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

TL;DR

This paper compares Schr"odinger and Planck oscillators in quantum statistical mechanics, revealing that including zero-point energy affects chemical potential, critical temperature, and physical properties like bulk modulus, with Schr"odinger oscillators providing more physically consistent results.

Contribution

It demonstrates that Schr"odinger oscillators yield more physically accurate results than Planck oscillators in quantum statistical models, especially regarding bulk modulus and particle divergence.

Findings

Schr"odinger oscillators have zero chemical potential at a specific temperature.

Planck oscillators always have negative chemical potential.

Schr"odinger oscillators produce positive bulk modulus, unlike Planck oscillators.

Abstract

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.