc}{\ppt x^2}      
- 
\frac{1 }{\bf \rdnv}   \frac{\ppt^2 \bf \rdnv}{\ppt x^2} \right)  
\\
\\
&& \phantom{xxyyyix}  \displaystyle                  
- \frac{1}{2}   {\bf \Im n_{cv}}  \left(  
\frac{1}{\bf n_{vv}}
{\frac{\ppt }{{\ppt x}|_c}} \left( {\bf \Im n_{cv}} \right)
+
\frac{1}{\bf n_{cc}}
{\frac{\ppt }{{\ppt x}|_v}} \left( {\bf \Im n_{cv}}  \right)
\right)
\\
\\
%%%%%%%%%% 2^ equaz
&&        \displaystyle         \frac{\ppt  }{\ppt t} \left({\bf \Im n_{cv}}\right)
=
-\frac{1}{2} \, \frac{1}{\bf \nc}  \left({\bf \Im n_{cv}}\right)
\left[  \frac{\ppt {\bf J_{cc}}}{\ppt x}  + {\frac{\ppt }{{\ppt x}|_v}}  \left({\bf \Re n_{cv}}\right)
      \right]
\\
\\
&& \phantom{xxyyyix}  \displaystyle 
=
-\frac{1}{2} \, \frac{1 }{\bf \nv} \left({\bf \Im n_{cv}}\right)
\left[  \frac{\ppt {\bf J_{vv}}}{\ppt x}  + {\frac{\ppt }{{\ppt x}|_c}}  \left({\bf \Re n_{cv}}\right)
      \right]   
\\ \\
&& \phantom{xxyyyix}  \displaystyle 
+   \frac{1}{2 \ze} {\bf \Re n_{cv}}
 \left( \frac{{\bf J_{vv}}^2}{{\bf \nv}^2} -  \frac{{\bf J_{cc}}^2}{{\bf \nc}^2}   \right) 
+   \frac{1}{\ze} {\bf \Re n_{cv}}  \left( V_v - V_c \right) 
\\ \\
&& \phantom{xxyyyix}  \displaystyle 
+
 \frac{\ze}{2} {\bf \Re n_{cv}}  \left(  
\frac{1 }{\bf \rdnc}   \frac{\ppt^2 \bf \rdnc}{\ppt x^2}      
- 
\frac{1 }{\bf \rdnv}   \frac{\ppt^2 \bf \rdnv}{\ppt x^2} \right)  
\\
\\
&& \phantom{xxyyyix}  \displaystyle                  
- \frac{1}{2}   {\bf \Re n_{cv}}  \left(  
\frac{1}{\bf n_{vv}}
{\frac{\ppt }{{\ppt x}|_c}} \left( {\bf \Im n_{cv}} \right)
+
\frac{1}{\bf n_{cc}}
{\frac{\ppt }{{\ppt x}|_v}} \left( {\bf \Im n_{cv}}  \right)
\right)
\end{array}
\right.
\nn
\end{equation}
\end{slide}

\oddsidemargin0.0cm
\begin{slide}
The previous equations are the 
\par \noindent \vskip-50pt
{\bf ZERO-TEMPERATURE QUANTUM
HYDRODYNAMIC EQUATIONS
for a TWO-BAND electron system}.
%\vskip10pt
%A difference from the one-band case: the system is closed only adding equations for the 
%interband terms $\bf n_{cv}$ and $\bf J_{cv}$.
%\vskip10pt
%For $p=0$ are the classical zero-temperature quantum hydrodynamic 
%equations.
%\vskip30pt

 In order to obtain a {\bf nonzero temperature}, we consider MIXED STATES.

A {\bf mixed quantum mechanical state} consists of a sequence of single states 
with occupation probabilities $\zl_j$, $j=0,1,2, \dots$
for the $j-$th single state described by the envelope function
% in the conduction (valence) band
$\psic^j$ ($\psiv^j$), 
%described by  the Kane system, 
{\bf for each index $j$.}

We look for solutions  $\psi^j_{c}(x,t)$ and $\psi^j_{v}(x,t)$, as before.
%%%%%%%%%%%%
We define for the mixed state the carrier densities and the current densities
$$
\begin{array}{lcl}
\nc(x) &=& \sumj \zl^j {n^j_c(x)}  \\
\nv(x) &=&  \sumj \zl^j {n^j_v(x)}  \\
\ncv(x) &=&  \sumj \zl^j {n^j_{cv}(x)}  \\
\jc(x) &=&  \sumj \zl^j {J^j_c(x)}       \\
\jv(x) &=&  \sumj \zl^j {J^j_v(x)}       \\
\jcv(x) &=&  \sumj \zl^j {J^j_{cv}(x)}       
\end{array}
$$
\end{slide}

\begin{slide}


\begin{itemize}
\item  
{\bf No problem} for the total quantum continuity equation. 
\item \vskip-20pt
The "one-band" part of the current densities takes the new form
\vskip-20pt

\begin{small}
\begin{equation}
%   equazione J_c
    \displaystyle         \frac{\ppt {\jc} }{\ppt t} +   
\displaystyle \frac{\ppt}{\ppt x} \left(  \frac{\jc^2} {\nc} + \nc \tc \right) +
% item 9
\nc  \frac{\ppt V_{c}}{\ppt x} -
% \nv  \frac{\ppt V_{v}}{\ppt x} 
\displaystyle \frac{\ze^2}{2} \nc  \frac{\ppt }{\ppt x} \left(  
\frac{1 }{ \rdnc}   \frac{\ppt^2  \rdnc}{\ppt x^2} \right) 
%
\nn \end{equation}
\end{small}
\end{itemize}
\vskip-20pt
where
$
\tc = \displaystyle
\frac{\ze^2}{4}
 \sumj \zl^j \frac{\nc^j}{\nc} \left( \frac{1}{{ n^j_c}}  {{ \frac{\ppt n^j_c}{\ppt x} }}
 -  \frac{1}{\nc} \frac{\ppt \nc}{\ppt x}    \right)^2  \\
\phantom{xxxx} +
\sumj \zl^j \frac{\nc^j}{\nc} \left( \frac{{ J^j_c}}{{ n^j_c}}  
        - \frac{\jc}{\nc} \right)^2 
$
\vskip2pt
and analogously for the valence electrons:  $\jv,\, \tv$.

For each band, $\tc$ and $\tv$ are the total temperature, sum of the {\bf current temperature}
$\tc^c$ and $\tv^c$, and the {\bf osmotic temperature} $\tc^{os}$ and $\tv^{os}$.
\end{slide}

\begin{slide}
\begin{itemize}
\item

More complicated are the nonzero temperature corrections in the "two-band" part of the current density
equations.

\begin{small}
\begin{eqnarray*}
\displaystyle  
{\bf T^c_{cv}} 
&= &\nc  \sumj \zl^j \frac{ \nc^j}{\nc}
\left(  
\frac{{ J^j_c}}{{ n^j_c}}  - \frac{\jc}{\nc} \right) \\
&& \phantom{xxxxx}
\left(
\frac{1}{{ n^j_c}}   \, \frac {\ppt } {{\ppt x}|_v} \left( { \Re n^{j}_{cv}} \right)   -   
 \frac{1}{\nc}\, \frac{\ppt }{{\ppt x}|_v} \left( {\Re n_{cv}} \right)
\right) \\
&& \displaystyle 
- \frac{\ze}{2}
\sumj \zl^j  {\nc^j} \left( \frac{1}{{ n^j_c}}  {{ \frac{\ppt n^j_c}{\ppt x} }}
 -   \frac{1}{\nc}  \frac{\ppt \nc}{\ppt x}  \right) \\
&& \phantom{xxxxx}
 \left( \frac{1}{{ n^j_c}}  {{ \frac{\ppt }{{\ppt x}|_v} \left( { \Im n^j_{cv}}\right) }}
 -   \frac{1}{\nc}   \frac{\ppt }{{\ppt x}|_v} \left( {\Im n_{cv}}\right)  \right) 
\end{eqnarray*}
\end{small}
and a similar term with $v$ instead of $c$.

%%%%%%%%%%%%
\item
The new corrections due to the interband coupling in the evolution of $\jcv$
are long and intricated.
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
{ \bf TWO-BAND QUANTUM HYDRODYNAMIC MODEL}
\end{center}
\vskip - 2pt
 
{\bf CONCLUSIONS}
\begin{enumerate}
\item
It is {\bf impossible} to get  a system where only the {\bf total quantum 
current density} appears.
\item The set of equations is {\bf not closed}.
\item  {\bf CLOSURE conditions}.
\begin{enumerate}
\item[2.1] {\bf Thermodynamic equilibrium}
\item[2.2]  {\bf Chapman-Enskog} expansion
\item[2.3]  {\bf MEP} (Maximum entropy principle)
\end{enumerate}
\item Numerical validation and comparison with other models and {\bf experimental data}.
\item {\bf Non-smooth} potentials.
\end{enumerate}
\end{slide}

\end{document}



\begin{slide}
%%%%%%%%%%%RIEPILOGO
\begin{equation}
\left\{
\begin{array}{ccl}
&&		\displaystyle       \frac{\ppt \nc }{\ppt t} 
                    	  = 
                           -   \frac{\ppt }{\ppt x}   \left(  \nc \frac{\ppt S_c }{\ppt x}  \right)
		\displaystyle    + p  \frac{\rdnc }{ \rdnv} \frac{\ppt \nv }{\ppt x} \cosvmc \\
&&  \phantom{xx}	\displaystyle 
- 2 p \frac{1}{\ze}  \rdnv \rdnc  \frac{\ppt S_v }{\ppt x} \sinvmc\\
\\
&&		\displaystyle      \frac{\ppt \nv }{\ppt t} 
	  =           -    \frac{\ppt  }{\ppt x}   \left(  \nv \frac{\ppt S_v }{\ppt x} \right)
		\displaystyle    + p  \frac{\rdnv }{\rdnc} \frac{\ppt \nc }{\ppt x} \coscmv  \\
&&  \phantom{xx}	\displaystyle 
- 2 p \frac{1}{\ze}  \rdnc \rdnv \frac{\ppt S_c }{\ppt x} \sincmv \\
\\
&&        \displaystyle         \frac{\ppt S_c }{\ppt t} +  
                             \frac{1}{2} \left( \frac{\ppt S_c}{\ppt x} \right)^2  +  V_{c}  
	\displaystyle                              -
                             \frac{\ze^2}{2} \left(  
\frac{1 }{\rdnc}   \frac{\ppt^2 \rdnc }{\ppt x^2}  \right)     \\
&&   \phantom{xx}	\displaystyle                            -
                            p \frac{\rdnv}{\rdnc}   \frac{\ppt S_v}{\ppt x} \cosvmc \\
&&   \phantom{xx}	\displaystyle      
- \ze p \frac{1}{2}  \frac{1 }{\rdnv\rdnc}  \frac{\ppt \nv}{\ppt x} \sinvmc = 0\\
\\
&&        \displaystyle                \frac{\ppt S_v }{\ppt t} +
                            \frac{1}{2}  \left( \frac{\ppt S_v}{\ppt x} \right)^2  +  V_{v}  
	\displaystyle  -
                             \frac{\ze^2}{2} \left(  
\frac{1 }{\rdnv}   \frac{\ppt^2 \rdnv}{\ppt x^2} \right)  \\
&&   \phantom{xx}	\displaystyle                        -
                            p \frac{\rdnc}{\rdnv} \frac{\ppt S_c}{\ppt x} \coscmv \\
&&  \phantom{xx}	\displaystyle      
- \ze p \frac{1}{2}  \frac{1 }{\rdnc\rdnv}  \frac{\ppt \nc}{\ppt x} \sincmv= 0 \,.
\end{array}
\right. \nn
\end{equation}
\end{slide}
222222222222222222222222222222222222222222222222222222222222222
\begin{slide}
\begin{small}
\begin{equation}
\left\{
\begin{array}{ccl}
&&        \displaystyle         \frac{\ppt {\bf J_{cc}} }{\ppt t} +   \frac{\ppt }{\ppt x} 
 \left( \frac{{\bf J_{cc}}^2}{{\bf \nc}} \right) +  {\bf \nc}    \frac{\ppt V_{c}}{\ppt x} 
	\displaystyle  -    \frac{\ze^2}{2} {\bf \nc} \frac{\ppt }{\ppt x}  \left(  
\frac{1 }{\bf \rdnc}   \frac{\ppt^2 \bf \rdnc}{\ppt x^2} \right)    
\\
\\
&&   \phantom{xxx}	\displaystyle                     
                            - 2 \ze p   \frac{{\bf J_{cc}}}{\bf \nc} \, {\frac{\ppt }{{\ppt x}|_v}} \left( {\bf \Re n_{cv}}\right) \\
\\
&&   \phantom{xxx}	\displaystyle      
+ {\ze^2} p \frac{1 }{\bf \nc}  \left( \frac{\ppt \bf \nc}{\ppt x} -  \frac{\ppt }{\ppt x} \right)  \, {\frac{\ppt }{{\ppt x}|_v}} 
%
\left( {\bf \Im n_{cv}}
\right)
\\
\\
%%%%%%%%%% 4^ equaz
&&        \displaystyle         \frac{\ppt {\bf J_{vv}} }{\ppt t} +   \frac{\ppt }{\ppt x} 
 \left( \frac{{\bf J_{vv}}^2}{{\bf \nc}} \right) +  {\bf \nv}    \frac{\ppt V_{v}}{\ppt x} 
	\displaystyle  -    \frac{\ze^2}{2} {\bf \nv} \frac{\ppt }{\ppt x}  \left(  
\frac{1 }{\bf \rdnv}   \frac{\ppt^2 \bf \rdnv}{\ppt x^2} \right)    
\\
\\
&&   \phantom{xxx}	\displaystyle                     
                            - 2 \ze p  \frac{{\bf J_{vv}}}{\bf \nv} \, {\frac{\ppt }{{\ppt x}|_c}} \left( {\bf \Re n_{cv}}\right) \\
\\
&&   \phantom{xxx}	\displaystyle     
+ {\ze^2} p \frac{1 }{\bf \nv}  \left( \frac{\ppt \bf \nv}{\ppt x}  -  \frac{\ppt }{\ppt x} \right) \, {\frac{\ppt }{{\ppt x}|_c}} 
%
\left( {\bf \Im n_{cv}}
\right)
\end{array}
\right.
\nn
\end{equation}
\end{small}
where
${\frac{\ppt }{{\ppt x}|_c}}$ $\left( {\frac{\ppt }{{\ppt x}|_v}} \right)$ is the partial derivative
with respect to $x$ which operates only on functions with index $c$ ($v$).
\begin{small}
$$
{\frac{\ppt }{{\ppt x}|_c}} \left( {\bf \Re n_{cv}}\right)= \frac12 
{\frac{\ppt }{{\ppt x}}} \left( {\bf \Re n_{cv}}\right) - \frac{i}{2 \ze} 
\left[ 
{\bf J_{cv}} - {\bf J_{vc}}\right]
$$
$$
{\frac{\ppt }{{\ppt x}|_v}} \left( {\bf \Re n_{cv}}\right)= \frac12 
{\frac{\ppt }{{\ppt x}}} \left( {\bf \Re n_{cv}}\right) + \frac{i}{2 \ze} 
\left[ 
{\bf J_{cv}} - {\bf J_{vc}}\right]
$$
\end{small}
etc. etc.

Evolution equations for $\bf n_{cv}$ and $\bf n_{vc}$: {\bf MORE
complicated!}
\end{slide}