Energy transfer between slowly moving atoms—the case of no crossing point

The classical path treatment of a collision between an excited atom and another atom with a nearly resonant energy level leads to coupled differential equations which have been discussed previously by many workers. Here we present an elementary procedure for solving this system of equations in the case of no crossing point and in the limit where $\Delta E\tau \gg \hslash$, with $\Delta E$ being the energy discrepancy and $\tau =\frac{b}{v}$ being a measure of the time of collision. Even though the method is designed to take advantage of the slowness of the collision, it is also shown to be exact if $\Delta E=0\mathrm{}$. The form of the solution is very similar to that of Vainshtein et al., but the present derivation avoids several of their approximations which individually are not as accurate as their final result. It is also pointed out that the coupled equations which arise for the energy-transfer problem also arise in connection with the interaction of laser pulses with atoms. Hence, the method has application to a wide class of multistate problems. A simple laser-interaction problem is discussed as a second example of the method.