Effect of Collisions on the Magnetoconductivity Tensor of a Quantum Plasma

The effect on the quantum magnetoconductivity tensor of the scattering of electrons by thermal phonons and lattice imperfections is taken into account by the introduction of a phenomenological relaxation time $\tau$. This is accomplished by introducing into the equation of motion of the single-particle density matrix $\rho$, a term ${\tau }^{-1\mathrm{}}\{\rho -{\rho }_0(H_K\mathrm{}\left(t\right)\left)\right\}$. $H_K\mathrm{}\left(t\right)\mathrm{}$ is the entire kinetic part of the Hamiltonian, including the time-dependent vector potential, and ${\rho }_0\left(H_K\mathrm{}\right(t\left)\right)$ is the equilibrium value of the density matrix appropraite to a system in which the Hamiltonian is held fixed for all time at the value which $H_K\mathrm{}\left(t\right)\mathrm{}$ has at time $t$. It is demonstrated that the results obtained in this work reduce to the known semiclassical conductivity tensor in the appropriate limit.