Investigating the Adiabatic Approximation in Quantum Mechanics through the Analysis of Two Coupled Harmonic Oscillators

The nature of the adiabatic approximation is investigated. This approximation is at the center of the Born-Oppenheimer approximation, commonly employed in the construction of molecular potentials. First, general behaviors of the approximation and some implications of these results are discussed. We show that the ground-state energy, calculated using the adiabatic approximation, will be lower than the calculated energy when no approximations are made. Next, a numerical and analytical application of the adiabatic approximation to systems comprising two coupled harmonic oscillators is described. These systems are used to demonstrate the results derived in the first part of the paper and to investigate how the accuracy of the approximation depends on the frequencies of the oscillators and the coupling strength.