In the usual development of quantum statistical mechanics the microcanonical ensemble is introduced in order to present the basic postulates of this subject with utmost clarity. However practical calculations of elementary problems using the microcanonical ensemble directly are seldom carried out. Either the method of the most probable distribution is introduced, or the canonical ensemble is introduced. Both of these methods require the development of further mathematical and physical concepts.It is shown here that very elementary methods can be used to develop the thermodynamic properties of monatomic crystals and the “corrected” Boltzmann monatomic gas directly from the basic concepts of the microcanonical ensemble.For more complex problems, it is shown that the microcanonical ensemble emphasizes the relation between basic concepts in statistical mechanics and certain unsolved problems in the theory of numbers.