  Through the H theorem, Bolzmann attempted to validate the foundations of
statistical mechanics. However, it is incompatible with the fundamental laws of
mechanics because its deduction requires the introduction of probability. In
this paper we attempt a justification of statistical mechanics without
deviating from the existing framework of quantum mechanics. We point out that
the principle of equal a priori probabilities is easily proven in the dual
space. The dual of the space of the quantum states is the space of the
observations. We then prove that time evolution of the operators of
observations obeys Boltzmann equation. This result implies that the difference
of the states from equal probability becomes unobservable as time elapses.
