content_image/370615.jpg content_image/370616.jpg
growth rate δ can be found for the filamentation instability ( K$_{z}$ = 0), without any further restriction, as 

<img src='content_image/370615.jpg'>

From Eq. (15) it can be shown that sufficiently large wave-numbers such that 

<img src='content_image/370616.jpg'>

are stable with respect to the filamentation instability. This stabilization effect can exist only in the quantum case ( H ̸ = 0). 

## V. CONCLUSION 

The essential results of this work are in Fig. 1, where the apparently distinct unstable quantum two-stream modes are shown to be connected. This achievement follows after con- sidering the two-dimensional k -space and including the Lorentz-force term in the pertinent quantum hydrodynamic equations. For arbitrary wave-vector orientation, exact analytic cal- culations are possible in the electrostatic approximation, which is fairly reasonable except for wave propagation near the axis normal to the beams direction. In this later (filamenta- tion) case, the exact growth rate is also acessible. In the general electromagnetic case, the quantum filamentation and quantum two-stream instabilities are viewed as a single, unified phenomenon. 

## Acknowledgments 

This work has been achieved under projects FIS 2006-05389 of the Spanish Ministerio de Educaci´ on y Ciencia and PAI08-0182-3162 of the Consejer´ıa de Educaci´ on y Ciencia de la Junta de Comunidades de Castilla-La Mancha. 

[1] PA Markowich, C Ringhofer, and C Schmeiser, Semiconductor Equations , Springer, New York (1990). 

