An electrostatic wave in a bounded geometry is studied for e-p-i quantum plasmas. The quantum hydrodynamic model is employed to find the dispersion relation of waveguide mode. The complex frequency leads to the growth rate and the phase speed in the cylindrical geometry. The numerical analysis of the growth rate and the phase speed of cylindrical waveguide is given graphically on varying the parameters. The growth rate and the phase are affected by the geometric dimensions, quantum parameter, streaming effects, and the number density.

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The oblique wave instability is studied extensively in cylindrical bounded quantum plasmas. This study emphasizes on role of quantum features of a plasma system like Fermi degenerate pressure, exchange-correlation potential and Bohm potential as well as the radius of cylindrical geometry on the instability. The growth rate is studied for two cases depending upon the ion gyro time scales. It is found that the growth rate for both cases decreases in a plasma column with the characteristics of wave vector and the parameters like plasma number density, poles of Bessel function and waveguide radius. On increasing these arguments, decreasing of the growth rates respond differently. Numerical analysis of this study is presented on the basis of astrophysical plasma parameters, however, it may be equally useful in the laboratory and discharge plasma technology for some typical set of parameters. Despite the fact of numerous possible applications in technology, the motivation may be extended to the fundamental aspects.

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