  Schr\"odinger suggested that thermodynamical functions cannot be based on the
gratuitous allegation that quantum-mechanical levels (typically the orthogonal
eigenstates of the Hamiltonian operator) are the only allowed states for a
quantum system [E. Schr\"odinger, Statistical Thermodynamics (Courier Dover,
Mineola, 1967)]. Different authors have interpreted this statement by
introducing density distributions on the space of quantum pure states with
weights obtained as functions of the expectation value of the Hamiltonian of
the system.
  In this work we focus on one of the best known of these distributions, and we
prove that, when considered in composite quantum systems, it defines partition
functions that do not factorize as products of partition functions of the
noninteracting subsystems, even in the thermodynamical regime. This implies
that it is not possible to define extensive thermodynamical magnitudes such as
the free energy, the internal energy or the thermodynamic entropy by using
these models. Therefore, we conclude that this distribution inspired by
Schr\"odinger's idea can not be used to construct an appropriate quantum
equilibrium thermodynamics.
