Dissipative system with spin-dependent interactions

An investigation is made of dissipative system with spin-dependent interaction potential. A general solution of the Schrödinger equation is obtained and the special cases $\alpha t\gg 1$ and $\alpha t\ll 1$ are discussed in more detail. It is shown that at time $t\gg \frac{1}{\alpha }$ there are two stationary states, one with energy $V_0$ and another with energy $-V_0$. It is also shown that, for a potential independent of the $z$ component of the spin operator, the existence of friction causes the $z$ component of the spin density to go to zero and the $x$ component and the $y$ component to be periodic in space regardless of the initial conditions. The wave function of an electron injected into a crystal is calculated. The result shows that the wave function has two parts, one oscillating in time and another exponentially decaying in time.