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arXiv:1909.12084

Schrödinger and Planck oscillators: not quite the same physics

Enrique N. Miranda

Published Sep 23, 2019• 0 citations
arXivHTMLPDF

Abstract

In the statistical mechanics of quantum harmonic oscillators, the zero-point energy can either be included (Schrödinger 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 μs of a Schrödinger oscillator set is calculated in the canonical formalism and this shows there is a temperature T0 for which μs=0; below this temperature, μs>0. When Planck oscillators are used instead, the chemical potential μp 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 Schrödinger oscillators with μs=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 Grüneisen 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 Schrödinger oscillators are used, the bulk modulus is always positive. Therefore, the different behavior of both types of oscillators would indicate that only Schrödinger oscillators lead to correct results.

DOI
10.48550/arxiv.1909.12084

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In the statistical mechanics of quantum harmonic oscillators, the zero-point energy can either be included (Schrödinger 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 μs of a Schrödinger oscillator set is calculated in the canonical formalism and this shows there is a temperature T0 for which μs=0; below this temperature, μs>0. When Planck oscillators are used instead, the chemical potential μp 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 Schrödinger oscillators with μs=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 Grüneisen 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 Schrödinger oscillators are used, the bulk modulus is always positive. Therefore, the different behavior of both types of oscillators would indicate that only Schrödinger oscillators lead to correct results.
For ideal Bose atoms in an isotropic harmonic trap, we consider thermodynamic variables obtained from microcanonical, canonical, and grand canonical ensembles, each with certain variables specified and other variables fluctuating. For the first two of these ensembles, we derive recursion relations that link partition functions for different dimensions. We discuss fluctuations in general, and obtain expressions for variances of the atom number N, the chemical potential \ensuremath{\mu}, and the temperature T for small, Gaussian fluctuations in the grand canonical ensemble. Then from our recursion relations and others given elsewhere, we obtain probability distributions for N, for ground-state occupation number n0{n}_{0}n0​, for \ensuremath{\mu}, and for T. Below the critical temperature, the shape of the distributions for N, n0,{n}_{0},n0​, and \ensuremath{\mu} are definitely not Gaussian for the grand canonical ensemble. For given temperature and small N, we find that the chemical potential values pertaining to the three ensembles differ. We compare the specific-heat function CN(T){C}_{N}(T)CN​(T) for the three ensembles and propose to use the minimum of dCN/dT{\mathrm{dC}}_{N}/dTdCN​/dT to define the critical temperature to facilitate comparisons with similar configurations of interacting atoms.
We have developed a theoretical formalism to introduce temperature as a parameter into the framework of non-relativistic quantum mechanics using the laws of classical thermodynamics and the canonical ensemble scheme of statistical mechanics. A self-consistent Hamiltonian has then been constructed for a given quantum many-body system which includes the effect of temperature in the form of correction terms added to the corresponding zero-temperature Hamiltonian of the system. Investigating some quantum mechanical systems with exact zero-temperature solutions including the particle-in-a-box model, the free particle, and the harmonic oscillator within our finite-temperature approach up to the first order of self-consistency has led to temperature-dependent Hamiltonians describing these systems above absolute zero without encountering any physically unacceptable brand of behavior for their wave functions and energy spectra. Results firmly support the view that a quantum mechanical system at a finite temperature behaves as if it is in a zero-temperature excited state.
It is proposed that the Schrodinger equation for a free point particle has non-linear corrections which depend on the mass of the particle. It is assumed that the corrections become extremely small when the mass is much smaller or much larger than a critical value (the critical value being related to but smaller than Planck mass). The corrections become significant when the mass is close to this critical value and could play a role in explaining wave-function collapse. It appears that such corrections are not ruled out by present day experimental tests of the Schrodinger equation. Corrections to the energy levels of a harmonic oscillator are calculated.
We consider a canonical ensemble of q oscillators. Using classical realization for q oscillator algebra,17 we calculate, for a small real q, the partition function and thermodynamic potentials F, E and S. We show that F reaches the minimum and E and S the maximum (as functions of the deformation parameter q) when q = 1 (for the classical oscillator). We argue about possible far-reaching consequences of this fact. As an application we obtain a first quantum correction to Planck's black body radiation law. We introduce the slightly deformed oscillator (SDO) model, which provides us with a significant amount of information about the system. When q = 1, our results are shown to coincide with classical results.
This thesis encompasses a number of problems related to the number fluctuations from the ground state of ideal particles in different statistical ensembles. In the microcanonical ensemble most of these problems may be solved using number theory. Given an energy E, the well-known problem of finding the number of ways of distributing N bosons over the excited levels of a one-dimensional harmonic spectrum, for instance, is equivalent to the number of restricted partitions of E. As a result, the number fluctuation from the ground state in the microcanonical ensemble for this system may be found analytically. When the particles are fermions instead of bosons, however, it is difficult to calculate the exact ground state number fluctuation because the fermionic ground state consists of many levels. By breaking up the energy spectrum into particle and hole sectors, and mapping the problem onto the classic number partitioning theory, we formulate a method of calculating the particle number fluctuation from the ground state in the microcanonical ensemble for fermions. The same quantity is calculated for particles interacting via an inverse-square pairwise interaction in one dimension. In the canonical ensemble, an analytical formula for the ground state number fluctuation is obtained by using the mapping of this system onto a system of noninteracting particles obeying the Haldane-Wu exclusion statistics. In the microcanonical ensemble, however, the result can be obtained only for a limited set of values of the interacting strength parameter. Usually, for a discrete set of a mean-field single-article quantum spectrum and in the micro canonical ensemble, there are many combinations of exciting particles from the ground state. The spectrum given by the logarithms of the prime number sequence, however, is a counterexample to this rule. Here, as a consequence of the fundamental theorem of arithmetic, there is a one-to-one correspondence between the microstate and the macrostate, resulting in the vanishing of number fluctuation for all excitations. The use of the canonical or grand canonical ensembles, on the other hand, gives a substantial number fluctuation from the ground state. For a related spectrum, that given by the logarithms of an integer n, the microcanonical number fluctuation is non-zero but the application of the other ensembles is still not valid. These two spectra are examples of systems where canonical and grand canonical ensembles averagings yield answers different from the microcanonical result. Some models in physics may be used to obtain formulae known in the theory of number partition. For the same problem of N ideal bosons in a one-dimensional harmonic oscillator potential mentioned earlier, it is well known that the asymptotic (N --> ∞) density of states is identical to the Hardy-Ramanujan formula for the number of partitions of an integer n. The same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for the number of partitions of n into a sum of 8th powers of a set of integers. By considering only the particle sector of the fermionic spectrum, a formula for the number of distinct partitions of n is obtained. For the s = 1 case and for finite N, the Erdos-Lehner formula for the restricted partitions, and a new formula for the distinct and restricted partitions are derived. As a diversion, we discuss the microcanonical entropy which may be uniquely defined in terms of the macrostate, or equivalently the many-body degeneracy of the state, at a given energy. The many-body degeneracy factor, however, is exceedingly difficult to calculate in general. It is thus desirable to find a different way to calculate the micro canonical entropy. It has been recently suggested that the microcanonical entropy may be accurately reproduced by including a logarithmic correction to the canonical entropy. This claim is readily tested using some of the models mentioned above, where the many-body degeneracy may be determined exactly. In addition, we also consider a system of N distinguishable particles in a d-dimensional harmonic energy spectrum. In this case the many-body degeneracy factor can be obtained analytically in a closed form.
As known all physical properties of solids are described well by the system of quantum linear harmonic oscillators. It is shown in the present paper that the system consisting of classical linear harmonic oscillators having temperature dependent masses or (and) frequencies has the same partition function as the system consisting of quantum linear harmonic oscillators having temperature independent masses and frequencies while the means of the square displacements of the positions of the oscillators from their mean positions for the system consisting of classical linear harmonic oscillators having: the temperature dependent masses; temperature dependent frequencies; and temperature dependent masses and frequencies differ from each other and from that of the system consisting of quantum linear harmonic oscillators, and hence, the system consisting of classical linear harmonic oscillators describes well the thermodynamic properties of the system consisting of quantum linear harmonic oscillators and solids.