  In the statistical mechanics of quantum harmonic oscillators, the zero-point
energy can either be included (Schr\"odinger oscillators) or omitted (Planck
oscillators). For the usual results, the type of oscillator makes no difference
but, looking more closely, it turns out that including or not this energy is
not without consequences. The chemical potential {\mu}s of a Schr\"odinger
oscillator set is calculated in the canonical formalism and this shows there is
a temperature T0 for which {\mu}s=0; below this temperature, {\mu}s>0. When
Planck oscillators are used instead, the chemical potential {\mu}p is negative
for all temperatures. If the problem is approached in phonon terms and the
system is considered to be in contact with a reservoir of particles (conditions
of the grand canonical ensemble), a sort of critical temperature Tc is found,
for which the number of particles in the system diverges. For Schr\"odinger
oscillators with {\mu}s=0, it turns out that Tc=T0, i.e. T0 is a reminder, in
the canonical ensemble, of the divergent behavior in the number of particles
when under the conditions of the grand canonical ensemble. Also, a modified
Einstein solid (MES) model is introduced. In this model the frequency of the
oscillators changes with the volume of the solid, and this change is
characterized by a certain value of the Gr\"uneisen parameter. The bulk modulus
of this solid can be calculated using Planck oscillators, and it becomes
negative for certain temperature and volume values, which is physically
incorrect. When Schr\"odinger oscillators are used, the bulk modulus is always
positive. Therefore, the different behavior of both types of oscillators would
indicate that only Schr\"odinger oscillators lead to correct results.
