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In the statistical mechanics of quantum harmonic oscillators, the zero-pointenergy can either be included (Schr\"odinger oscillators) or omitted (Planckoscillators). For the usual results, the type of oscillator makes no differencebut, looking more closely, it turns out that including or not this energy isnot without consequences. The chemical potential {\mu}s of a Schr\"odingeroscillator set is calculated in the canonical formalism and this shows there isa temperature T0 for which {\mu}s=0; below this temperature, {\mu}s>0. WhenPlanck oscillators are used instead, the chemical potential {\mu}p is negativefor all temperatures. If the problem is approached in phonon terms and thesystem is considered to be in contact with a reservoir of particles (conditionsof the grand canonical ensemble), a sort of critical temperature Tc is found,for which the number of particles in the system diverges. For Schr\"odingeroscillators with {\mu}s=0, it turns out that Tc=T0, i.e. T0 is a reminder, inthe canonical ensemble, of the divergent behavior in the number of particleswhen under the conditions of the grand canonical ensemble. Also, a modifiedEinstein solid (MES) model is introduced. In this model the frequency of theoscillators changes with the volume of the solid, and this change ischaracterized by a certain value of the Gr\"uneisen parameter. The bulk modulusof this solid can be calculated using Planck oscillators, and it becomesnegative for certain temperature and volume values, which is physicallyincorrect. When Schr\"odinger oscillators are used, the bulk modulus is alwayspositive. Therefore, the different behavior of both types of oscillators wouldindicate that only Schr\"odinger oscillators lead to correct results.

Schr\"odinger and Planck oscillators: not quite the same physics