A quantum theory of conductivity is constructed for semiconductor objects such as quantum wells, wires, and dots. The mean values of current and charge densities induced by a weak electromagnetic field are calculated. It is shown that the mean values of current and charge densities consist of two parts, the first of which is expressed in terms of the electric field and the second is expressed in terms of derivatives of the electric field with respect to spatial coordinates. Appropriate expressions are derived for the conductivity tensor that depends on coordinates; these expressions can be applied to any spatially inhomogeneous systems. The results obtained can be used in the theory of secondary radiation from objects of reduced dimension in the cases of monochromatic or pulsed irradiation.
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