Schrödinger and Planck oscillators: not quite the same physics for a modified Einstein solid

In the statistical mechanics of quantum harmonic oscillators, the zero-point energy can either be included (Schrodinger 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. A simple model is introduced that we called the modified Einstein solid (MES). In this model the frequency of the oscillators change with the volume of the solid, and this change is characterized by a certain value of the Gruneisen parameter. The specific heat is the same as in the standard Einstein model, but the pressure, equation of state and bulk modulus can be evaluated in the MES. Using Planck oscillators, the pressure shows an anomalous behavior in terms of the volume, and the bulk modulus becomes negative for certain temperature and volume values, which is physically incorrect. When Schrodinger oscillators are used, the bulk modulus is always positive. Therefore, the different behavior of both types of oscillators indicates that only Schrodinger oscillators lead to correct results.