Validity of the Born-Oppenheimer Approximation

Solutions of the Schr\"odinger equation for several simple models in which a light particle, mass $m$, is bound to a heavy particle, mass $M$, which in turn is bound to another heavy particle which may be fixed are considered as expansions in the parameter $\frac{1}{\ensuremath{\beta}}\ensuremath{\equiv}{(\frac{m}{M})}^{\frac{1}{2}}$. A characteristic of such systems is that when the light particle is far from the heavy particle, it centers on the mean position of the heavy particle and when it is close to the heavy particle, it centers on the heavy particle. The classical prediction that the change in centering takes place when the motional frequencies of the light and heavy particles are equal is borne out in quantum mechanics except that here the transition takes place over an extended region as shown by the detailed investigation of models. General arguments are presented which show how low and high momenta of the light particle contribute, respectively, to centering on the mean position and on the changing position of the heavy particle.The effect of the centering on expectation values of quantities dependent on the density of the light particle at the heavy one is of the order of the ratio of an average radius of the heavy particle motion to an average radius of the light particle motion. Further corrections due to incomplete centering are of the order $\frac{1}{\ensuremath{\beta}}$ times this ratio.