  A condition on the Hamiltonian of a time-dependent quantum mechanical system
is derived which, if satisfied, implies optimal adiabaticity (defined below).
The condition is expressed in terms of the Hamiltonian and in terms of the
evolution operator related to it. Since the latter depends in a complicated way
on the Hamiltonian, it is not yet clear how the condition can be used to
extract useful information about the optimal Hamiltonian. The condition is
tested on an exactly-soluble time-dependent problem (a spin in a magnetic
field), where perfectly adiabatic evolution can be easily identified.
