growth rate δ can be found for the filamentation instability ( K$_{z}$ = 0), without any further restriction, as δ 2 = 1 2   [ ( 1 + c $^{2}$K2 x v 2 0 + H $^{2}$K4 x 4 ) 2 + K 2 x ( $^{4}$− H $^{2}$K2 $_{x}$(1 + c $^{2}$K2 x v 2 0 ) ) ] 1 / 2 − $^{1}$− c $^{2}$K2 x v 2 0 − H $^{2}$K4 x 4   . (15) H $^{2}$K2 x 4 ( 1 + c $^{2}$K2 x v 2 0 ) ≥ 1 (16) From Eq. (15) it can be shown that sufficiently large wave-numbers such that are stable with respect to the filamentation instability. This stabilization effect can exist only in the quantum case ( H = 0). 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. 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). 8 Acknowledgments V. CONCLUSION