When dealing with physical systems, especially at the atomic or subatomic level, the choice between classical and quantum mechanical treatment depends on the scale of the system and the nature of the phenomena involved.
Classical mechanics is generally applicable to macroscopic objects with large masses and well-defined trajectories, while quantum mechanics is essential for describing the behavior of particles at the atomic and subatomic scales.
If the relevant energy scales are on the order of or smaller than thermal energy (k₃T), or if you are comparing the system to k₃T, a quantum mechanical description may be necessary.
The comparison of the energy of the system (AE) to k₃T (where k₃ is the Boltzmann constant and T is the temperature) is a criterion to determine whether quantum mechanics is more appropriate than classical mechanics.